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\begin{document}
\title[]{Projective Representations
of Generalized Symmetric Groups}%
\author{Alun O Morris and Huw I Jones}%
\address{Institute of
Mathematical and Physical Sciences\\
 University of Wales\\ Aberystwyth, Ceredigion SY23 3BZ, Wales, U.K.}%
\email{alun@morus25.fsnet.co.uk}%

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\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 50 \rms (2003), Article~B50b\hfill}
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\section{\bf Introduction}\label{int}

The representation theory of generalized symmetric groups has been
of interest over a long period dating back to the classical work
of W. Specht \cite{specht1},\cite{specht2}  and M.Osima
\cite{osima} --- an exposition of this work and other references may
be found in \cite{jk81}. Furthermore, the projective
representations of these groups have been considered by a number
of authors, much of the this work was not published or was
published in journals not readily accessible in the western world.
The first comprehensive work on the projective representations of
the generalized symmetric groups was due to E. W. Read \cite{Re77}
which was followed later by an improvement in the work of M.
Saeed-ul-Islam, see, for example, \cite{Sae84'}. Of equal
interest has been the representation theory of the hyperoctahedral
groups, which are a special case of the generalized symmetric
groups. The projective representations of these groups was
considered by M. Munir in his thesis \cite{Mu88} which elaborated
on the earlier work of E. W. Read and M. Saeed-ul-Islam and also
by J. Stembridge \cite{Ste92} who gave an independent development
which was more complete and satisfactory in many respects. This
approach later influenced that used by H. I. Jones in his thesis
\cite{Jo93} where the use of Clifford algebras was emphasized.

More recently, the generalized symmetric groups have become far
more predominant in the context of complex reflection groups and
the corresponding cyclotomic Hecke algebras where they and their
subgroups form the infinite family $G(m,p,n)$, see for example
\cite{bm},\cite{bmr1} and \cite{bmr2}. In view of this interest,
it was thought worthwhile to present this work which is based on
the earlier work of H. I. Jones which has not been published. As
this article is also meant to be partially expository, a great
deal of the background material is also presented.

There are eight non-equivalent $2$-cocycles for the generalized
symmetric group\break $G(m,1,n)$, which will be denoted by $B_{n}^{m}$
in this paper. Thus, in addition to the ordinary irreducible
representations, there are seven other classes of projective
representations to be considered. However, the position is not too
complicated in that all of the non-equivalent irreducible
projective representations can be expressed in terms of certain
'building blocks'. These are the ordinary and spin representations
of the symmetric group $S_{n}$, that is, the generalized symmetric
group $G(1,1,n)$, which are well known and date back to the early
work of  F.G. Frobenius and A. Young (see \cite{jk81}) and I.
Schur \cite{Sch11} respectively. Also, required are basic spin
representations $P, Q$ and $R$ of $B_{n}^{m}$ for certain
$2$-cocycles. All of these can be constructed in a uniform way
using Clifford algebras and the basic spin representations of the
orthogonal groups. Thus, we will present all of the required
information for constructing these building blocks.

The paper is organised as follows. In Section 2 we present all of
the background information and notation required later, there are
short subsections on partitions, the projective representations of
groups, the method of J. R. Stembridge on Clifford theory (A. H.
Clifford) for $\mathbb{Z}_{2}^{2}$-quotients \cite{Ste92} and
Clifford algebras (W. K. Clifford) and their representations.
Section 3 contains all of the information required about the
generalized symmetric groups $B_{n}^{m}$; a presentation, classes
of conjugate elements and its linear characters are given. In
Section 4, the main aim is to construct the three classes of basic
spin representations $P, Q$ and $R$ of $B_{n}^{m}$ mentioned above
and some additional information required later --- these are mainly
based on the authors earlier work, \cite{Mo76}, \cite{Mo80},
\cite{Mo03}. For the sake of completeness we also include a brief
description of the elegant construction of the irreducible spin
representations of the symmetric groups given by M. L. Nazarov
\cite{Na90}. The final section then contains the construction of
the irreducible projective representations for the eight
$2$-cocycles. In this section, we follow J. R. Stembridge's work
in the special case $B_{n}^{2}$. Our results are not as complete
as his and an indication of proof only is given in some cases. A
detailed description, including the construction of the
irreducible representations for three closely connected subgroups
will appear later.

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\markboth{\SMALL ALUN O MORRIS AND HUW I JONES}{\SMALL PROJECTIVE 
REPRESENTATIONS OF GENERALIZED SYMMETRIC GROUPS}



\section{\bf Background and Notation}\label{bn}
\subsection{\bf Partitions}
The notation follows \cite{Mac95}. Let $\lambda =
(\lambda_{1},\lambda_{2}, \ldots, \lambda_{k})$ be a partition of
$n$, then $l(\lambda) = k$ is the {\bf length} of $\lambda$ and
$|\lambda| = n$ is the {\it weight} of $\lambda$. The {\bf
conjugate} of $\lambda$ is denoted by $\lambda^{\prime}$. A
partition $\lambda$ is called an {\bf even(odd) partition} if the
number of even parts in $\lambda$ is even(odd). A partition is
sometimes written as $\lambda = (1^{a_{1}}2^{a_{2}} \ldots
n^{a_{n}}),\; 0 \leq a_{i} \leq n$ indicating that $a_{i}$ parts
of $\lambda$ are equal to $i,\; 1 \leq i \leq n$, $|\lambda| =
\sum_{i=1}^{n}ia_{i}$ and $l(\lambda)=\sum_{i=1}^{n}a_{i}$.

Let $P(n)$ denote the set of all partitions of $n$, then $DP(n) =
\{\lambda \in P(n)\; |\; \lambda_{1} > \lambda_{2} > \cdots
> \lambda_{k} > 0 \}$ is the set of all partitions of $n$ into
distinct parts,  $DP^{+}(n) = \{ \lambda \in DP(n) \; | \;
|\lambda|-l(\lambda) \; \mbox{is even} \}$, $DP^{-}(n) = \{
\lambda \in DP(n) \;|\; |\lambda|-l(\lambda) \; \mbox{is odd} \}$,
$OP(n) = \{ \lambda = (1^{\alpha_{1}}3^{\alpha_{3}}
\ldots)\}\;\mbox{is the set of all partitions of $n$ into odd
parts}$, $EP(n) = \{ \lambda = (2^{\alpha_{2}}4^{\alpha_{4}}
\ldots)\}\; \mbox{is the set of all partitions of $n$ into even
parts}$ and $SCP(n) = \{\lambda \in P(n) \;| \; \lambda =
\lambda^{\prime} \}\; \mbox{is the set of self-conjugate
partitions of $n$}.$

An $m$-{\bf partition} of $n$ is a partition comprising of m
partitions $(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$
such that $\lambda_{(i)} \in P(n_{i})$ , $1 \leq i \leq m$ and
$\sum_{i=1}^{m}n_{i} = n$. The partition $\lambda_{(i)}$ is
written as $(\lambda_{i1},\lambda_{i2}, \ldots ,
\lambda_{ik_{i}})$, where $k_{i} = l(\lambda_{(i)})$ for $ 1 \leq
i \leq m$. The {\bf conjugate } of $(\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)})$ is the $m$-partition
$(\lambda_{(1)}^{\prime};\lambda_{(2)}^{\prime}; \ldots;
\lambda_{(m)}^{\prime})$. An $m$-partition is said to be {\bf
even(odd)} if the total number of even parts of
$(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$ is
even(odd). An $m$-partition is sometimes written in the form
$$((1^{\alpha_{11}}2^{\alpha_{12}} \ldots);
(1^{\alpha_{21}}2^{\alpha_{22}}\ldots);\ldots;
(1^{\alpha_{m1}}2^{\alpha_{m2}} \ldots ));$$
$l(\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)})=l(\lambda_{(1)})+l(\lambda_{(2)})+\cdots+
l(\lambda_{(m)})$ is the {\bf length of}
$(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$ and
$|(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})|=
|(\lambda_{(1)})|+|(\lambda_{(2)}|+\cdots+ |(\lambda_{(m)})|$ is
the {\bf weight of } $(\lambda_{(1)};\lambda_{(2)}; \ldots;$
\newline $ \lambda_{(m)}).$ We note that
$l(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})=
\Sigma_{i=1}^{m}\Sigma_{j=1}^{n}a_{ij}$.

\subsection{Projective representations}
We present some basic background material on the projective
representations of groups which is required later.

Let $G$ be a group with identity $1$ of order $|G|$, ${\mathbb C}$
the field of complex numbers, ${\mathbb C}^{\times}={\mathbb
C}\setminus\{0\}$ and $GL(n,{\mathbb C})$ the group of invertible
$n \times n$ matrices over ${\mathbb C}$.

A {\bf projective representation of degree n} of $G$ is a map $P:G
\rightarrow GL(n,{\mathbb C})$ such that for $g,h \in G$
$$P(g)P(h)={\alpha}(g,h)P(gh)$$
and $P(1)=I_{n},$ where $I_{n}$ is the identity $n \times n$
matrix and ${\alpha}(g,h) \in {\mathbb C}^{\times}$. Since
multiplication in $G$ and $GL(n,{\mathbb C})$ is associative, it
follows that
\begin{equation}
\alpha(g,h)\alpha(gh,k)=\alpha(g,hk)\alpha(h,k)
\end{equation}
for all $g,h,k \in G$. A map $\alpha : G \times G \rightarrow
{\mathbb C}^{\times}$ which satisfies (2.1) is called a {\bf
2-cocycle}({\bf factor set}) of $G$ in ${\mathbb C}$ and we shall
say that $P$ is a projective representation with $2$-cocycle
$\alpha$.

Projective representations $P$ and $Q$ of degree $n$ with
$2$-cocycles ${\alpha}$ and $\beta$ respectively are said to be
{\bf projectively equivalent} if there exists a map $\mu : G
\rightarrow {\mathbb C}^{\times}$ and a matrix $S \in
GL(n,{\mathbb C})$ such that
$$Q(g)=\mu(g)S^{-1}P(g)S$$
for all $g \in G$. If $P$ and $Q$ are projectively equivalent, it
follows that
\begin{equation}
\beta(g,h)=\frac{\mu(g)\mu(h)}{\mu(gh)}\alpha(g,h)
\end{equation}
for all $g,h \in G$. The corresponding $2$-cocycles $\beta$ and
$\alpha$ are then said to be {\bf equivalent}.

Let $H^{2}(G,{\mathbb C}^{\times})$ denote the set of equivalence
classes of $2$-cocycles; then $H^{2}(G,{\mathbb C}^{\times})$ is
an abelian group which is called the {\bf Schur multiplier} of
$G$. The Schur multiplier gives a measure of the number of
different classes of projectively inequivalent representations
which a group $G$ possesses. If $G$ is a finite group, then
$H^{2}(G,{\mathbb C}^{\times})$ is a finite abelian group.

All projective representations of $G$ may be obtained from
ordinary representations of a larger group; thus the problem of
determining all the projective representations of a group $G$ is
essentially reduced to that of determining ordinary
representations of a larger finite group.

A {\bf central extension} $(H,\phi)$ of a group $G$ is a group $H$
together with a homomorphism $\phi : H \rightarrow G$ such that
$ker\phi \subset Z(H)$, where $Z(H)$ is the centre of $H$, that
is,
$$1 \rightarrow ker\phi \rightarrow H \stackrel{\phi}{\rightarrow} G \rightarrow \{1\}$$
is exact. Let $A=ker\phi$, and let $\{\gamma(g)\enspace |\enspace
g \in G \}$ be a set of coset representatives of $A$ in $H$ which
are in $1-1$ correspondence with the elements of $G$; thus
$$H=\bigcup_{g \in G}A\gamma(g).$$
Then, for all $g,h \in G$, let $a(g,h)$ be the unique element in
$A$ such that
$$\gamma(g)\gamma(h)=a(g,h)\gamma(gh).$$
The associative law in $H$ and $G$ implies that
\begin{equation}
a(g,h)a(gh,k)=a(g,hk)a(h,k)
\end{equation}
for all $g,h,k \in G$. Now, let $\gamma$ be a linear character of
the abelian group $A$ and put
$$\alpha(g,h)=\gamma(a(g,h))$$
for all $g,h \in G$, then (2.3) implies that $\alpha$ is a
$2$-cocycle of $G$.

Now, let $T$ be an ordinary irreducible representation of $H$ of
degree $n$ put $P(g)=T(\gamma(g))$ for all $g \in G$, then $P$ is
a projective representation of $G$ with 2-cocycle $\alpha$. A
projective representation $P$ of $G$ arising from an irreducible
ordinary representation $T$ of $H$ in this way is said to be {\bf
linearized} by the ordinary representation $T$.

If $G$ is a finite group, then there exists a central extension
$H$ of $G$ with kernel $H^{2}(G,{\mathbb C}^{\times})$ which
linearizes every projective representation of $G$. Such a group
$H$ is called a {\bf representation group} of $G$; this implies
that every finite group has at least one representation group.
Thus, the problem of determining all the irreducible projective
representations of $G$ for all possible $2$-cocycles is reduced to
determining all the ordinary irreducible representations of a
representation group $H$.


In practice, we shall see that it will be sufficient to determine
a complete set of irreducible projective representations of a
group $G$ for a {\bf fixed $2$-cocycle} $\alpha$ whose values are
roots of unity. In that case, we can calculate in terms of a
subgroup of the representation group of $G$ which will be called a
$\alpha$-{\bf covering group} of $G$.

Let $\alpha $ be a $2$-cocycle such that $\{\alpha\}$ has order
$n$ and let $\omega$ be a primitive $n$-th root of unity, then
$\alpha(g,h)=\omega^{\eta(g,h)}$ for some $0 \leq \eta(g,h) < n$.
Suppose that $\{\nu(g)\enspace|\enspace  g \in G \}$ is a set of
distinct symbols in one-one correspondence with the elements of
$G$. Let $G(\alpha) =\{(\alpha^{j},\nu(g)) \enspace |\enspace
 0 \leq j < n,g \in G \}$, then it is easily
verified that $G(\alpha)$ is a group with composition defined by
$$(\alpha^{j},\nu(g)) (\alpha^{k},\nu(h)) = (\alpha^{j+k+\eta(g,h)},\nu(gh))$$
for all $g,h \in G, 0 \leq j,k < n$.

If now $P$ is a projective representation of $G$ of degree $n$
with $2$-cocycle $\alpha$, then define $T:G(\alpha) \rightarrow
GL(n,{\mathbb C})$ by
$$T(\alpha^{j},\nu(g))= \omega^{j}P(g),$$
then $T$ is an ordinary representation of $G(\alpha)$. That is,
$P$ has been lifted to an ordinary representation of $G(\alpha)$.
Such a group $G(\alpha)$ is called an $\alpha$-{\bf covering
group} of the group $G$.

In the case of the generalized symmetric group, the $2$-cocycles
are of order two, thus we shall then refer to the $G(\alpha)$ as
{\bf double covers}. As we are basically working with ordinary
representations of the $G(\alpha)$, we can apply all the usual
results from representation theory. However, we shall be
interested in the non-ordinary projective representations, namely
the ones in which the central element $-1 \in G(\alpha)$ is
represented faithfully, we refer to these as {\bf spin
representations} of $G$ with $2$-cocycle $\alpha$.

If $\mathcal{C}$ denotes a class of conjugate elements in $G$, let
$\mathcal{C}(\alpha) \in G(\alpha)$ denote the inverse image in
$G(\alpha)$. If any $g \in \mathcal{C}(\alpha)$ is conjugate to
$-g$, then $\mathcal{C}(\alpha)$ is a class of conjugate elements
in $G(\alpha)$, otherwise $\mathcal{C}(\alpha)$ splits into two
classes. The spin character will only be non-zero on the splitting
classes; thus it will be necessary to determine the splitting
classes for each $2$-cocycle.

\subsection{Clifford theory for $\mathbb{Z}_{2}^{2}$-quotients}

Let $G$ be a group with a subgroup $H$ of index $2$ and let $\eta$
be a linear character of $G$ defined by
\begin{eqnarray*}
\eta(g) & = & \left\{\begin{array}{ll}
  1 & \mbox{if} \; g \in H \\
  -1 & \mbox{if} \; g \not\in H. \\
\end{array}\right.
\end{eqnarray*}
If $T$ is an irreducible representation of $G$ with character
$\chi$, then $\eta \otimes T$ is also an irreducible
representation of $G$, if these representations are equivalent,
then we say that $T$ is {\bf self-associate}, but if not, they are
said to be $\eta$-{\bf associate}, and are denoted by $T_{+}$ and
$T_{-}$, their characters are denoted by $\chi_{+}$ and
$\chi_{-}$; clearly $\chi_{-}(g) = \eta(g)\chi_{+}(g)$ for all $g
\in G$. If $T$ is self-associate, then the unique(up to sign)
matrix $S$ such that $$ T(g)S = \eta(g)ST(g)$$ for all $g \in G$,
is called the $\eta$-{\bf associator} of $T$. If $T$ is
self-associate, then $T|_{H}$ decomposes into two inequivalent
irreducible representations of $H$ of equal degree, say $T_{1}$
and $T_{2}$ with characters $\chi_{1}$ and $\chi_{2}$
respectively, then the {\bf difference character}
$\Delta^{\eta}\chi$, is defined by $$\Delta^{\eta}\chi(g) =
trST(g) = \chi_{1}(g)- \chi_{2}(g)$$ for all $g \in H$. Knowledge
of the difference character then gives the corresponding
characters of H,
$$ \frac{1}{2}(\chi \pm \Delta^{\eta}\chi).$$

All the above results are classic \cite{jk81} and date back to A.
H. Clifford. Recently, J. R. Stembridge \cite{Ste92} has extended
this detailed analysis to the case where $G/H \cong \mathbb{Z}_{2}
\times \mathbb{Z}_{2}$; we briefly recall his results. Let $L =
\{1,\eta,\sigma,\eta\sigma\}$ be the four corresponding linear
characters of $G$. If $T$ is an irreducible representation of $G$,
then $\nu \otimes T$ for all $\nu \in L$ is also an irreducible
representation of $G$. As before, the question is whether these
are equivalent or not. Let $L_{T} = \{\nu \in L | \nu \otimes T\
\sim T\}$. Then, the following proposition gives the behavior of
$T$ on restriction to $H$.
\begin{prop} Let $T$ be an irreducible representation of degree $d$ of $G$.

(i)  If $L_{T}= \{1\}$, then $T_{H}$ is an irreducible
representation of degree $d$ of $H$.

(ii)   If $L_{T}= \{1,\nu\}$, where $\nu \in L,\; \nu \neq 1$,
then $T_{H}$ is the direct sum of two inequivalent irreducible
representation of degree $d/2$ of $H$.

(iii)   If $L_{T}= L$, and $R,S$ are the $\eta,\sigma$-associators
of $T$ respectively,  then

      \indent (a) if $RS=SR$, then $T_{H}$ is the direct sum of four inequivalent
irreducible representation of degree $d/4$ of $H$,

      \indent (b) if $RS=-SR$, then $T_{H}$ is the direct sum of two copies of one
irreducible representation of degree $d/2$ of $H$.
\end{prop}

As in the above, knowledge of the difference characters enables
one to write out the irreducible representations of $H$, the only
additional case which needs to be considered is (iii)(b); in that
case, the four irreducible characters are
$$ \frac{1}{4}(\chi  \pm \Delta^{\eta}\chi \pm \Delta^{\sigma}\chi \pm
\Delta^{\eta\sigma}\chi),$$ where an even number of the $-$ signs
occur.



\subsection{\bf Clifford algebras and their representations.}
Let $C(n)$ be the Clifford algebra generated by
$1,e_{1},\ldots,e_{n}$ subject to the relations
$$ e_{j}^{2}=1, \enspace e_{j}e_{k}=-e_{k}e_{j}, \enspace 1 \leq
j,k \leq n, \enspace j \neq k .$$ If $Pin(n)$ is defined to be the
set of invertible elements $s$ of $C(n)$ such that
${(s\alpha(s^{t}))}^{2}=1$, where $\alpha$ is the natural
${\mathbb Z}_{2}$-grading on $C(n)$ and $^{t}$  is the transpose,
then we have the short exact sequence
\begin{equation}\begin{array}{cccccccccc} 1 & \longrightarrow &
{\mathbb Z}_{2} & \longrightarrow & Pin(n) & \stackrel{\rho_{n}}
{\longrightarrow} & O(n) & \longrightarrow & 1 &
\end{array},\end{equation} where $\rho_{n}$ is defined by
$\rho_{n}(s)e_{j}=\alpha(s)e_{j}s^{-1}, \enspace \mbox{for all}
\enspace s \in Pin(n), \enspace 1 \leq j \leq n.$

In fact, the Schur multiplier of $O(n)$ is given by
\begin{eqnarray}
H^{2}(O_{n},{\mathbb C}^{*})= {\mathbb Z}_{2}.
\end{eqnarray}
Furthermore, if $Spin(n) = \rho_{n}^{-1}(SO(n))$, then we also
have the classical double covering of the special orthogonal
(rotation) group $SO(n)$
\begin{equation}\begin{array}{cccccccccc} 1 &
\longrightarrow & {\mathbb Z}_{2} & \longrightarrow & Spin(n) &
\stackrel{\rho_{n}} {\longrightarrow} & SO(n) & \longrightarrow &
1 & \end{array},\end{equation} Clearly, $Spin(n)$ is of index $2$
in $Pin(n)$; let $\eta$ denote the corresponding linear character
of $Pin(n)$.

We now construct the so-called {\bf basic spin representation} of
Clifford algebras. Let
\[ E = \left( \begin{array}{lr}
               1 & 0 \\
               0 & 1
              \end{array} \right), \enspace
I = \left( \begin{array}{lr}
               0 & 1 \\
               1 & 0
              \end{array} \right), \enspace
J = \left( \begin{array}{lr}
               0 & i \\
               -i & 0
              \end{array} \right), \enspace
K = \left( \begin{array}{lr}
               1 & 0 \\
               0 & -1
              \end{array} \right) \]
then
\[ I^{2} = K^{2} = E, \enspace J^{2} = E, \enspace JI = -IJ = iK, \]
\[ KI = -IK = iJ, \enspace KJ = -JK = I. \]
Then, if $n = 2\mu$ is even, we define an isomorphism $P_{n} :
C_{n} \rightarrow {\mathbb C}(2^{\mu})$ by
\begin{equation}
\left\{
\begin{array}{ccc}
P_{n}(e_{2j-1}) & = & M_{2j-1} := K^{\otimes(j-1)} \otimes I \otimes E^{\otimes(\mu -j)}\\
P_{n}(e_{2j}) & = & M_{2j} := K^{\otimes(j-1)} \otimes J \otimes
E^{\otimes(\mu -j)}
\end{array}
\right. \end{equation}
for $1 \leq j \leq \mu$ and if $n = 2\mu +
1$ is odd, we define an isomorphism $P_{n,+} : C_{n} \rightarrow
{\mathbb C}(2^{\mu})$ by
\begin{equation}
\left\{
\begin{array}{ccc}
P_{n,+}(e_{j}) & = & P_{n}(e_{j}) \\
P_{n,+}(e_{2\mu + 1}) & = & M_{n} = K^{\otimes\mu}
\end{array}
\right. \end{equation}
for $1 \leq j \leq 2\mu$. Furthermore, for
$1 \leq j \leq n$, put
\[ P_{n,-}(e_{j}) = -P_{n,+}(e_{j})  \]
Then we note that
\begin{equation}
M_{j}^{2} = I, \enspace M_{j}M_{k} = - M_{k}M_{j} \enspace
\mbox{for} \enspace 1 \leq j,k \leq n.
\end{equation}

Then, if $n$ is even, $P_{n}$ is the unique irreducible complex
representation of degree $2^{n/2}$ of $C_{n}$ and if $n$ is odd,
$P_{n,+}$ and $P_{n,-}$ are the two inequivalent irreducible
complex representations of degree $2^{n/2}$ of $C_{n}$ which are
clearly $\eta$-associate representations. From now on, we denote
these by $P, P_{\pm}$. We shall refer to these as the {\bf basic
spin representations} of the Clifford algebra. It is easily
checked that an $\eta-$associator of $P$ is $K^{\otimes\mu}$. In
\cite{Mo80}, it was proved that the basic spin representation of a
Clifford algebra $C(n)$ is irreducible when restricted to the
orthogonal group, or to be more precise, to its double cover
$Pin(n)$. This restricted representation is called the {\bf basic
spin representation} of the orthogonal group.

We now define a twisted outer product of spin representations. Let
$m$ and $n$ be positive integers such that $m + n =l$. We show how
to construct irreducible spin representations of $Pin(m,n)$ by
taking a product of an irreducible spin representation of $Pin(m)$
with an irreducible spin representation of $Pin(n)$.

Let $P_{1}$ and $P_{2}$ be irreducible spin representations of
$Pin(m)$ and $Pin(n)$ respectively of degrees $d_{1}$ and $d_{2}$
respectively. Then the {\bf twisted product} $P_{1}\hat{\otimes}
P_{2}$ is a spin representation of the twisted product $Pin(m,n)
\cong Pin(m) \hat{\otimes} Pin(n)$ (see \cite{Mo80}) defined as
follows; there are 3 cases to be considered.

\noindent {\bf Case 1:} If $P_{1}$ and $P_{2}$ are
$\eta$-associate spin representations of $Pin(m)$ and $Pin(n)$
respectively, then put
\begin{eqnarray*}
(P_{1} \hat{\otimes} P_{2})(\tau,\sigma) & = & E \otimes
P_{1}(\tau) \otimes P_{2}(\sigma)
\enspace \mbox{if} \enspace \tau \in Spin(m), \sigma \in Spin(n), \\
(P_{1} \hat{\otimes} P_{2})(\tau,1) & = & I \otimes P_{1}(\tau)
\otimes I_{d_{2}}
\enspace \mbox{if} \enspace \tau \in Pin(m) \setminus Spin(m), \\
(P_{1} \hat{\otimes} P_{2})(1 ,\sigma) & = & J \otimes I_{d_{2}}
\otimes P_{2}(\sigma) \enspace \mbox{if} \enspace \sigma \in
Pin(n) \setminus Spin(n);
\end{eqnarray*}
the relation $ IJ = -JI$ ensures that $P_{1} \hat{\otimes} P_{2}$
is a spin representation of $Pin(m) \hat{\otimes} Pin(n)$ of
degree $2d_{1}d_{2}$. Furthermore, $P_{1} \hat{\otimes} P_{2}$ is
self-associate, since $tr(I) = tr(J) = 0$ and so $P_{1}
\hat{\otimes} P_{2}$ and $\eta \otimes (P_{1} \hat{\otimes}
P_{2})$ have equal characters.

\noindent {\bf Case 2:} If $P_{1}$ is a self-associate spin
representation of $Pin(m)$ with $\eta$-associator $S_{1}$ and
$P_{2}$ is an $\eta$-associate spin representation of $Pin(n)$,
then
\[ S_{1}P_{1}(\sigma) = \left\{ \begin{array}{ll}
         P_{1}(\sigma)S_{1} & \mbox{if} \enspace \sigma \in Spin(m) \\
         -P_{1}(\sigma)S_{1} & \mbox{if} \enspace \sigma \in Pin(m) \setminus Spin(m).
            \end{array} \right. \]
Now, define
\begin{eqnarray*}
(P_{1} \hat{\otimes} P_{2})_{\pm}(\tau,\sigma) & = & P_{1}(\tau)
\otimes P_{2\pm}(\sigma) \enspace \mbox{if}
\enspace \tau \in Spin(m), \sigma \in Spin(n), \\
(P_{1} \hat{\otimes} P_{2})_{\pm}(\tau,1) & = & P_{1}(\tau)
\otimes I_{d_{2}} \enspace \mbox{if}
\enspace \tau \in Pin(m) \setminus Spin(m), \\
(P_{1} \hat{\otimes} P_{2})_{\pm}(1, \sigma) & = & S_{1} \otimes
P_{2\pm}(\sigma) \enspace \mbox{if} \enspace \sigma \in Pin(n)
\setminus Spin(n).
\end{eqnarray*}
Then $(P_{1} \hat{\otimes} P_{2})_{\pm}$ are $\eta$-associate
irreducible spin representations of $Pin(m) \hat{\otimes} Pin(n)$
of degree $d_{1}d_{2}$.

\noindent {\bf Case 3:} If $P_{1}$ and $P_{2}$ are both
self-associate representations, then define $(P_{1} \hat{\otimes}
P_{2})_{\pm}$ as in Case 2, but replacing $P_{2\pm}$ by $P_{2}$,
then $(P_{1} \hat{\otimes} P_{2})_{+}$ and $(P_{1} \hat{\otimes}
P_{2})_{-}$ are equivalent irreducible spin representations of
$Pin(m) \hat{\otimes} Pin(n)$, thus $(P_{1} \hat{\otimes}
P_{2})_{+}$ is a self-associate spin representation of degree
$d_{1}d_{2}$ in this case.

If we let $\chi_{P_{1}}, \chi_{P_{2}}$ and $\chi_{P_{1}
\hat{\otimes} P_{2}}$ denote the characters of $P_{1},P_{2}$ and
$(P_{1} \hat{\otimes} P_{2})$ respectively, and $\Delta_{P_{1}},
\Delta_{P_{2}}$ and $\Delta_{P_{1} \hat{\otimes} P_{2}}$ denote
the difference characters if $P_{1},P_{2}$ or $P_{1} \hat{\otimes}
P_{2}$ are self-associate, then as a consequence of the above we
have the following proposition.
\begin{prop}
If $P_{1}$ and $P_{2}$ are spin representations of $Pin(m)$ and
$Pin(n)$ respectively and

(i) if $P_{1}$ and $P_{2}$ are $\eta$-associate representations
then
\[ \chi_{P_{1} \hat{\otimes} P_{2}}(\tau, \sigma) = \left\{ \begin{array}{ll}
2\chi_{P_{1}}(\tau)\chi_{P_{2}}(\sigma) & \mbox{if} \enspace \tau \in Spin(m), \sigma \in Spin(n)  \\
                               0       & \mbox{otherwise}.
                  \end{array} \right. \]

(ii) if one of $P_{1}$ or $P_{2}$ is self-associate, then
\[ \chi_{P_{1} \hat{\otimes} P_{2}}(\tau, \sigma) = \left\{ \begin{array}{ll}
\chi_{P_{1}}(\tau)\chi_{P_{2}}(\sigma) & \mbox{if} \enspace \tau \in Spin(m), \sigma \in Spin(n)  \\
\Delta_{P_{1}}(\tau)\chi_{P_{2}}(\sigma) & \mbox{if} \enspace \tau
\in Pin(m) \setminus Spin(m),
 \sigma \in Pin(n) \setminus Spin(n)  \\
                               0       & \mbox{otherwise}.
                  \end{array} \right. \]
\end{prop}

The above can be generalized, that is, we can define the twisted
product $P_{1} \hat{\otimes} \cdots \hat{\otimes} P_{t}$, where
$\hat{\otimes}$ is an associative 'multiplication'.

Let $m_{1}, \ldots , m_{t}$ be positive integers such that $m_{1}
+ \cdots + m_{t} = l$ and for $1 \leq j \leq t$, let $P_{j}$ be an
irreducible spin representation of $Pin(m_{j})$ of degree $d_{j}$.
For simplicity, we assume that $P_{j}, \enspace 1 \leq j \leq r
\leq t$, are self-associate representations and that the remaining
$s = t - r$ representations $P_{j}$ are $\eta$-associate
representations. Let $\pm S_{j}, \enspace 1 \leq j \leq r $, be
the $\eta$-associators of the representations $P_{j}$, then
\begin{equation}
P_{j}(\sigma_{j}) = \left\{ \begin{array}{ll}
          S_{j}P_{j}(\sigma_{j}) & \mbox{if} \enspace \sigma_{j} \in Spin(m_{j}) \\
          -S_{j}P_{j}(\sigma_{j}) & \mbox{if} \enspace \sigma_{j} \not\in Spin(m_{j}).
              \end{array} \right.
\end{equation}
Let $\sigma_{j}$ also denote the element $1 \otimes \cdots \otimes
1 \otimes \sigma_{j} \otimes 1 \otimes \cdots \otimes 1$ in
$Pin(m_{1}) \hat{\otimes} \cdots \hat{\otimes} Pin(m_{t})$, with
$\sigma_{j}$ in the $j$-th position, where $\sigma_{j} \in
Pin(m_{j}), \enspace 1 \leq j \leq t$. If $\sigma_{j} \in
Spin(m_{j})$, $ 1 \leq j \leq t,$ put
\begin{equation}
P(\sigma_{j}) = I_{2^{\lfloor s/2\rfloor}} \otimes I_{d_{1}}
\otimes \cdots \otimes I_{d_{j-1}} \otimes P_{j}(\sigma_{j})
\otimes I_{d_{j+1}} \otimes \cdots \otimes I_{d_{t}}
\end{equation}
and if $\sigma_{j} \not \in Spin(m_{j})$, put
\begin{equation}
P(\sigma_{j}) = \left\{ \begin{array}{l} I_{2^{\lfloor
s/2\rfloor}} \otimes S_{1} \otimes \cdots \otimes S_{j-1} \otimes
P_{j}(\sigma_{j}) \otimes I_{d_{j+1}} \otimes \cdots \otimes
I_{d_{t}} \enspace \mbox{if} \enspace
1 \leq j \leq r,\\
M_{j-r} \otimes S_{1} \otimes \cdots \otimes S_{r} \otimes
I_{d_{r+1}} \otimes \cdots \otimes I_{d_{j-1}}
P_{j}(\sigma_{j}) \otimes I_{d_{j+1}} \otimes \cdots \otimes I_{d_{t}} \\
\enspace \mbox{if} \enspace r+1 \leq j \leq r+s=t.
\end{array} \right.
\end{equation}
The relations (2.3) ensure that $P$ is a spin representation of
$$Pin(m_{1}, \ldots ,m_{t}) \cong Pin(m_{1})
\hat{\otimes} \cdots \hat{\otimes} Pin(m_{t}).$$ 
The degree of $P$
is $2^{\lfloor s/2\rfloor}d_{1} \cdots d_{t}$.

The character of this representation was also calculated in
\cite{Mo80} to give the following proposition.
\begin{prop}
Let $\zeta$ be the character of $P$ and $\zeta_{j}, \enspace 1
\leq j \leq t$, be the characters of $P_{j}$.

(i)  If $\sigma_{j} \in Spin(m_{j}), \enspace 1 \leq j \leq t$,
then
\[ \zeta(\sigma_{1} \cdots \sigma_{t}) = 2^{\lfloor s/2\rfloor}\zeta_{1}(\sigma_{1}) \cdots
\zeta_{t}(\sigma_{t}). \]

(ii)  If $s$ is odd and $\Delta_{j}$ is the difference character
of the self-associate representations $P_{j}, \enspace 1 \leq j
\leq r$, and if $\sigma_{j} \in Spin(m_{j}), \enspace 1 \leq j
\leq r, \sigma_{j} \not \in Spin(m_{j}), \enspace r+1 \leq j \leq
t,$ then
\[ \zeta(\sigma_{1} \cdots \sigma_{t}) = \pm(2i)^{[s/2]}\Delta_{1}(\sigma_{1}) \cdots
\Delta_{r}(\sigma_{r})\zeta_{r+1}(\sigma_{r+1}) \cdots
\zeta_{t}(\sigma_{t}). \]

(iii)  In all other cases
\[ \zeta(\sigma_{1} \cdots \sigma_{t}) = 0 \]
\end{prop}

The above proposition can be applied in particular to the special
case where the $P_{i}$ are the basic spin representations of
$Pin(m_{i})$. Then, the assumption that the first $r$ of the
representations are self-associate is equivalent to assuming that
the $m_{i}$ are even for $1 \leq i \leq r$ and that the $m_{i}$
are odd for $r+1 \leq i \leq t$. The degree of the representation
$P$ will therefore be $2^{\lfloor s/2\rfloor}2^{m_{1}/2} \cdots
2^{(m_{r+1}-1)/2} \cdots 2^{(m_{t}-1)/2} = 2^{\lfloor
s/2\rfloor}2^{(l-s)/2} =2^{\lfloor l/2\rfloor}$. Furthermore, the
explicit formulae of Proposition 2.3 could be used to give more
explicit values for the characters in terms of the eigenvalues of
the elements $\sigma_{1}, \ldots, \sigma_{t}$. This will not be
done at this point, it is postponed for consideration later when
these results are applied to certain reflection groups.



\section{The Generalized Symmetric Group ${\mathbb Z}_{m}^{n} \rtimes S_{n}$}

\subsection{\bf Presentation}
If ${\mathbb Z}_{m}$ is the cyclic group of order $m$ and $S_{n}$
is the symmetric group of order $n!$, the {\it generalized
symmetric group} is the {\it wreath product} ${\mathbb Z}_{m}\wr
S_{n}$ or the semi-direct product ${\mathbb Z}_{m}^{n}\rtimes
S_{n}$. This group is of order $m^{n}n!$; in the sequel, it is
denoted by $B_{n}^{m}$ (when $m=1$, we have the symmetric group
$S_{n}$ or the Weyl group of type $A_{n-1}$ and when $m=2$, we
have the hyperoctahedral group or the Weyl group of type $B_{n}$).

If $S_{n}$ is considered as a permutation group acting on the set
$\{1,2, \ldots, n\}$, then $S_{n}$ is generated by $s_{i},\:1 \leq
i \leq n-1$ with relations
$$  s_{i}^{2}=1,
     {(s_{i}s_{i+1})}^{3}=1,\: 1 \leq i \leq n-2, {(s_{i}s_{j})}^{2}=1,
      |i-j|\geq 2,\:1 \leq i,j \leq n-1,$$where $s_{i}$ is the
transposition $(i,i+1), 1 \leq i \leq n-1$ The group $B_{n}^{m}$
can be considered as the group generated by $s_{i},\:1 \leq i \leq
n-1,\; w_{j},\: 1 \leq j \leq n$ with relations
$$s_{i}^{2}=1,w_{j}^{m}=1;
     {(s_{i}s_{i+1})}^{3}=1,\: 1 \leq i \leq n-2,\:s_{i}w_{i}=w_{i+1}s_{i},\:
      s_{i}w_{j}=w_{j}s_{i}, j \ne i,i+1 $$
      $${(s_{i}s_{j})}^{2}=1, |i-j|\geq 2,\:1 \leq i,j \leq n-1,
      \:
      w_{i}w_{j}=w_{j}w_{i}, i \ne j,\:2 \leq i \leq n-1 \:.$$

Comparing this with the presentation of $S_n$ , we see the natural
embedding of $S_{n}$ in $B_{n}^{m}$; also $w_{i}$ may be regarded
as the mapping which takes $i$ onto $\zeta i$, with $\{1,2,
\ldots, i-1,i+1, \ldots ,n\}$ fixed, where $\zeta$ is a primitive
$m$-th root of unity. It can be verified that
$w_{j}=s_{j-1}s_{j-2} \cdots s_{1}w_{1}s_{1} \cdots
s_{j-2}s_{j-1}$ for $1 \leq j \leq n$. That is, $B_{n}^{m}$ is the
permutation group acting on the set $\{1,2, \dots, n\}$, but also
with the 'sign' changes $w_{i}$ which are written as $w_{i}=$
\begin{tiny}$\left(\begin{array}{@{\hspace{0.1cm}}c}i\\\zeta
i\end{array}\right).$\end{tiny}
\subsection{\bf Classes of conjugate elements}
The classes of conjugate elements of $S_{n}$ are parameterized by
the partitions $(1^{{n}_{1}}2^{{n}_{2}}
 \ldots n^{{n}_{n}})$ of $n$, where ${n}_{i}\geq 0,\: 1 \leq i \leq n$.

The classes of $B_{n}^{m}$ are defined similarly in terms of
$m$-partitions (see, for example, \cite{jk81}).  The elements of
$B_{n}^{m}$ permute the set $\{1,2, \ldots ,n\}$ and multiply each
of the elements of this set by a power of $\zeta$. Thus the
elements of $B_{n}^{m}$ are of the form
$$x = \left( \begin{array}{cccc}
1&2& \ldots &n \\
\zeta^{k_{1}}b_{1} & \zeta^{{k}_{2}}b_{2} & \ldots &
\zeta^{{k}_{n}}b_{n}\end{array} \right),$$ where $\{b_1,b_2,
\ldots ,b_n\}$ is a permutation of the set $\{1,2, \ldots ,n\}$
and $1 \leq k_{i} \leq m,\: 1 \leq i \leq n$. Any element of
$B_{n}^{m}$ can be uniquely expressed as a product of disjoint
cycles $x= \prod_{i=1}^{t}\theta_{i}$. where $$\theta_{i} = \left(
\begin{array}{cccc}
b_{i_{1}}&b_{i_{2}}& \ldots &b_{i_{l_{i}}} \\
\zeta^{k_{i_{1}}}b_{i_{2}} & \zeta^{k_{i_{2}}}b_{i_{3}} & \ldots &
\zeta^{k_{i_{l_{i}}}}b_{i_{1}}\end{array} \right),$$ where
$\sum_{i=1}^{t}l_{i} = n$; put $f(\theta_{i}) =
\sum_{j=1}^{l_{i}}k_{i_{j}}$.

Then the classes of conjugate elements of $B_{n}^{m}$ correspond
to the $m$-partitions of $n$
$$(1^{{a}_{11}}2^{{a}_{12}} \ldots n^{{a}_{1n}}; 1^{{a}_{21}}2^{{a}_{22}}
 \ldots n^{{a}_{2n}}; \ldots ; 1^{{a}_{m1}}2^{{a}_{m2}} \ldots n^{{a}_{mn}}),$$
where $\sum_{i=1}^{n}a_{ij}=n_{j}\:1 \leq j \leq m$, where
$a_{pq}$ denotes the number of cycles $\theta_{i}$ in the above
decomposition of $\sigma$ of length $q$ such that $f(\theta_i)
\equiv p-1 \pmod{m}$.  The order of this class is
\begin{equation}
\frac{m^{n}n!}{\prod_{p,q}a_{pq}!(qm)^{a_{pq}}}
\end{equation}

We have, by definition, the short exact sequence
\begin{equation}
\begin{array}{cccccccccc}
1 & \longrightarrow & {\mathbb Z}_{m}^{n} & \longrightarrow &
B_{n}^{m} & \stackrel{\upsilon_{n}} {\longrightarrow} & S_{n} &
\longrightarrow & 1 & \end{array},\end{equation} where
$\upsilon_{n}$ is defined by $\upsilon_{n}(s_{i})=s_{i}, \:
\upsilon_{n}(w_{i})=1\;\mbox{for all} \; 1 \leq i \leq n,$ where
${\mathbb Z}_{m}^{n} = {\mathbb Z}_{m}\otimes \ldots \otimes
{\mathbb Z}_{m}$, ($n$ copies), where the $i$-th copy of ${\mathbb
Z}_{m}$ should be regarded as the cyclic group generated by
$w_{i}$. In the case where $m$ is even, there is a corresponding
short exact sequence \begin{equation}\begin{array}{cccccccccc} 1 &
\longrightarrow & {\mathbb Z}_{m/2}^{n} & \longrightarrow &
B_{n}^{m} & \stackrel{\tau_{n}} {\longrightarrow} & B_{n}^{2} &
\longrightarrow & 1 & \end{array},\end{equation} where $\tau_{n}$
is defined by $\tau_{n}(s_{i})=s_{i}, \enspace
\tau_{n}(w_{i})=w_{i}\: \mbox{for all} \enspace 1 \leq j \leq n,$
where now the $i$-th copy of ${\mathbb Z}_{m/2}$ should be
regarded as the cyclic group generated by $w_{i}^{2}$.

Under the homomorphism $\upsilon_{n}$ the class
$$(1^{{a}_{11}}2^{{a}_{12}} \ldots n^{{a}_{1n}};
1^{{a}_{21}}2^{{a}_{22}}
 \ldots n^{{a}_{2n}}; \ldots ; 1^{{a}_{m1}}2^{{a}_{m2}} \ldots
 n^{{a}_{mn}}),$$
of $B_{n}^{m}$  fuses to the class
$(1^{\sum_{i=1}^{m}a_{i1}}2^{\sum_{i=1}^{m}a_{i2}}
 \ldots n^{\sum_{i=1}^{m}a_{in}})$ of $S_{n}$ and under the homomorphism $\tau_{n}$ this class
 fuses to the class
$$(1^{\sum_{\stackrel{i=1}{i \:
odd}}^{m}a_{i1}}2^{\sum_{\stackrel{i=1}{i \: odd}}^{m}a_{i2}}
 \ldots n^{\sum_{\stackrel{i=1}{i \: odd}}^{m}a_{in}};1^{\sum_{\stackrel{i=1}{i\: even}}^{m}
 a_{i1}}2^{\sum_{\stackrel{i=1}{i \:even}}^{m}a_{i2}}
 \ldots n^{\sum_{\stackrel{i=1}{i \:even}}^{m}a_{in}})$$ of
 $B_{n}^{2}$.

These two isomorphisms will allow us to use known results about
the spin representations of the symmetric group $S_{n}$ and the
hyperoctahedral group $B_{n}^{2}$ to determine the spin
representations of $B_{n}^{m}$.

 The group ${B}_{n}^{m}$ has a total of $2m$ linear characters
defined by

\begin{equation}\left\{
\begin{array}{cc}
\sigma_{k}(s_{i}) = 1, 1 \leq i \leq n-1 &
\sigma_{k}(w_{j})=\zeta^{k}, 1 \leq j \leq n \\
\eta(s_{i})=-1, 1 \leq i \leq n-1 &
\eta(w_{j})=1, 1 \leq j \leq n \\
\epsilon_{k}(s_{i})=-1, 1 \leq i \leq n-1 &
\epsilon_{k}(w_{j})=\zeta^{k}, 1 \leq j \leq n,
\end{array}
\right.\end{equation}

\noindent where $1 \leq k \leq m-1$, together with the identity
character. In the special case $k=m/2$, we write $\epsilon$ for
$\epsilon_{m/2}$ and $\sigma$ for $\sigma_{m/2}$. The values of
these characters for an element in the class
$(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$ are as
follows
\begin{eqnarray*}
 \eta(\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}) & = & (-1)^{\sum_{i=1}^{m}l(\lambda_{(i)})}
,\\
\sigma(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}) & = &
(-1)^{n - \sum_{i=2,i \;even \;}^{m}l(\lambda_{(i)})},\\
 \epsilon(\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}) & = & (-1)^{n - \sum_{i=1,i \;odd
\;}^{m-1}l(\lambda_{(i)})}.
\end{eqnarray*}

Then, we prove the following lemma which describes the kernels of
some of the characters. The descriptions are given in terms of the
classes of conjugate elements of $B_{n}^{m}$.
\begin{lem}
(i) $ker \enspace \eta = \{ x \in (\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}) \enspace | \enspace
(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$ is even\},

(ii) $ker \enspace \sigma = \{ x \in (\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}) \enspace |\enspace \sum_{i=2\;i \;
even}^{m}\sum_{j=1}^{n}a_{ij}$ is even \},

(iii) $ker \enspace \epsilon=\eta \sigma = \{ x \in
(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}) \enspace
|\enspace (\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$ is
even and \linebreak $\sum_{i=2\;i \;
even}^{m}\sum_{j=1}^{n}a_{ij}$ is even , or $\enspace
(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$ is odd and
$\sum_{i=2\;i \; even}^{m}\sum_{j=1}^{n}a_{ij}$ is odd \}.
\end{lem}

{\it Proof.} (i) Since $\eta(w_{j})=1 \;\mbox{for}\; 1 \leq j \leq
n$ and $\eta(s_{j})=-1 \;\mbox{for}\; 1 \leq j \leq n$,  then for
$x \in ker \enspace \eta$, the total number of the generators
$s_{i}$ in any expression for $x$ must be even, that is, the
number of even cycles in this expression must be even, thus
$[\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}]$ is even.

(ii) Since $\sigma(w_{j})=-1 \;\mbox{for} \;1 \leq j \leq n$ and
$\sigma(s_{j})=1 \;\mbox{for}\; 1 \leq j \leq n$, then for $x \in
ker \enspace \sigma$, the total number of the generators $w_{i}$
in any expression for $x$ must be even. In an expression $x=
\prod_{i=1}^{t}\theta_{i}$ of $x$ as a product of cycles, the
cycles $\theta_{i}$ for which $f(\theta_{i}$ is even (odd) give
rise to an even (odd) number of $w_{j}$. Thus, for $\sigma(x) =
1$, we require an even number of cycles $\theta_{i}$ with
$f(\theta_{i})$ odd. This can only occur if $\sum_{i=2\;i \;
even}^{m}\sum_{j=1}^{n}a_{ij}$ is even.

(iii) Since $\epsilon(w_{j})=-1 \;\mbox{for}\; 1 \leq j \leq n$
and $\epsilon(s_{j})=-1 \;\mbox{for}\; 1 \leq j \leq n$, then for
$x \in ker \enspace \sigma$, the total number of the generators
$w_{i}$ and $s_{i}$ in any expression for $x$ must be even. Then,
for similar reasons to those in the proof of (i) and (ii), there
are two possible cases. Thus, either both
$(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$ and
$\sum_{i=2\;i \; even}^{m}\sum_{j=1}^{n}a_{ij}$ are even or both
are odd which results in the required conclusion.

If we now let $M = ker \; \eta \bigcap ker \; \sigma \bigcap ker
\; \epsilon$, then $M=  \{ x \in (\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}) \; | \;
(\lambda_{(1)};\lambda_{(2)};\linebreak \ldots; \lambda_{(m)})$ is
even and $\sum_{i=2\;i \; even}^{m}\sum_{j=1}^{n}a_{ij}$ is even
\}. Then, the following lemma can be proved.

\begin{lem} If $m$ is even, the following is a short exact sequence
$$\begin{array}{cccccccccc}
1 & \longrightarrow & M & \longrightarrow & B_{n}^{m} &
\longrightarrow & {\mathbb Z}_{2} \times {\mathbb Z}_{2} &
\longrightarrow & 1  \end{array}.$$
\end{lem}

{\it Proof.}  Define $\phi : B_{n}^{m}  \longrightarrow
 {\mathbb
Z}_{2} \times {\mathbb Z}_{2}$ by
$$ \phi(x) = (1,-1)^{k_{1}}(-1,1)^{k_{2}},$$
where $k_{1}$ and $k_{2}$ are the number of the $s_{i}$ and
$w_{i}$ respectively in any expression for $x$ in terms of the
generators of $B_{n}^{m}$. Then $\phi$ is well-defined. Clearly,
the map $\phi$ is surjective and it only remains to determine $ker
\phi$.

For $x \in ker \phi$, then it is necessary for both $k_{1}$ and
$k_{2}$ to be even. It now suffices to check against the
calculation of all the kernels in Lemma 3.1 to verify that $ker
\phi$ is indeed the subgroup M.


\section{A Covering Group $\tilde{B}_{n}^{m}$ of $B_{n}^{m}$ and its Basic Spin Representations}

The Schur multiplier of $B_{n}^{m}$ was obtained in \cite{dm74}
\begin{eqnarray}
H^{2}(B_{n}^{m},{\mathbb C}^{*})= \left\{ \begin{array}{ll}
{\mathbb Z}_{2}=\{\gamma\}             & \mbox{if $m$ is odd, $n \geq 4$}, \\
{\mathbb Z}_{2}\times{\mathbb Z}_{2}\times{\mathbb Z}_{2}=\{(\gamma,\lambda,\mu)\} &
\mbox{if $m$ is even, $n \geq 4$},\\
{\mathbb Z}_{2}\times{\mathbb Z}_{2}=\{(\lambda,\mu)\} & \mbox{if $m$ is even, $n =3$}, \\
{\mathbb Z}_{2}=\{\mu\}  & \mbox{if $m$ is even, $n=2$},  \\
\{1\}  & \mbox{otherwise},
\end{array} \right.
\end{eqnarray}
where $\gamma=\lambda=\mu=\pm 1$.

This means that if $m$ is even $B_{n}^{m}$ has eight $2$-cocycles
$\{(\gamma,\lambda, \mu)| \gamma^{2}=\lambda^{2}=\mu^{2}=1\}$ and
two $2$-cocycles if $m$ is odd, $\{(\gamma)|\gamma^{2}=1\}$. A
corresponding representation group is denoted by
$\tilde{B}_{n}^{m}$ which has a presentation
\begin{eqnarray}
\tilde{B}_{n}^{m}& = & <t_{i},\:1 \leq i \leq n-1,\; u_{j},\: 1 \leq j \leq n \enspace| \enspace t_{i}^{2}=1,u_{j}^{m}=1 \nonumber \\
     &   & {(t_{i}t_{i+1})}^{3}=1,\: 1 \leq i \leq n-2,\:t_{i}u_{i}=u_{i+1}t_{i},\: t_{i}u_{j}=\lambda u_{j}t_{i}, j \ne i,i+1 \nonumber \\
     &   & {(t_{i}t_{j})}^{2}=\gamma1, |i-j|\geq 2,\:1 \leq i,j \leq n-1, \\
     &   & u_{i}u_{j}=\mu u_{j}u_{i}, i \ne j,\:2 \leq i \leq n-1\rangle, \nonumber
\end{eqnarray}
where $$\gamma^{2} = \lambda^{(2,m)} = \mu^{(2,m)} = 1$$ and
$\gamma,\lambda,\mu$ commute with each other and with the
$t_{i},u_{j}$.

For simplicity, from now on, we will fix a 2-cocycle
$[\gamma,\lambda, \mu] \in (\gamma,\lambda, \mu)$, with
$\gamma^{2} = \lambda^{(2,m)} = \mu^{(2,m)} = 1$ and with the
convention that $\lambda = \mu = 1$ if $m$ is odd; $\gamma = 1$ if
$m$ is even and $n=3$; $\gamma = \lambda =1$ if $m$ is even and
$n=2$; and $\gamma = \lambda = \mu =1$ if $n=1$. Thus, the
2-cocycles will be denoted by $[\pm1,\pm1,\pm1]$; we note that
only the 2-cocycles $[\pm1,1,1]$ appear in the case $m$ odd (and
in particular for the group $S_{n}$).

The splitting classes for spin representations of ${B}_{n}^{m}$
for all 2-cocycles were first given by Read \cite{Re76} (who in
\cite{Re77} was the first to determine all the irreducible spin
representations of ${B}_{n}^{m}$ for all 2-cocycles). Later,
Stembridge \cite{Ste92} did the same for the hyperoctahedral
groups, the special case $m=2$. He showed that the splitting
classes are given as in Table 1. This table is broken into four
columns according to the four possible values of $\eta$ and
$\sigma$. The entry indicates the splitting classes of $B_{n}$
corresponding to the $2$-cocycle. For example, for the $2$-cocycle
$[1,-1,-1]$, the splitting classes $(\lambda, \mu)$ of $B_{n}$ for
which $\eta = -1, \sigma = -1$ are of the form $(DOP;DEP)$, that
is, $\lambda$ has distinct odd parts and $\mu$ has distinct even
parts.

\begin{table}[h]
\begin{center}
\begin{tiny}
\begin{tabular}{|c|c|c|c|c|} \hline
$2$-cocycle & $\eta = 1, \sigma = 1$ & $\eta = -1, \sigma = 1$ &
$\eta = 1, \sigma = -1$ & $\eta = -1, \sigma = -1$ \\
\hline $[1,-1,1]$ & $(P;P)$ & $(EP;\emptyset)$ &
$(DOP;DOP)$ & $(\emptyset;EP)$\\
$[-1,1,1]$ & $(OP;OP)$ & $(DP;DP)$ & $(OP;OP)$ & $(DP;DP)$
\\$[-1,-1,1]$ & $(OP;OP)$ &
$(DEP;\emptyset)$ & $(DP;DP)$ & $(\emptyset;DEP)$\\$[1,1,-1]$ &
$(OP;\emptyset)$ & $\emptyset$ & $(\emptyset;DP)$ &
$(\emptyset;DP)$\\$[1,-1,-1]$ & $(OP;\emptyset)$ &
$(\emptyset;DP)$ & $(\emptyset;OP)$ & $(DOP;DEP)$\\$[-1,1,-1]$ &
$(OP;EP)$ & $\emptyset$ & $(\emptyset;DOP)$ &
$(\emptyset;P)$\\
$[-1,-1,-1]$ & $(OP;EP)$ & $(\emptyset;P)$ & $(\emptyset;P)$ &
$(OP;EP)$\\ \hline
\end{tabular}
\end{tiny}
\caption{Splitting classes for $B_{n}^{2}$}
\end{center}
\end{table}

We now obtain splitting classes for the group ${B}_{n}^{m}$ for
all the $2$-cocycles. Indeed, the table in the case $m$ even can
be obtained directly from Table 1 using the homomorphism
$B_{n}^{m} \stackrel{\tau_{n}} {\longrightarrow} B_{n}^{2}$ given
in (3.3). Alternatively, these results can be proved directly
without invoking those obtained by Stembridge. Reinterpreting the
results of Read \cite{Re76} in our notation, shows that our
results are consistent with those obtained very much earlier by
him. We again note that only the second row of Table 2 is relevant
in the case $m$ odd.

\begin{table}[h]
\begin{center}
\begin{tiny}
\begin{tabular}{|c|c|c|c|c|} \hline
$2$-cocycle & $\eta = 1, \sigma = 1$ & $\eta = -1, \sigma = 1$ &
$\eta = 1, \sigma = -1$ & $\eta = -1, \sigma = -1$ \\
\hline $[1,-1,1]$ & $(P;\ldots ;P)$ &
$(EP;\emptyset;\ldots;EP;\emptyset)$ &
$(DOP;\dots;DOP)$ & $(\emptyset;EP;\ldots;\emptyset;EP)$\\
$[-1,1,1]$ & $(OP;\ldots ;OP)$ & $(DP;\ldots ;DP)$ & $(OP;\ldots
;OP)$ & $(DP;\ldots ;DP)$ \\$[-1,-1,1]$ & $(OP;\ldots ;OP)$ &
$(DEP;\emptyset;\ldots;DEP;\emptyset)$ & $(DP;\dots;DP)$ &
$(\emptyset;DEP;\ldots;\emptyset;DEP)$\\$[1,1,-1]$ &
$(OP;\emptyset;\ldots ;OP;\emptyset)$ & $\emptyset$ &
$(\emptyset;DP;\dots;\emptyset;DP)$ &
$(\emptyset;DP;\dots;\emptyset;DP)$\\$[1,-1,-1]$ &
$(OP;\emptyset;\ldots ;OP;\emptyset)$ &
$(\emptyset;DP;\dots;\emptyset;DP)$ &
$(\emptyset;OP;\dots;\emptyset;OP)$ &
$(DOP;DEP;\dots;DOP;DEP)$\\$[-1,1,-1]$ & $(OP;EP;\ldots ;OP;EP)$ &
$\emptyset$ & $(\emptyset;DOP;\dots;\emptyset;DOP)$ &
$(\emptyset;P;\dots;\emptyset;P)$\\
$[-1,-1,-1]$ & $(OP;EP;\ldots ;OP;EP)$ &
$(\emptyset;P;\dots;\emptyset;P)$ &
$(\emptyset;P;\dots;\emptyset;P)$ & $(OP;EP;\dots;OP;EP)$\\ \hline
\end{tabular}
\end{tiny}
\caption{Splitting classes for $B_{n}^{m}$}
\end{center}
\end{table}
For example, for the 2-cocycle $[-1,1,1]$, the splitting classes
of ${B}_{n}^{m}$ (or of $\tilde{B}_{n}^{m}$) in the notation of
this paper are classes of the $m$-partition form $(OP,OP, \ldots,
OP)$ and $(DP,DP, \ldots ,DP)$.

\subsection{\bf Basic spin representations of generalized
symmetric groups.} Let $W(\Phi)$ be the irreducible finite
reflection group of rank $l$ with root system $\Phi$ and simple
system $\Pi = \{\alpha_{1}, \ldots, \alpha_{l}\}$ and let
$\tau_{j} = \tau_{\alpha_{j}}$ be the reflection corresponding to
$\alpha_{j} \in \Pi$. Then the group $W(\Phi)$ is generated by the
simple reflections $\tau_{j},\enspace 1 \leq j \leq l$ subject to
the relations
$$\tau_{j}^{2} = 1, 1 \leq j \leq l,\enspace  (\tau_{j}\tau_{k})^{m_{jk}}
 = 1, 1 \leq j,k \leq l, j \neq k,$$
where $m_{jk}$ are positive integers such that $m_{kj} = m_{jk}$.

If the group $W(\Phi)$ of rank $l$ is embedded in the orthogonal
group
 $O(n)$; say $\phi : W(\Phi) \hookrightarrow O(n)$ is an embedding of $W(\Phi)$
into an orthogonal group $O(n)$, for some $n$, then let
$M_{\phi}(\Phi) = \rho_{n}^{-1}(W(\Phi))$. Then we have the
following (see \cite{ABS64} and \cite{Mo80} for the details
including notation)
\begin{equation}\begin{array}{ccccccccc} 1 & \longrightarrow &
{\mathbb Z}_{2} & \longrightarrow & Pin(n) & \stackrel{\rho_{n}}
{\longrightarrow} & O(n) & \longrightarrow & 1  \\
  &             &                 &             &        &     & \Big\uparrow \phi &    &  \\
1 & \longrightarrow & {\mathbb Z}_{2} & \longrightarrow &
M_{\phi}(\Phi) &
\stackrel{\rho_{n}}{\longrightarrow} & W(\Phi) & \longrightarrow & 1. \\
\end{array} \end{equation}
It is clear that the lower sequence in (4.3) is also an exact
sequence, but to show that $M_{\phi}(\Phi)$ is a covering group of
$W(\Phi)$, it is necessary to show that $M_{\phi}(\Phi)$ is a stem
extension of $W(\Phi)$, that is, to verify that
$${\mathbb Z}_{2} \subset Z(M_{\phi}(\Phi))\cap(M_{\phi}(\Phi))^{\prime}.$$
This will ensure that the basic spin representation of $O(n)$ will
still be a non-trivial spin representation, that is, not
projectively equivalent to an ordinary representation, on
restriction to the subgroup $W(\Phi)$. Furthermore, it was shown
in \cite{Mo80}, that if $n=l$ the basic spin representations $P,
P_{\pm}$ remain irreducible on restriction to the finite
irreducible reflection groups $W(\Phi)$, where $rank(\Phi)=l$.

This is now used to construct a number of basic spin
representations of $B_{n}^{m}$ for certain $2$-cocycles. We first
consider the natural embedding $\eta : W(\Phi) \hookrightarrow
O(l)$, where rank$\Phi = l$. In this case, put
$$ M(\Phi) = \phi_{l}^{-1}(W(\Phi)).$$
Then, a presentation of $M(\Phi)$ is obtained.  We have that
$$\rho_{l}(\alpha_{j}) =\tau_{j}, \enspace 1 \leq j \leq l.$$
and if we let $r_{j} =  \alpha_{j}/{\parallel \alpha_{j}
\parallel}, \enspace 1 \leq j \leq l,$
then we also have
$$\rho_{l}(r_{j}) =\tau_{j} = \tau_{r_{j}}, \enspace 1 \leq j \leq l.$$
If, in addition, $z \in Pin(l)$, is such that $\rho_{l}(z) =
 I_{l}$, then $z \in {\mathbb
Z}_{2}, \mbox{that is}, \enspace z^{2}=1$. Then, we have that the
group $M(\Phi)$ is generated by $r_{j}, 1 \leq j \leq l, \enspace
z$ subject to the relations $$(r_{j}r_{k})^{m_{jk}} =
z^{m_{jk}-1},\enspace 1 \leq j,k \leq l, \enspace z^{2} =1,
zr_{j}=r_{j}z, \enspace 1 \leq j \leq l.$$

We apply these results in particular to the reflection groups of
type $A_{n-1}$ (the symmetric group $S_{n}$), $B_{n}$ (the
hyperoctahedral group $B_{n}^{2}$) and $I_{2}(2)$ (the dihedral
group of order $4$).

\noindent {\bf Type $\mbox{\boldmath $A_{n-1}$}$}. In order to
apply the above, we use an embedding of the root system $A_{n-1}$
in ${\mathbb R}^{n-1}$ where the simple system is given by
$$\{\alpha_{j} =\sqrt{j-1}e_{j-1} - \sqrt{j+1}e_{j}, \; 1 \leq j \leq n-1 \},$$
(rather than the usual one)
 then
$$P(s_{j}) = \frac{1}{\sqrt{2j}}(\sqrt{j-1}M_{j-1} - \sqrt{j+1}M_{j}), \; 1 \leq j \leq n-1 $$
is the irreducible basic spin representation of $S_{n}$ if $n$ is
odd and $P_{\pm}$ are the two associate basic spin representation
of $W(A_{n-1})$ if $n$ is even. In the above, the generators
$\tau_{j}$ have been replaced by the corresponding ones in this
setting. In fact, we obtain the presentation
\begin{eqnarray*}
\tilde{A}_{n-1}& = & \langle t_{i},\;1 \leq i \leq n-1,z\; | \;
t_{i}^{2}=1,\;z^{2}=1,\; {(t_{i}t_{i+1})}^{3}=1,
\; 1 \leq i \leq n-2, \\
     &   & {(t_{i}t_{j})}^{2}=z,\; t_{i}z=zt_{i},\; |i-j|\geq 2,\;1 \leq i,j \leq
     n-1\rangle
\end{eqnarray*}
and thus, this representation, as was shown in \cite{Mo80} is the
irreducible basic spin representation of $S_{n}$ for the 2-cocycle
[-1,1,1]. Furthermore, the value of its character was determined
as given in the following proposition.
\begin{prop}
Let $\psi, (\psi_{\pm})$ be the character of the basic spin
representation $P, (P_{\pm})$.

(i) If $x \in(\rho), \rho \in OP(n)$,then
$$ \psi(x) =
\begin{array}{c} 2^{\lfloor\frac{1}{2}(l(\rho)-1)\rfloor}.\end{array}
$$

(ii) If $x \in (n)$, then $$ \psi_{\pm}(x) = \begin{array}{cc} \pm
i^{\frac{1}{2}(n-2)}\sqrt{n/2}&
 \mbox{if n is even}.\end{array}$$

(iii) $$\begin{array}{cc} \psi(x) = 0 &
\mbox{otherwise.}\end{array}$$

\end{prop}

These representations can in turn be lifted to give an irreducible
basic spin representation of $B_{n}^{m}$ again denoted by $P$
which corresponds to the 2-cocycle $[-1,1,1]$ using the
homomorphism $\upsilon_{n}$ defined in (3.2). This results in the
following proposition.

\begin{prop}
Let $\psi, (\psi_{\pm})$ be the character of the basic spin
representation $P, (P_{\pm})$.

(i) If $x \in (\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}),\; \lambda_{(i)}\in (OP(|\lambda_{(i)}|),\;1 \leq i
\leq m$, then
$$ \psi(x) =
\begin{array}{c}
2^{\lfloor\frac{1}{2}(l(\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)})-1)\rfloor}. \end{array}$$

(ii) If $ x \in (\emptyset; \ldots;\emptyset;n;\emptyset;
\ldots;\emptyset)$, then $$\psi_{\pm}(x) = \begin{array}{cc}\pm
i^{\frac{1}{2}(n-2)}\sqrt{n/2}& \mbox{if n is even},\end{array}$$
where $n$ can be in any one of $m$ possible positions.

(iii) $$ \psi(x) = \begin{array}{cc} 0 &
\mbox{otherwise}.\end{array}$$

\end{prop}

It was I. Schur \cite{Sch11} who first showed that the irreducible
representations for this 2-cocycle correspond to partitions
$\lambda \in DP(n)$. These were constructed in a remarkable way by
M. L. Nazarov \cite{Na90} which is a generalization of the above
construction which corresponds to the partition $(n)$. We briefly
recall his results.

Let $ \lambda \in DP(n)$, the {\bf shifted diagram} for $\lambda$
is $$ D_{\lambda} = \{(i,j) \in \mathbb{Z}^{2}\;|\; 1 \leq i \leq
l(\lambda);\; i \leq j \leq \lambda_{i}+i-1\}.$$ This is
represented graphically where a point $(i,j) \in \mathbb{Z}^{2}$
is represented by the unit square in the plane $\mathbb{R}^{2}$
with centre $(i,j)$, the coordinates $i$ and $j$ increasing  from
top to bottom and from left to right respectively. A {\bf shifted
tableau} of shape $\lambda$ is a bijection $\Delta:D_{\lambda}
\rightarrow \{1,2, \ldots,n\}$; a bijection is represented as a
filling of the squares of $D_{\lambda}$ with the numbers $1,2,
\ldots,n$, each of these numbers being used once only. A shifted
tableau $\Delta$ is {\bf standard} if the numbers increase down
its columns and across its rows. Now, let $\mathcal{S}_{\lambda}$
denote the set of all standard shifted tableaux of shape
$\lambda$. Let $\Delta \in \mathcal{S}_{\lambda}$ and let $k \in
\{1,2, \ldots, n\}$ be fixed. Let $k$ and $k+1$ have the
coordinates $(i,j)$ and $(i^{\prime},j^{\prime})$ in $\Delta$. Put
$a=j-i+1,\;b=j^{\prime}-i^{\prime}+1$. Consider $\Delta$ and
$s_{k}\Delta$, then $s_{k}\Delta \in \mathcal{S}_{\lambda}$ or
$s_{k}\Delta \not\in \mathcal{S}_{\lambda}$. Assume $a<b$,
otherwise work with $s_{k}\Delta$, even if $s_{k}\Delta \not\in
\mathcal{S}_{\lambda}$. Put
$$f(a,b) = \frac{\sqrt{2b(b-1)}}{(a-b)(a+b-1)},$$
$x=(-1)^{b+k}f(a,b)$, $y=(-1)^{a+k}f(b,a)$, $z=
\frac{\sqrt{1-x^{2}-y^{2}}}{2}$ and $u=\sqrt(1-x^{2})$. Let $$ A =
\left(
\begin{array}{cc}
  x & z \\
  z & y \\
\end{array}
\right), \; B = \left(
\begin{array}{cc}
  -y & z \\
  z & -x \\
\end{array}
\right)\; \mbox{and} \; C = \left(
\begin{array}{cc}
  x & u \\
  u & -x \\
\end{array}
\right).$$

Let $h$ be the number of rows in $\Delta$ occupied by $1,2,\ldots,
k+1$.Then, if $\Delta,\Delta^{\prime} \in \mathcal{S}_{\lambda}$,
put $$X_{\pm}^{\langle\lambda\rangle}(s_{k}) =
\left\{\begin{array}{ll} A \otimes M_{k-h+1} + B \otimes M_{k-h} &
\mbox{if}\; 1 < a < b
\\C \otimes M_{k-h+1} & \mbox{if} \; 1=a < b \end{array} \right.$$
and if $\Delta^{\prime} \not\in \mathcal{S}_{\lambda}$, replace
the matrices $A,B$ and $C$ by the element which appears in the
$(2,2)$-position. This is repeated for all the tableaux $\Delta
\in \mathcal{S}_{\lambda}$ and we obtain the following
proposition.
\begin{prop}
The $X^{\langle\lambda\rangle},\;\lambda \in
DP^{+}(n),\;X_{\pm}^{\langle\lambda\rangle},\;\lambda \in
DP^{-}(n)$ form a complete set of irreducible spin representations
of $S_{n}$ of degree $2^{\lfloor
\frac{n-l(\lambda)}{2}\rfloor}g_{\lambda}$, where $g_{\lambda}$ is
the number of shifted standard tableaux of shape $\lambda$.
\end{prop}
We note that the $\eta$-associator of $X^{\langle\lambda\rangle}$
is $id \otimes K^{\otimes\mu}$, where $\mu = \lfloor n/2 \rfloor$.

\noindent {\bf Type $\mbox{\boldmath $B_{n}$}$}. In order to apply
the above, we use an embedding of the root system $B_{n}$ in
${\mathbb R}^{n}$ where the simple system is given by
$$\{\alpha_{j} =e_{j-1} - e_{j}, \; 1 \leq j \leq n-1, \;\alpha_{n}=e_{n} \},$$
then
$$Q(s_{j}) =\frac{1}{\sqrt{2}} (M_{j-1} - M_{j}), \; 1 \leq j \leq n-1,\: Q(w_{1}) = M_{n} $$
is the irreducible basic spin representation of $W(B_{n})$ if $n$
is even and $Q_{\pm}$ are the two associate basic spin
representation of $W(B_{n})$ if $n$ is odd. Here, we have replaced
the notation $P$ $(P_{\pm})$ by $Q$ $(Q_{\pm})$ for obvious
reasons.

In this case, we obtain the presentation
\begin{eqnarray*}
\tilde{B}_{n}& = & \langle t_{i},\:1 \leq i \leq n-1,\; u_{j},\: 1
\leq j \leq n \;|
\; t_{i}^{2}=1,u_{j}^{2}=1, \; {(t_{i}t_{i+1})}^{3}=1, \\
     &   &  1 \leq i \leq n-2,\;
     t_{i}u_{i}=u_{i+1}t_{i},\; t_{i}u_{j}=\lambda u_{j}t_{i},
     j \ne i,i+1, \; {(t_{i}t_{j})}^{2}=\gamma, \\
     &   &  |i-j|\geq 2,\;1 \leq i,j \leq n-1, \;
     u_{i}u_{j}=\mu u_{j}u_{i}, i \ne j,\;2 \leq i \leq n-1\rangle,
\end{eqnarray*}
where $\gamma^{2} = \lambda^{(2,m)} = \mu^{(2,m)} = 1$ and
$\gamma,\lambda,\mu$ commute with each other and with the
$t_{i},u_{j}$ (note that $u_{n-i}=t_{n-i}u_{n-i+1}t_{n-i},\; 1
\leq i \leq n-1$) for the covering group of $B_{n}$. From this we
deduce that this representation is the irreducible basic spin
representation for the 2-cocycle $[-1,-1,-1]$. Furthermore, the
value of its character can be determined \cite{Mo80} as given in
the following proposition.

\begin{prop}
Let $\chi, (\chi_{\pm})$ be the character of the basic spin
representation $Q, (Q_{\pm})$.

(i) If $x \in(\rho;\varrho),\;(\rho;\varrho) \in
(OP(|\rho|);EP(|\varrho|))$, then
$$ \chi(x) = \left\{\begin{array}{cc}
2^{\frac{1}{2}(l(\rho;\varrho))}& \mbox{if n is even}\\
2^{\frac{1}{2}(l(\rho;\varrho)-1)} & \mbox{if n is odd}
\end{array}\\\right.$$

(ii) If $x \in (\emptyset;\varrho),\;\varrho \in P(n)$, then
$$ \chi_{\pm}(x) =
\begin{array}{cc}\pm i^{\frac{1}{2}(n-1)}2^{\frac{1}{2}(l(\varrho)-1)} &
 \mbox{if n is odd},\end{array}$$

(iii) $$ \chi(x) = \begin{array}{cc}0 &
\mbox{otherwise.}\end{array}$$

\end{prop}

In the same way as above, these representations and characters are
now lifted to $B_{n}^{m}$ using the homomorphism $\tau_{n}$
defined in (3.3) to give the following proposition.

\begin{prop}
Let $\chi, (\chi_{\pm})$ be the character of the basic spin
representation $Q, (Q_{\pm})$.

(i) If $x \in(\rho_{1};\varrho_{1};
\ldots;\rho_{m/2};\varrho_{m/2}),\; \rho_{i}\in OP;\varrho_{i} \in
EP,\; 1 \leq i \leq m/2$, then
$$ \chi(x) = \left\{\begin{array}{cc}
2^{\frac{1}{2}(\sum_{i=1}^{m/2}l(\rho_{i};\varrho_{i}))}& \mbox{if n is even}\\
2^{\frac{1}{2}(\sum_{i=1}^{m/2}l(\rho_{i};\varrho_{i})-1)} &
\mbox{if n is odd}
\end{array}\\\right.$$

(ii) If $x \in (\emptyset,\varrho_{1},
\ldots,\emptyset,\varrho_{m/2} ),\;\varrho_{i} \in P, \; 1 \leq i
\leq m/2$, then
$$ \chi_{\pm}(x) =
\begin{array}{cc}\pm i^{\frac{1}{2}(n-1)}2^{\frac{1}{2}(\sum_{i=1}^{m/2}l(\varrho_{i})-1)} &
 \mbox{if n is odd},\end{array}$$

(iii) $$ \chi(x) = \begin{array}{cc}0 &
\mbox{otherwise.}\end{array}$$

\end{prop}

In \cite{Mo03} it was shown how to determine an irreducible basic
spin representation of $B_{n}^{m}$ for the 2-cocycle $[1,-1,1]$.
In fact, we use the embedding $B_{n}^{m} \hookrightarrow O(2)$
given by
$$(\sigma \oplus \eta)(s_{i}) = \left( \begin{array}{rr}
1 & 0 \\ 0 & -1 \\ \end{array} \right), \enspace 1 \leq i \leq
l-1, \enspace (\sigma \oplus \eta)(w_{l}) = \left(
\begin{array}{rr} -1 & 0 \\ 0 & 1 \\ \end{array} \right).$$ Now,
if we use the exact sequence
$$\begin{array}{ccccccccc}
1 & \longrightarrow & {\mathbb Z}_{2} & \longrightarrow & Pin(2) &
\stackrel{\rho_{2}}
{\longrightarrow} & O(2) & \longrightarrow & 1 \\
 & & & & & & \Big\uparrow \sigma \oplus \eta & & \\
 & & & & & & B_{n}^{m}/M & & \\
\end{array} $$
we ultimately, by putting
$$ R(s_{i}) = M_{2}= \left( \begin{array}{rr} 0 & i \\ -i & 0
\end{array} \right),
 \: 1 \leq i \leq n-1, \: R(w_{l})=M_{1}=
  \left( \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right),$$
obtain an irreducible spin representation $R$ of degree $2$ of
$B_{n}^{m}$ corresponding to the 2-cocycle $[1,-1,1]$. The $\sigma
-,\eta - $ and $\epsilon-$associators of this representation are
$J,I$ and $K$ respectively.

The character of this representation is given by the following
proposition.
\begin{prop}
If $\xi$ denotes the character of $R$, then
\begin{eqnarray*}
\xi(x) & = & \left\{\begin{array}{cl} 2 & \mbox{if} \; x \in M \\0
& \mbox{otherwise.} \end{array}\right.
\end{eqnarray*}
\end{prop}

We have now constructed irreducible basic spin representations
$P,Q$ and $R$ of $B_{n}^{m}$ corresponding to the 2-cocycles
$[-1,1,1],[-1,-1,-1]$ and $[1,-1,1]$ (respectively) of degrees
$2^{\lfloor\frac{1}{2}(n-1)\rfloor}$, $2^{\frac{1}{2}n}$ or
$2^{\frac{1}{2}(n-1)}$ according as $n$ is even or odd and $2$
(respectively). These are now shown to be absolutely fundamental
in the construction of the irreducible projective representations
for the remaining $2$-cocycles. But before proceeding to show how
this is done, following Stembridge \cite{Ste92}, we apply
Proposition 2.1 to obtain a general result which proves to be
extremely helpful in many of these cases.

The $2$-cocycle of $R$ is $\alpha = [1,-1,1]$, let P be any
projective representation of $B_{n}^{m}$ with $2$-cocycle $\beta$,
then $R \otimes P$ is a projective representation of $B_{n}^{m}$
with $2$-cocycle $\alpha\beta$. Then we have the following
proposition.

\begin{prop} Let $P$ be an irreducible projective representation of
degree $d$ of $B_{n}^{m}$ with $2$-cocycle $\beta$.

(i)  If $L_{P}= \{1\}$, then $R \otimes P$ is an irreducible
representation of degree $2d$ of $B_{n}^{m}$ with 2-cocycle
$\alpha\beta$.

(ii)   If $L_{P}= \{1,\nu\}$, where $\nu \in L$, then $R \otimes
P$ is the direct sum of two inequivalent irreducible projective
representation of degree $d$ of $B_{n}^{m}$ with 2-cocycle
$\alpha\beta$.

(iii)   If $L_{P}= L$, and $U,V$ are the $\eta,\sigma$-associators
of $P$ respectively,  then

       (a) if $UV=-VU$, then $R \otimes
P$ is the direct sum of four inequivalent irreducible projective
representation of degree $d/2$ of $B_{n}^{m}$ with 2-cocycle
$\alpha\beta$.

       (b) if $UV=VU$, then $R \otimes
P$ is the direct sum of two equivalent irreducible projective
representation of degree $d$ of $B_{n}^{m}$ with 2-cocycle
$\alpha\beta$.
\end{prop}

\section{\bf Irreducible spin representations of generalized symmetric
groups.}

\subsection{The 2-cocycle [1,1,1] --- ordinary representations}

We first review the construction of the irreducible ordinary
representations of the generalized symmetric groups, these are the
ones corresponding to the $2$-cocycle $[1,1,1]$, see \cite{jk81},
but also for a treatment which is more in line with our
requirements, see the work of M. Saeed-ul Islam \cite{Sae87}. As
this work is not easily available a review of his presentation is
given below. Furthermore, H. Can \cite{Ca96} has given a
description of the construction of the corresponding Specht
modules and also in \cite{Ca98}, he gives a description of these
in the context of complex reflection groups. For recent work on
the calculation of the characters from a combinatorial point of
view, see \cite{AM02}.

Let $X^{[\lambda]}$ denote the irreducible representation of
$S_{n}$ corresponding to the partition $\lambda$ of $n$, let
$\chi^{[\lambda]}$ denote the corresponding irreducible character.
The irreducible representations of $B_{n}^{m}$ are indexed by
$m$-partitions $(\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)})$ of $n$, the corresponding representations and
characters will be denoted by $X^{[\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}]}$ and $\chi^{[\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}]}$ respectively.

If we let $p_{0}=0$ and $p_{i} = \sum_{j=1}^{i}k_{j}, \; 1 \leq i
\leq m$, then $B_{k_{i}}^{m}$ is the generalized symmetric group
acting on the set of $k_{i}$ elements $P_{i}=\{p_{i-1}+1, \ldots,
p_{i} \}, \;1 \leq i \leq m$, where $\sum_{i=1}^{m}k_{i}=n$. Then
let $B_{(k_{1}, \ldots, k_{m})}^{m} = B_{k_{1}}^{m} \times \cdots
\times B_{k_{m}}^{m}$ be the corresponding generalized Young
subgroup. Recall that we have defined earlier the linear
characters $\sigma_{k}, \; 1 \leq k \leq m-1$ by
$$\sigma_{k}(s_{i}) = 1, \;1 \leq i \leq n-1; \;
\sigma_{k}(w_{j})=\zeta^{k}, 1 \leq j \leq n.$$

The representation $X^{[\lambda;\emptyset; \ldots; \emptyset]}$ is
obtained by lifting $X^{[\lambda]}$ from $S_{n}$ to $B_{n}^{m}$,
we define $X^{[\emptyset;\ldots; \emptyset;\lambda;\emptyset;
\ldots; \emptyset]}$, where $\lambda$ is in the $(k+1)$-th
position, $ 1 \leq k \leq m-1$ to be $\sigma_{k} \otimes
X^{[\lambda;\emptyset; \ldots; \emptyset]}$. If
$|\lambda_{i}|=k_{i} \; 1 \leq i \leq m$, define
$$X^{[\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}]}= (X^{[\lambda_{(1)};\emptyset; \ldots;
\emptyset]} \otimes X^{[\emptyset;\lambda_{(2)}; \ldots;
\emptyset]} \otimes \cdots \otimes X^{[\emptyset;\emptyset;
\ldots; \lambda_{(m)}]})\uparrow B_{n}^{m},$$ inducing  from
$B_{(k_{1}, \ldots, k_{m})}^{m}$ to $B_{n}^{m}$. If we let
$$\chi_{(k_{1},\ldots,k_{m})} = 1 \otimes \sigma_{1} \otimes
\cdots \otimes \sigma_{m-1},$$ then we have
$$X^{[\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}]}= (\chi_{(k_{1},\ldots,k_{m})} \otimes
X^{[\lambda_{(1)};\emptyset; \ldots; \emptyset]} \otimes
X^{[\lambda_{(2)};\emptyset; \ldots; \emptyset]} \otimes \cdots
\otimes X^{[\lambda_{(m)};\emptyset; \ldots; \emptyset]})\uparrow
B_{n}^{m}.$$

We can now prove the following lemma and theorem.
\begin{lem} $\chi_{(k_{1},\ldots,k_{m})}$ is a linear character
of $B_{(k_{1}, \ldots, k_{m})}^{m}$ such that
\newline $\chi_{(k_{1},\ldots,k_{m})}^{g}(x) \neq
\chi_{(\tilde{k}_{1},\ldots,\tilde{k}_{m})}(x)$ for some $x \in
\ker \upsilon_{n}$ and for all $g \in  B_{n}^{m}$ unless
$(k_{1},\ldots,k_{m})=(\tilde{k}_{1},\ldots,\tilde{k}_{m})$ in
which case this holds for all $g \in B_{n}^{m}\backslash
B_{(k_{1}, \ldots, k_{m})}^{m}$.
\end{lem}

{\it Proof.} If $(\tilde{k}_{1},\ldots,\tilde{k}_{m}) \neq
(k_{1},\ldots,k_{m})$, assume without loss of generality that
$\tilde{k}_{i} > k_{i}$ for some $i$, that is, $P_{i} \subset
\tilde{P}_{i}$. If $g \in B_{n}^{m}$ is such that
$\upsilon_{n}(g)P_{i}=P_{i}$, if $j \in \tilde{P}_{i}\backslash
P_{i}$, then $\upsilon_{n}(j)\in P_{l}, \; l \neq i$ and we put $x
=$ \begin{tiny}
$(\begin{array}{@{\hspace{0.1cm}}c}1\\1\end{array}) \cdots
(\begin{array}{@{\hspace{0.1cm}}c}j\\\zeta j\end{array})\cdots
(\begin{array}{@{\hspace{0.1cm}}r}n\\n\end{array})$.\end{tiny}

If $\upsilon_{n}(g)P_{i} \neq P_{i}$, then there exists $ j \in
P_{i} \subset \tilde{P}_{i}$ such that  $\upsilon_{n}(j)\in P_{l},
\;1 \leq l \leq m, \; l \neq i$ and for this $j$, we define $x$ as
above.

In each case, $\chi_{(\tilde{k}_{1},\ldots,\tilde{k}_{m})}(x)=
\zeta^{i}$, but $$\chi_{(k_{1},\ldots,k_{m})}^{g}(x) =
\chi_{(k_{1},\ldots,k_{m})}(\upsilon_{n}(g)x\upsilon_{n}(g)^{-1})
= \zeta^{l}, \:l \neq i.$$ If
$(\tilde{k}_{1},\ldots,\tilde{k}_{m}) = (k_{1},\ldots,k_{m})$ and
$g \in B_{n}^{m}\backslash B_{(k_{1}, \ldots, k_{m})}^{m}$, then
there exists at least one index $i,\; 1 \leq i \leq m$ and an
integer $j \in P_{j}$ such that $\upsilon_{n}(j)\in P_{l},\;  1
\leq l \leq m, \; l \neq i$. Once again we define $x \in \ker
\upsilon_{n}$ as above for this particular $j$. Clearly,
$$\chi_{(k_{1},\ldots,k_{m})}^{g}(x) = \zeta^{l}\neq \zeta^{i} =
\chi_{(\tilde{k}_{1},\ldots,\tilde{k}_{m})}(x),$$ which completes
the proof of the lemma.

\begin{thm}
A complete set of inequivalent irreducible (ordinary)
representations  of $B_{n}^{m}$ is given by the
$X^{[\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}]}$, where
$(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$ is any
$m$-partition of $n$.
\end{thm}
{\it Proof.} Let $(k_{1},\ldots,k_{m})$ and
$(\tilde{k}_{1},\ldots,\tilde{k}_{m})$ be two arbitrary $m$-tuples
of $n$ and let \newline $X^{[\lambda_{(1)},\lambda_{(2)}, \ldots,
\lambda_{(m)}]}=X^{[\lambda_{(1)};\emptyset; \ldots; \emptyset]}
\otimes X^{[\emptyset;\lambda_{(2)}; \ldots; \emptyset]} \otimes
\cdots \otimes X^{[\emptyset;\emptyset; \ldots; \lambda_{(m)}]}$
and $X^{[\tilde{\lambda}_{(1)},\tilde{\lambda}_{(2)}, \ldots,
\tilde{\lambda}_{(m)}]}=X^{[\tilde{\lambda_{(1)}};\emptyset;
\ldots; \emptyset]} \otimes X^{[\emptyset;\tilde{\lambda_{(2)}};
\ldots; \emptyset]} \otimes \cdots \otimes
X^{[\emptyset;\emptyset; \ldots; \tilde{\lambda_{(m)}}]}$ be two
corresponding representations of $B_{(k_{1}, \ldots, k_{m})}^{m}$
and $B_{(\tilde{k}_{1},\ldots,\tilde{k}_{m})}^{m}$ with characters
$\chi^{[\lambda_{(1)},\lambda_{(2)}, \ldots, \lambda_{(m)}]}$ and
$\chi^{[\tilde{\lambda}_{(1)},\tilde{\lambda}_{(2)}, \ldots,
\tilde{\lambda}_{(m)}]}$ respectively as defined above. We will
prove that
$$(\chi^{[\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}]},\chi^{[\tilde{\lambda}_{(1)};\tilde{\lambda}_{(2)};
\ldots; \tilde{\lambda}_{(m)}]})_{B_{n}^{m}} = 0$$ unless $k_{i} =
\tilde{k}_{i}, \; 1 \leq i \leq m$, in which case it is equal to
$1$.

By Frobenius' reciprocity theorem and Mackey's subgroup theorem,
the above inner product is equal to
\begin{eqnarray*}
& &(\chi^{[\lambda_{(1)},\lambda_{(2)}, \ldots,
\lambda_{(m)}]},(\chi^{[\tilde{\lambda}_{(1)};\tilde{\lambda}_{(2)};
\ldots; \tilde{\lambda}_{(m)}]})\downarrow B_{(k_{1}, \ldots,
k_{m})}^{m} )_{B_{(k_{1}, \ldots, k_{m})}^{m}}
\\& = &\sum_{x}(\chi^{[\lambda_{(1)},\lambda_{(2)},
\ldots,
\lambda_{(m)}]},(\chi^{[\tilde{\lambda}_{(1)},\tilde{\lambda}_{(2)},
\ldots, \tilde{\lambda}_{(m)}]}\downarrow H_{x})\uparrow
B_{(k_{1}, \ldots, k_{m})}^{m} )_{B_{(k_{1}, \ldots,
k_{m})}^{m}}\\
&= &\sum_{x}(\chi^{[\lambda_{(1)},\lambda_{(2)}, \ldots,
\lambda_{(m)}]}\downarrow
H_{x},(\chi^{[\tilde{\lambda}_{(1)},\tilde{\lambda}_{(2)}, \ldots,
\tilde{\lambda}_{(m)}]})^{x}\downarrow H_{x})_{H_{x}},
\end{eqnarray*}
where $H_{x} = B_{(k_{1}, \ldots, k_{m})}^{m} \bigcap
x^{-1}B_{(\tilde{k}_{1},\ldots,\tilde{k}_{m})}^{m}x$,
$(\chi^{[\tilde{\lambda}_{(1)},\tilde{\lambda}_{(2)}, \ldots,
\tilde{\lambda}_{(m)}]})^{x}(x^{-1}gx) =$ \newline $
(\chi^{[\tilde{\lambda}_{(1)},\tilde{\lambda}_{(2)}, \ldots,
\tilde{\lambda}_{(m)}]})(g)$ for all $g \in
B_{(\tilde{k}_{1},\ldots,\tilde{k}_{m})}^{m}$ and $x$ ranges over
the double coset representatives of the generalized Young
subgroups $B_{(k_{1}, \ldots, k_{m})}^{m}$ and
$B_{(\tilde{k}_{1},\ldots,\tilde{k}_{m})}^{m}$ in $B_{n}^{m}$.

We now show that each term in the above summation is zero except
in the case noted above. If for some $x$,
$$\sum_{x}\chi^{[\lambda_{(1)},\lambda_{(2)}, \ldots,
\lambda_{(m)}]}\downarrow
H_{x}=\sum_{x}(\chi^{[\tilde{\lambda}_{(1)},\tilde{\lambda}_{(2)},
\ldots, \tilde{\lambda}_{(m)}]})^{x}\downarrow H_{x}$$ have an
irreducible component in common, then so do
$$\sum_{x}\chi^{[\lambda_{(1)},\lambda_{(2)}, \ldots,
\lambda_{(m)}]}\downarrow \ker
\upsilon_{n}=\sum_{x}(\chi^{[\tilde{\lambda}_{(1)},\tilde{\lambda}_{(2)},
\ldots, \tilde{\lambda}_{(m)}]})^{x}\downarrow \ker
\upsilon_{n},$$ since $\ker \upsilon_{n} \subset H_{x}$. Then,
using the alternative form for $X^{[\lambda_{(1)},\lambda_{(2)},
\ldots, \lambda_{(m)}]}$ given above, we see that these
respectively coincide with $\chi_{(k_{1},\ldots,k_{m})}\downarrow
\ker\upsilon_{n}$ and
$\chi_{(\tilde{k}_{1},\ldots,\tilde{k}_{m})}^{x}\downarrow \ker
\upsilon_{n}$. Since both of these are irreducible, they are equal
on $\ker \upsilon_{n}$ and so, by Lemma 5.1, we have
$(k_{1},\ldots,k_{m}) = (\tilde{k}_{1},\ldots,\tilde{k}_{m})$ and
$x \in B_{(k_{1}, \ldots, k_{m})}^{m}$. Now, using this
information, an elementary inner product calculation shows that
$$(\chi^{[\lambda_{(1)},\lambda_{(2)}, \ldots,
\lambda_{(m)}]},\chi^{[\tilde{\lambda}_{(1)},\tilde{\lambda}_{(2)},
\ldots, \tilde{\lambda}_{(m)}]})_{B_{(k_{1}, \ldots, k_{m})}^{m}}
= (\chi^{\lambda_{1}} \cdots
\chi^{\lambda_{m}},\chi^{\tilde{\lambda}_{1}} \cdots
\chi^{\tilde{\lambda}_{m}})_{S_{(k_{1}, \ldots, k_{m})}},$$ which
is non-zero only if these two characters are equal and we have the
desired result.

\subsection{The 2-cocycle [-1,1,1]}

The approach in this section follows closely that of the previous
section and thus the proof is only outlined, but now the
irreducible spin representations of $S_{n}$ are used in place of
the ordinary representations.

As constructed in Proposition 4.3, if $\lambda \in DP(n)^{+}$,
$X^{\langle\lambda\rangle}$ are the irreducible spin
representation of $S_{n}$ and if $\lambda \in DP(n)^{-}$,
$X_{\pm}^{\langle\lambda\rangle}$ are the two $\eta$-associate
irreducible spin representations. The corresponding spin
characters are denoted by
$\chi^{\langle\lambda\rangle},\;\chi_{\pm}^{\langle\lambda\rangle}$.

We show that the irreducible representations of $B_{n}^{m}$ for
the 2-cocycle [-1,1,1] are indexed by $m$-partitions
$(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$ of $n$,
where $\lambda_{(i)} \in DP(|\lambda_{(i)}|),\: 1 \leq i \leq m$,
the corresponding representations and characters will be denoted
by \newline $X^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle}$
$(X_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle})$and
$\chi^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle}$
$(\chi_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle})$(if they are \newline $\eta$-associate)
respectively.

The representation $X^{\langle\lambda;\emptyset; \ldots;
\emptyset\rangle}$ is obtained by lifting
$X^{\langle\lambda\rangle}$ from $S_{n}$ to $B_{n}^{m}$, we define
$X^{\langle\emptyset;\ldots; \emptyset;\lambda;\emptyset; \ldots;
\emptyset\rangle}$, where $\lambda$ is in the $(k+1)$-th position,
$ 1 \leq k \leq m-1$ to be $\sigma_{k} \otimes
X^{\langle\lambda;\emptyset; \ldots; \emptyset\rangle}$. If
$|\lambda_{i}|=k_{i} \: 1 \leq i \leq m$, define
$$X^{\langle\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}\rangle}= (X^{\langle\lambda_{(1)};\emptyset;
\ldots; \emptyset\rangle} \hat{\otimes}
X^{\langle\emptyset;\lambda_{(2)}; \ldots; \emptyset\rangle}
\hat{\otimes} \cdots \hat{\otimes} X^{\langle\emptyset;\emptyset;
\ldots; \lambda_{(m)}\rangle})\uparrow B_{n}^{m},$$ inducing  from
$B_{(k_{1}, \ldots, k_{m})}^{m}$ to $B_{n}^{m}$, where
$\hat{\otimes}$ is the twisted tensor product
\cite{Mo80},\cite{hh92}. Then we have
$$X^{\langle\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}\rangle}= (\chi_{(k_{1},\ldots,k_{m})}
\otimes X^{\langle\lambda_{(1)};\emptyset; \ldots;
\emptyset\rangle} \hat{\otimes} X^{\langle\lambda_{(2)};\emptyset;
\ldots; \emptyset\rangle} \hat{\otimes} \cdots \hat{\otimes}
X^{\langle\lambda_{(m)};\emptyset; \ldots;
\emptyset\rangle})\uparrow B_{n}^{m}.$$ There are similar
statements for the $\eta$-associate representations and
characters.

The representation $X^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle}$ can be written down explicitly using
formulae (2.11) and (2.12) by simply replacing the representation
$P_{j}$ by the representation $X^{\langle\emptyset;\ldots;
\emptyset;\lambda_{j};\emptyset; \ldots; \emptyset\rangle}$.
Furthermore, using Proposition 2.3 we get a far more explicit
formula for the character
$\chi^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle}$, in fact this formula is obtained by a
slight modification of the one in Proposition 2.3 and will not
thus be repeated.

\begin{thm}
A complete set of inequivalent irreducible spin representations of
$B_{n}^{m}$ for the 2-cocycle $[-1,1,1]$ is given by the
$$X^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle}  \:
 \mbox{if $n - l(\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)})$ is even}$$ and
$$X_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}\rangle}
\: \mbox{if $n - l(\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)})$ is odd},$$ where $(\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)})$ is an $m$-partition of $n$, with
$\lambda_{(i)} \in DP(|\lambda_{(i)}|), \: 1 \leq i \leq m$.
\end{thm}

The details of the proof will not be given in that it now follows
very closely the structure of the proof of Theorem 5.2 in the
previous section. The only major difference will be in the
character calculations.

{\bf From now on in this paper we assume that $m$ is even as the
remaining 2-cocycles only exist in this case.}

\subsection{The 2-cocycle [-1,-1,-1]}

Let $Q ,\;Q_{\pm}$ be the basic spin representation of $B_{n}^{m}$
for the 2-cocycle [-1,-1,-1] constructed in Section 4.1. Then, for
the $m$-partition of $n$
$(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;$ $ \ldots;
\lambda_{(m-1)};\emptyset)$, put $$Q
X^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset]} = Q \otimes
X^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset]}\; (n \;even)$$ and
$$Q_{\pm}X^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset]} = Q_{\pm} \otimes
X^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset]} \;(n \; odd).$$ Then, we prove the
following theorem.

\begin{thm}
A complete set of inequivalent irreducible spin representations of
$B_{n}^{m}$ for the 2-cocycle $[-1,-1,-1]$ is given by the
\newline $QX^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}\; (n \;even)$ and
$Q_{\pm}X^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]} \;(n \;odd)$, \newline where
$[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset]$ is an $m$-partition of $n$.
\end{thm}

{\bf Proof.} We shall give a proof in the case $n$ even only, the
odd case is dealt with similarly.

If we consider the representation
$X^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset]}$, for any element of cycle type
$(P(|\lambda_{(1)}|),0,P(|\lambda_{(3)}|),0,\ldots,P(|\lambda_{(m-1)}|),0)$
in $B_{n}^{m}$, by dividing each partition into its odd and even
parts, we obtain classes of $B_{n}^{m}$ of type
$(OP(|\rho_{1}|),EP(|\varrho_{1}|), \dots, OP(|\rho_{m/2}|),$
 $EP(|\varrho_{m/2}| ))$, where $|\rho_{i}|+|\varrho_{i}| =
|\lambda_{(2i-1)}|,\; 1 \leq i(odd) \leq m-1$. If $\zeta$ is the
character of $Q$, then by Proposition 4.4, we have that
$\zeta(\rho_{1},\varrho_{1}, \ldots,\rho_{m/2},\varrho_{m/2})$ are
nonzero on the classes of type $(OP(|\rho_{1}|),EP(|\varrho_{1}|),
\dots,$
 $OP(|\rho_{m/2}|),EP(|\varrho_{m/2}| ))$, thus it follows that the
characters
$(\zeta\chi^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]})$ are linearly independent.

Conversely, from Table 2 in Section 4 we see that the splitting
classes of $B_{n}^{m}$ for the $2$-cocycle $[-1,-1,-1]$ are of the
form $(OP,EP, \dots,OP,EP)$ and $(P,\emptyset,P,\emptyset, \ldots,
P,\emptyset)$; it can be shown that the latter only occurs for $n$
odd. Thus, for the case $n$ even, the above characters span the
space of spin characters. It only remains to show that these
characters are irreducible.

For, the case $n$ even, using Proposition 4.4, we have that

\begin{eqnarray*}
\lefteqn{||\zeta\chi^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}||^{2} }\\
& = & \sum_{\stackrel{\rho_{i}\in OP,\varrho_{i}\in EP}{1 \leq i
\leq m/2}} \frac{1}{z_{\rho_{1},\varrho_{1},
\ldots,\rho_{m/2},\varrho_{m/2}}}|\zeta\chi^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}(\rho_{1},\varrho_{1},
\ldots,\rho_{m/2},\varrho_{m/2})|^{2} \\
& = & \sum_{\stackrel{\rho_{i}\in OP,\varrho_{i}\in EP}{1 \leq i
\leq m/2}} \frac{1}{z_{\rho_{1},\varrho_{1},
\ldots,\rho_{m/2},\varrho_{m/2}}}2^{l(\rho,\varrho))}|\chi^{\lambda_{(1)}}(\rho_{1}\cup\varrho_{1})\cdots
\chi^{\lambda_{m-1}}(\rho_{m/2}\cup\varrho_{m/2})|^{2} \\
& = & \sum_{\stackrel{\rho_{i}\in OP,\varrho_{i}\in EP}{1 \leq i
\leq m/2}} \frac{1}{z_{\rho_{1},\varrho_{1}},
\cdots,z_{\rho_{m/2},\varrho_{m/2}}}\prod_{i=1}^{m/2}2^{l(\rho_{i},\varrho_{i}))}
|\prod_{i(odd)}^{m/2}\chi^{\lambda_{(2i-1)}}(\rho_{i}\cup\varrho_{i})|^{2}
= 1
\end{eqnarray*}
using the corresponding result, Theorem 9.2, in \cite{Ste92} and
where $z_{\rho_{1},\varrho_{1}, \ldots,\rho_{m/2},\varrho_{m/2}}$
is the order of the centralizer of that class in $B_{n}^{m}$.

\subsection{The 2-cocycle [1,-1,-1]}

In this case, a similar process to that used in the previous
section is applied to the representations
\begin{align*}
QX^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle} &= Q \otimes
X^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset\rangle},\\
QX_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle} &= Q \otimes
X_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle},\\
Q_{\pm}X^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle} &= Q_{\pm} \otimes
X^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset\rangle}
\end{align*} or 
$$
Q_{\pm}X_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle} = Q_{\pm} \otimes
X_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle},$$ 
where $
(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset)$ is an $m$-partition of $n$, with
$\lambda_{(i)} \in DP(|\lambda_{(i)}|)$,\break $ 1 \leq i \leq m-1$ as
the case may be. These representations have $2$-cocycle
$[1,-1,-1]$.

We prove the following theorem.

\begin{thm}
A complete set of inequivalent irreducible spin representations of
$B_{n}^{m}$ for the 2-cocycle $[1,-1,-1]$ is given by the

(i) if n is even
$$QX^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle} \:
 \mbox{if $n - l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset)$ is even}$$ and
$$QX_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset\rangle} \: \mbox{if $n -
l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset)$ is odd},$$ and

(ii) if n is odd
$$Q_{\pm}X^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle} \:
 \mbox{if $n - l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset)$ is even}$$ and
$$Q_{\pm}X_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset\rangle} \: \mbox{if $n -
l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset)$ is odd},$$ where
$(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset)$ is an $m$-partition of $n$, with
$\lambda_{(i)} \in DP(|\lambda_{(i)}|), \: 1 \leq i \leq m-1$.
\end{thm}

The proof follows along the same lines as the one given in the
previous section, we note that the splitting classes in this case
are 
\begin{gather*}
(OP;\emptyset;\ldots ;OP;\emptyset),\
(DOP;DEP;\dots;DOP;DEP),\\ (\emptyset;OP;\dots;\emptyset;OP)
\text{ and } 
(\emptyset;DP;\dots;\emptyset;DP),
\end{gather*} the latter two again only
occur in the case $n$ odd. In the even case, we use the well known
one-one correspondence between the sets $OP(n)$ and $DP(n)$ (which
is used in the case of Schur's theory for irreducible spin
representations osf $S_{n}$) and the clear one-one correspondence
between the sets $DP(n)$ and $DOP(k),DEP(n-k)$ (separate the odd
and even parts), for all values of $k$.


\subsection{The 2-cocycle [1,-1,1]}

The following lemma is required.
\begin{lem}
If $X^{[\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}]}$ is
an ordinary representation of $B_{n}^{m}$ with character \newline
$\chi^{[\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}]}$
corresponding to the $m$-partition $(\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)})$ of $n$, then
\begin{eqnarray*}
 (i) \;\eta \chi^{[\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}]}& = &
\chi^{[\lambda^{\prime}_{(1)};\lambda^{\prime}_{(2)}; \ldots;
\lambda^{\prime}_{(m)}]},  \\ (ii) \;\sigma
\chi^{[\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}]} & = &
\chi^{[\lambda_{(\frac{m}{2}+1)};
\ldots;\lambda_{(m)};\lambda_{(1)}; \ldots;
\lambda_{(\frac{m}{2})}]},\\(iii)\; \varepsilon
\chi^{[\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}]} & = &
\chi^{[\lambda^{\prime}_{(\frac{m}{2}+1)};
\ldots;\lambda^{\prime}_{(m)};\lambda^{\prime}_{(1)}; \ldots;
\lambda^{\prime}_{(\frac{m}{2})}]},
\end{eqnarray*}
where $\eta,\sigma,\varepsilon=\zeta\sigma$ are linear characters
of $B_{n}^{m}$.
\end{lem}
{\bf Proof}  Using the well-known fact \cite{jk81} that
$\eta\chi^{[\lambda]} =\chi^{[\lambda^{\prime}]}$, where
$\chi^{[\lambda^{\prime}]}$ is the character of $S_{n}$
$\eta-$associate to $\chi^{[\lambda]}$ and noting that
$$\chi^{[\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}]}=   \chi^{[\lambda_{(1)};\emptyset; \ldots;
\emptyset]} \otimes ( \sigma_{1} \otimes
X^{[\lambda_{(2)};\emptyset; \ldots; \emptyset]}) \otimes \cdots
\otimes (\sigma_{m-1} \otimes X^{[\lambda_{(m)};\emptyset; \ldots;
\emptyset]})$$ then (i) follows. Furthermore, since by definition
$$ \sigma \chi^{[\lambda;\emptyset; \ldots;
\emptyset]} = \sigma_{m/2} \otimes \chi^{[\lambda;\emptyset;
\ldots; \emptyset]} = \chi^{[\emptyset; \dots ;
\emptyset;\lambda;\emptyset; \ldots; \emptyset]},$$ (ii) also
follows. (iii) is now a direct consequence of (i) and (ii).

Recall that the subgroup $M$ of $B_{n}^{m}$ is defined by $M = ker
\; \eta \bigcap ker \; \sigma \bigcap ker \; \varepsilon$, thus we
have the following corollary.
\begin{cor}
$$\chi_{M}^{[\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}]} =
\chi_{M}^{[\lambda^{\prime}_{(1)};\lambda^{\prime}_{(2)}; \ldots;
\lambda^{\prime}_{(m)}]} = \chi_{M}^{[\lambda_{(\frac{m}{2}+1)};
\ldots;\lambda_{(m)};\lambda_{(1)}; \ldots;
\lambda_{(\frac{m}{2})}]}$$ $$ =
\chi_{M}^{[\lambda^{\prime}_{(\frac{m}{2}+1)};
\ldots;\lambda^{\prime}_{(m)};\lambda^{\prime}_{(1)}; \ldots;
\lambda^{\prime}_{(\frac{m}{2})}]}.$$
\end{cor}

If we now define $\xi^{[\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}]} = \xi \otimes \chi^{[\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}]}$, where $\xi$ is the character of the
irreducible basic spin representation $R$ of $B_{n}^{m}$ for the
$2$-cocycle $[1,-1,1]$, then it follows from Proposition 4.5 that
$$\xi^{[\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}]} =
\xi^{[\lambda^{\prime}_{(1)};\lambda^{\prime}_{(2)}; \ldots;
\lambda^{\prime}_{(m)}]} = \xi^{[\lambda_{(\frac{m}{2}+1)};
\ldots;\lambda_{(m)};\lambda_{(1)}; \ldots;
\lambda_{(\frac{m}{2})}]}$$ $$ =
\xi^{[\lambda^{\prime}_{(\frac{m}{2}+1)};
\ldots;\lambda^{\prime}_{(m)};\lambda^{\prime}_{(1)}; \ldots;
\lambda^{\prime}_{(\frac{m}{2})}]}.$$

If we now put $RX^{[\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}]} = R \otimes X^{[\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}]}$ for each $m$-partition
$(\lambda_{(1)};\lambda_{(2)}; \ldots;$ $ \lambda_{(m)})$ of $n$,
then the $RX^{[\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}]}$ are spin representations of $B_{n}^{m}$ with
2-cocycle $[1,-1,1]$. We use Proposition 4.7 to show that the
irreducible spin representations of $B_{n}^{m}$ for this 2-cocycle
appear as constituents of these.

\begin{thm}
The representation $RX^{[\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}]}$ is

(i)  is a sum of two equivalent irreducible representations if
$\lambda_{(i)}=\lambda_{(\frac{m}{2} + i)}\in SCP, \: 1 \leq i
\leq \frac{m}{2}$ and $n \equiv 0 \pmod{4}$,

(ii)  is a sum of four inequivalent irreducible spin
representations of equal degrees if
$\lambda_{(i)}=\lambda_{(\frac{m}{2} + i)}\in SCP, \: 1 \leq i
\leq \frac{m}{2}$ and $n \equiv 2 \pmod{4}$,

(iii)  is a sum of two inequivalent representations of equal
degrees if $\lambda_{(i)}\in SCP, \: 1 \leq i \leq m$ or
$\lambda_{(i)}=\lambda_{(\frac{m}{2} + i)}$ or
$\lambda_{(i)}=\lambda_{(\frac{m}{2} + i)}^{\prime},$
 $1 \leq i \leq
\frac{m}{2}$ but not $ \lambda_{(i)}=\lambda_{(\frac{m}{2} +
i)}\in SCP, \: 1 \leq i \leq \frac{m}{2}$.

In all other cases, the four representations
\newline $RX^{[\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}]},
RX^{[\lambda^{\prime}_{(1)};\lambda^{\prime}_{(2)}; \ldots;
\lambda^{\prime}_{(m)}]}, RX^{[\lambda_{(\frac{m}{2}+1)};
\ldots;\lambda_{(m)};\lambda_{(1)}; \ldots;
\lambda_{(\frac{m}{2})}]},
RX^{[\lambda^{\prime}_{(\frac{m}{2}+1)};
\ldots;\lambda^{\prime}_{(m)};\lambda^{\prime}_{(1)}; \ldots;
\lambda^{\prime}_{(\frac{m}{2})}]}$ are equivalent irreducible
spin representations of $B_{n}^{m}$.
\end{thm}


\subsection{The 2-cocycle [-1,-1,1]}

The procedure in this case follows that of the previous section,
but now we consider the representations
\newline $RX^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}\rangle} = R
\otimes X^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle}$
($RX_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle} = R \otimes
X_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle}$)
\newline if $n - l(\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)})$ is odd (even), where
$(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$ is an
$m$-partition of $n$, with $\lambda_{(i)} \in DP(|\lambda_{(i)}|),
\: 1 \leq i \leq m$. Then the
$RX^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle}$,
$RX_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle}$ are spin representations of $B_{n}^{m}$
with 2-cocycle $[-1,-1,1]$.

In this case the following lemma is required.
\begin{lem}
If $\chi^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle},\chi_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}\rangle}$ are the spin character of
$B_{n}^{m}$
 then
\begin{eqnarray*}
 (i) \;\eta \chi_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle} & = &
\chi_{\mp}^{\langle\lambda^{\prime}_{(1)};\lambda^{\prime}_{(2)};
\ldots; \lambda^{\prime}_{(m)}\rangle},\\
\;\eta\chi^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle} & = &
\chi^{\langle\lambda^{\prime}_{(1)};\lambda^{\prime}_{(2)};
\ldots; \lambda^{\prime}_{(m)}\rangle},\\
(ii) \;\sigma\chi^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle} & = &
\chi^{\langle\lambda_{(\frac{m}{2}+1)};
\ldots;\lambda_{(m)};\lambda_{(1)}; \ldots;
\lambda_{(\frac{m}{2})}\rangle}, \\
\;\sigma\chi_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle} & = &
\chi_{\pm}^{\langle\lambda_{(\frac{m}{2}+1)};
\ldots;\lambda_{(m)};\lambda_{(1)}; \ldots;
\lambda_{(\frac{m}{2})}\rangle},\\
(iii)\; \varepsilon
\chi_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle} & = &
\chi_{\mp}^{\langle\lambda^{\prime}_{(\frac{m}{2}+1)};
\ldots;\lambda^{\prime}_{(m)};\lambda^{\prime}_{(1)}; \ldots;
\lambda^{\prime}_{(\frac{m}{2})}\rangle},\\
\; \varepsilon \chi^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle} & = &
\chi^{\langle\lambda^{\prime}_{(\frac{m}{2}+1)};
\ldots;\lambda^{\prime}_{(m)};\lambda^{\prime}_{(1)}; \ldots;
\lambda^{\prime}_{(\frac{m}{2})}\rangle},
\end{eqnarray*}
where $\zeta,\sigma,\varepsilon = \zeta\sigma$ are linear
characters of $B_{n}^{m}$.
\end{lem}
{\bf Proof}  From \cite{Sch11} we have that
$\eta\chi_{\pm}^{\langle\lambda\rangle}
=\chi_{\mp}^{\langle\lambda\rangle}$, and noting that
$$\chi^{\langle\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}\rangle}=
\chi^{\langle\lambda_{(1)};\emptyset; \ldots; \emptyset\rangle}
\otimes ( \sigma_{1} \otimes X^{\langle\lambda_{(2)};\emptyset;
\ldots; \emptyset\rangle}) \otimes \cdots \otimes (\sigma_{m-1}
\otimes X^{\langle\lambda_{(m)};\emptyset; \ldots;
\emptyset\rangle})$$ then (i) follows. Furthermore, (ii) follows
using the same argument as in Lemma 5.6 (ii) and (iii) is now a
direct consequence of (i) and (ii).

\begin{cor}
$$\chi_{\pm M}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}\rangle} =
\chi_{\mp
M}^{\langle\lambda^{\prime}_{(1)};\lambda^{\prime}_{(2)}; \ldots;
\lambda^{\prime}_{(m)}\rangle} = \chi_{\pm
M}^{\langle\lambda_{(\frac{m}{2}+1)};
\ldots;\lambda_{(m)};\lambda_{(1)}; \ldots;
\lambda_{(\frac{m}{2})}\rangle}$$ $$ = \chi_{\mp
M}^{\langle\lambda^{\prime}_{(\frac{m}{2}+1)};
\ldots;\lambda^{\prime}_{(m)};\lambda^{\prime}_{(1)}; \ldots;
\lambda^{\prime}_{(\frac{m}{2})}\rangle}.$$
\end{cor}

If we now define $\xi^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle} = \xi \otimes
\chi^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle}$, where $\xi$ is the character of the
irreducible basic spin representation $R$ of $B_{n}^{m}$ for the
$2$-cocycle $[1,-1,1]$, then it follows from Proposition 4.6 that
$$\xi_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}\rangle} =
\xi_{\mp}^{\langle\lambda^{\prime}_{(1)};\lambda^{\prime}_{(2)};
\ldots; \lambda^{\prime}_{(m)}\rangle} =
\xi_{\pm}^{\langle\lambda_{(\frac{m}{2}+1)};
\ldots;\lambda_{(m)};\lambda_{(1)}; \ldots;
\lambda_{(\frac{m}{2})}\rangle}$$ $$ =
\xi_{\mp}^{\langle\lambda^{\prime}_{(\frac{m}{2}+1)};
\ldots;\lambda^{\prime}_{(m)};\lambda^{\prime}_{(1)}; \ldots;
\lambda^{\prime}_{(\frac{m}{2})}\rangle}.$$

\begin{thm}
The representation $RX^{\langle\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)}\rangle}$ is

(i)  is a sum of two irreducible equivalent representations if
$\lambda_{(i)}=\lambda_{(\frac{m}{2} + i)}, \: 1 \leq i \leq
\frac{m}{2}$ and $l(\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)})$ is even ,

(ii)  is a sum of four inequivalent irreducible representations of
equal degrees if $\lambda_{(i)}=\lambda_{(\frac{m}{2} + i)}, \: 1
\leq i \leq \frac{m}{2}$ and $l(\lambda_{(1)};\lambda_{(2)};
\ldots; \lambda_{(m)})$ is odd,

(iii)  is a sum of two inequivalent representations of equal
degrees if 
$$n - l(\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)})$$ 
is even, but $\lambda_{(i)} \neq
\lambda_{(\frac{m}{2} + i)},$ for some $ 1 \leq i \leq
\frac{m}{2}$.

If $n - l(\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)})$ is
odd, the four representations
$$RX_{\pm}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots; \lambda_{(m)}\rangle},
RX_{\mp}^{\langle\lambda_{(1)};\lambda_{(2)}; \ldots;
\lambda_{(m)}\rangle},$$
$$ RX_{\pm}^{\langle\lambda_{(\frac{m}{2}+1)};
\ldots;\lambda_{(m)};\lambda_{(1)}; \ldots;
\lambda_{(\frac{m}{2})}\rangle},
RX_{\mp}^{\langle\lambda_{(\frac{m}{2}+1)};
\ldots;\lambda_{(m)};\lambda_{(1)}; \ldots;
\lambda_{(\frac{m}{2})}\rangle}$$ are equivalent irreducible spin
representations of $B_{n}^{m}$.
\end{thm}

\subsection{The 2-cocycle [-1,1,-1]}
In this case, for the $m$-partition of $n$
$(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;\ldots;$
\newline $\lambda_{(m-1)};\emptyset)$, if $n$ is even, put $$RQ
X^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset]} = R \otimes
QX^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset]}$$ and if $n$ is odd, put
$$RQ_{\pm}X^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset]} = R \otimes
Q_{\pm}X^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}.$$ Then, these representations
have factor set $[-1,1,-1]$ and we prove the following theorem.
\begin{thm}
The representation
$RQX^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset]}$ is \newline(i)  is a sum of two
equivalent irreducible representations if $\lambda_{(i)} \in SCP,
\: 1 \leq i (odd)\leq m $ and n is even
\newline(ii)  is a sum of two inequivalent representations of equal
degrees if $\lambda_{(i)}\in SCP, \: 1 \leq i (odd)\leq m$ and n
is odd or $\lambda_{(i)}\not\in SCP, $ for some $1 \leq i
(odd)\leq m$ and n is even. \newline If $\lambda_{(i)}\not\in SCP,
\: 1 \leq i (odd)\leq m$ and n is odd, the representations
$RQ_{\pm}X^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]},$ $
RQ_{\pm}X^{[\lambda_{(1)}^{\prime};\emptyset;\lambda_{(3)}^{\prime};\emptyset;
\ldots; \lambda_{(m-1)}^{\prime};\emptyset]},$ $
RQ_{\pm}X^{[\lambda_{(\frac{m}{2})};\emptyset;\ldots;\lambda_{(m-1)};\emptyset;\ldots;\lambda_{(1)};\emptyset;
\ldots; \lambda_{(\frac{m}{2}-1)};\emptyset]}$,\newline  $
RQ_{\pm}X^{[\lambda_{(\frac{m}{2})}^{\prime};\emptyset;\ldots;\lambda_{(m-1)}^{\prime};\emptyset;\ldots;\lambda_{(1)}^{\prime};\emptyset;
\ldots; \lambda_{(\frac{m}{2}-1)}^{\prime};\emptyset]}$ are
equivalent irreducible spin representations of $B_{n}^{m}$.
\end{thm}

For the proof of this theorem, the lemma below will be needed in a
similar way to the preceding sections; we denote the character of
$QX^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset]}$ by \newline
$\zeta\chi^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}$.
\begin{lem}
If $n$ is even, then \newline$\begin{array}{rcl}
 (i) \;\eta (\zeta\chi^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}) \kern-9pt& = &\kern-9pt
\zeta\chi^{[\lambda^{\prime}_{(1)};\emptyset;\lambda^{\prime}_{(3)};\emptyset;
\ldots; \lambda^{\prime}_{(m-1)};\emptyset]},\\ (ii) \;\sigma
(\zeta\chi^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}) \kern-9pt& = &\kern-9pt
\zeta\chi^{[\lambda^{\prime}_{(1)};\emptyset;\lambda^{\prime}_{(3)};\emptyset;
\ldots; \lambda^{\prime}_{(m-1)};\emptyset]}, \\
(iii)\; \varepsilon
(\zeta\chi^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}) \kern-9pt& = &\kern-9pt
\zeta\chi^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}.
\end{array}$
\newline
If $n$ is odd, then \newline$\begin{array}{rcl}
 (i)\ \eta (\zeta_{\pm}\chi^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}) \kern-9pt& = &\kern-9pt
\zeta_{\mp}\chi^{[\lambda^{\prime}_{(1)};\emptyset;\lambda^{\prime}_{(3)};\emptyset;
\ldots; \lambda^{\prime}_{(m-1)};\emptyset]},\\ 
(ii)\ \sigma
(\zeta_{\pm}\chi^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}) \kern-9pt& = &\kern-9pt
\left\{\begin{array}{@{}l@{}l}\zeta_{\pm}\chi^{[\lambda_{(\frac{m}{2}+1)};\emptyset;\ldots;
\lambda_{(m-1)} ;\emptyset;\lambda_{1};\emptyset;
\ldots;\lambda_{(\frac{m}{2}-1)};\emptyset]}&\mbox{if}\;m \equiv 0
\pmod{4}
\\\zeta_{\pm}\chi^{[\emptyset;\lambda_{(\frac{m}{2}+2)};\ldots;
\lambda_{(m-1)} ;\emptyset;\lambda_{1};\emptyset;
\ldots;\emptyset;\lambda_{(\frac{m}{2})}]}&\mbox{if}\; m \equiv 2
\pmod{4}
\end{array}\right.\\
(iii)\ \varepsilon
(\zeta_{\pm}\chi^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}) \kern-9pt& = &\kern-9pt
\left\{\begin{array}{@{}l@{}l}\zeta_{\pm}\chi^{[\lambda^{\prime}_{(\frac{m}{2}+1)};\emptyset;\ldots;
\lambda^{\prime}_{(m-1)}
;\emptyset;\lambda^{\prime}_{1};\emptyset;
\ldots;\lambda^{\prime}_{(\frac{m}{2}-1)};\emptyset]}&\mbox{if}\;m
\equiv 0 \pmod{4}
\\\zeta_{\pm}\chi^{[\emptyset;\lambda^{\prime}_{(\frac{m}{2}+2)};\ldots;
\lambda^{\prime}_{(m-1)}
;\emptyset;\lambda^{\prime}_{1};\emptyset;
\ldots;\emptyset;\lambda^{\prime}_{(\frac{m}{2})}]}&\mbox{if}\; m
\equiv 2 \pmod{4}
\end{array}\right.
 \\
\end{array}$\newline 
where $\zeta,\sigma,\varepsilon = \zeta\sigma$
are linear characters of $B_{n}^{m}$.
\end{lem}
{}From this lemma, we obtain, as before in Section 5.5, the
following corollary.
\begin{cor}
If $n$ is even, then
$$\zeta\chi_{M}^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}  =
\zeta\chi_{M}^{[\lambda^{\prime}_{(1)};\emptyset;\lambda^{\prime}_{(3)};\emptyset;
\ldots; \lambda^{\prime}_{(m-1)};\emptyset]}.$$ If $n$ is odd,
then
$$\zeta\chi_{M}^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}  =
\zeta\chi_{M}^{[\lambda^{\prime}_{(1)};\emptyset;\lambda^{\prime}_{(3)};\emptyset;
\ldots; \lambda^{\prime}_{(m-1)};\emptyset]}$$ $$ =
\left\{\begin{array}{@{}l@{}l}\zeta_{\pm}\chi^{[\lambda_{(\frac{m}{2}+1)};\emptyset;\ldots;
\lambda_{(m-1)} ;\emptyset;\lambda_{1};\emptyset;
\ldots;\lambda_{(\frac{m}{2}-1)};\emptyset]}&\mbox{if}\;m \equiv 0
\pmod{4}
\\\zeta_{\pm}\chi^{[\emptyset;\lambda_{(\frac{m}{2}+2)};\ldots;
\lambda_{(m-1)} ;\emptyset;\lambda_{1};\emptyset;
\ldots;\emptyset;\lambda_{(\frac{m}{2})}]}&\mbox{if}\; m \equiv 2
\pmod{4}
\end{array}\right.$$
$$ = \left\{\begin{array}{@{}l@{}l}\zeta_{\pm}\chi^{[\lambda^{\prime}_{(\frac{m}{2}+1)};\emptyset;\ldots;
\lambda^{\prime}_{(m-1)}
;\emptyset;\lambda^{\prime}_{1};\emptyset;
\ldots;\lambda^{\prime}_{(\frac{m}{2}-1)};\emptyset]}&\mbox{if}\;m
\equiv 0 \pmod{4}
\\\zeta_{\pm}\chi^{[\emptyset;\lambda^{\prime}_{(\frac{m}{2}+2)};\ldots;
\lambda^{\prime}_{(m-1)}
;\emptyset;\lambda^{\prime}_{1};\emptyset;
\ldots;\emptyset;\lambda^{\prime}_{(\frac{m}{2})}]}&\mbox{if}\; m
\equiv 2 \pmod{4}
\end{array}\right..$$
\end{cor}

In turn, this leads to the corresponding results for the
characters
$\xi\zeta\chi_{M}^{[\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]}$ and in turn to the proof of
Theorem 5.12.


\subsection{The 2-cocycle [1,1,-1]}
In this case, let
$(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;\ldots;$
 $\lambda_{(m-1)};\emptyset)$ be the $m$-partition of $n$ with $\lambda_{(i)} \in DP(|\lambda_{(i)}|),
\: 1 \leq i \leq m$. \newline If $n$ is even and $n -
l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset)$ is even,  put
$$RQX^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle} = R \otimes
QX^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}\;$$ and if $n -
l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset)$ is odd,  put
$$RQX_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle} = R \otimes
QX_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle};$$ and if $n$ is odd and
$n - l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset)$ is even,  put
$$RQ_{\pm}X^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle} = R \otimes
Q_{\pm}X^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}\;$$ and if $n -
l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset)$ is odd,  put
$$RQ_{\pm}X_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle} = R \otimes
Q_{\pm}X_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}.$$ These representations
have $2$-cocycle $[1,1,-1]$.

\begin{thm}
The representation
$RQX^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}$ is

(i)  is a sum of two equivalent irreducible representations if  $n
- l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};$ \newline $\emptyset)$ is even and n is even ,

(ii)  is a sum of two inequivalent representations of equal
degrees if  $n -
l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;$
\newline $ \lambda_{(m-1)};\emptyset)$ is even and n is odd or if $n
- l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset; \ldots;
\lambda_{(m-1)};\emptyset)$ is odd and n is even.

If  $n - l(\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset)$ is odd and n is odd, the
representations \newline $R
Q_{\pm}X_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}$ and
$RQ_{\pm}X_{\pm}^{\langle\lambda_{(\frac{m}{2})};\emptyset;\ldots;
\lambda_{(m-1)};\emptyset;\ldots;\lambda_{(1)};\emptyset; \ldots;
\lambda_{(\frac{m}{2}-1)};\emptyset\rangle}$ are equivalent
irreducible spin representations of $B_{n}^{m}$.
\end{thm}
In this case, we merely state the corresponding lemma to Lemma
5.13.
\begin{lem}
If $n$ is even, then \newline$\begin{array}{rcl}
 (i) \;\eta (\zeta\chi_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}) \kern-9pt& = \kern-9pt&
\zeta\chi_{\mp}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset]},\\ (ii) \;\sigma
(\zeta\chi_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}) \kern-9pt& = &\kern-9pt
\zeta\chi_{\mp}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}, \\
(iii)\; \varepsilon
(\zeta\chi_{\pm}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}) \kern-9pt& = &\kern-9pt
\zeta\chi_{\mp}^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}.
\end{array}$
\newline
If $n$ is odd, then \newline$\begin{array}{rcl}
 (i)\ \eta (\zeta_{\pm}\chi^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}) \kern-9pt& = &\kern-9pt
\zeta_{\mp}\chi^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle},\\ 
(ii)\ \sigma
(\zeta_{\pm}\chi^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}) \kern-9pt& = &\kern-9pt
\left\{\begin{array}{@{}l@{}l}\zeta_{\mp}\chi^{\langle\lambda_{(\frac{m}{2}+1)};
\emptyset;\ldots; \lambda_{(m-1)}
;\emptyset;\lambda_{1};\emptyset;
\ldots;\lambda_{(\frac{m}{2}-1)};\emptyset\rangle}&\mbox{if}\;m
\equiv 0 \pmod{4}
\\\zeta_{\mp}\chi^{\langle\emptyset;\lambda_{(\frac{m}{2}+2)};\ldots;
\lambda_{(m-1)} ;\emptyset;\lambda_{1};\emptyset;
\ldots;\emptyset;\lambda_{(\frac{m}{2})}\rangle}&\mbox{if}\; m
\equiv 2 \pmod{4}
\end{array}\right.\\
(iii)\ \varepsilon
(\zeta_{\pm}\chi^{\langle\lambda_{(1)};\emptyset;\lambda_{(3)};\emptyset;
\ldots; \lambda_{(m-1)};\emptyset\rangle}) \kern-9pt& = &\kern-9pt
\left\{\begin{array}{@{}l@{}l}\zeta_{\mp}\chi^{\langle\lambda_{(\frac{m}{2}+1)};
\emptyset;\ldots; \lambda_{(m-1)}
;\emptyset;\lambda_{1};\emptyset;
\ldots;\lambda_{(\frac{m}{2}-1)};\emptyset\rangle}&\mbox{if}\;m
\equiv 0 \pmod{4}
\\\zeta_{\mp}\chi^{\langle\emptyset;\lambda_{(\frac{m}{2}+2)};\ldots;
\lambda^{\prime}_{(m-1)} ;\emptyset;\lambda_{1};\emptyset;
\ldots;\emptyset;\lambda_{(\frac{m}{2})}\rangle}&\mbox{if}\; m
\equiv 2 \pmod{4}
\end{array}\right.
 \\
\end{array}$\newline 
where $\zeta,\sigma,\varepsilon = \zeta\sigma$
are linear characters of $B_{n}^{m}$.
\end{lem}



% ----------------------------------------------------------------

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\end{small}
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\end{document}

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