Séminaire Lotharingien de Combinatoire 50 (2003), Article B50b

PROJECTIVE REPRESENTATIONS OF GENERALIZED SYMMETRIC GROUPS

ALUN O MORRIS AND HUW I JONES

1. Introduction

The representation theory of generalized symmetric groups has been of interest over a long period dating back to the classical work of W. Specht [28],[29] and M.Osima [22] — an exposition of this work and other references may be found in [12]. 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 [24] which was followed later by an improvement in the work of M. Saeed-ul-Islam, see, for example, [26]. 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 [20] which elaborated on the earlier work of E. W. Read and M. Saeed-ul-Islam and also by J. Stembridge [31] 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 [13] 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 [3],[4] and [5]. 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 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 [12]) and I. Schur [30] 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 ℤ_{2}^{2}-quotients [31] 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, [17], [18], [19]. 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 [21]. 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.

2. Background and Notation

2.1. Partitions. The notation follows [14]. Let λ = (λ_{1}, λ_{2}, …, λ_{k}) be a partition of n, then l(λ) = k is the length of λ and |λ| = n is the weight of λ. The conjugate of λ is denoted by λ′. A partition λ is called an even(odd) partition if the number of even parts in λ is even(odd). A partition is sometimes written as λ = (1^{a_{1}}2^{a_{2}} … n^{a_{n}}), 0 ≤ a_{i} ≤ n indicating that a_{i} parts of λ are equal to i, 1 ≤ i ≤ n, |λ| = ∑_{i=1}^{n} ia_{i} and l(λ) = ∑_{i=1}^{n} a_{i}.

Let P (n) denote the set of all partitions of n, then DP (n) = {λ ∈ P (n) | λ_{1} > λ_{2} > ⋯ > λ_{k} > 0} is the set of all partitions of n into distinct parts, DP^{+}(n) = {λ ∈ DP (n) | |λ| − l(λ) is even}, DP^{−}(n) = {λ ∈ DP (n) | |λ| − l(λ) is odd}, OP (n) = {λ = (1^{α_{1}}3^{α_{3}} …)} is the set of all partitions of n into odd parts, EP (n) = {λ = (2^{α_{2}}4^{α_{4}} …)} is the set of all partitions of n into even parts and SCP (n) = {λ ∈ P (n) | λ = λ′} is the set of self-conjugate partitions of n.

An m-partition of n is a partition comprising of m partitions (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) such that λ_{(i)} ∈ P (n_{i}) , 1 ≤ i ≤ m and ∑_{i=1}^{m} n_{i} = n. The partition λ_{(i)} is written as (λ_{i1}, λ_{i2}, …, λ_{ik_{i}}), where k_{i} = l(λ_{(i)}) for 1 ≤ i ≤ m. The conjugate of (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is the m-partition (λ′_{(1)}; λ′_{(2)}; …; λ′_{(m)}). An m-partition is said to be even(odd) if the total number of even parts of (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is even(odd). An m-partition is sometimes written in the form

((1^{α_{11}}2^{α_{12}} …); (1^{α_{21}}2^{α_{22}} …); …; (1^{α_{m1}}2^{α_{m2}} …));

l(λ_{(1)}; λ_{(2)}; …; λ_{(m)}) = l(λ_{(1)}) + l(λ_{(2)}) + ⋯ + l(λ_{(m)}) is the length of (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) and |(λ_{(1)}; λ_{(2)}; …; λ_{(m)})| = |(λ_{(1)})|+|(λ_{(2)}|+⋯+|(λ_{(m)})| is the weight of (λ_{(1)}; λ_{(2)}; …; λ_{(m)}). We note that l(λ_{(1)}; λ_{(2)}; …; λ_{(m)}) = Σ_{i=1}^{m}Σ_{j=1}^{n}a_{ij}.

2.2. 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|, ℂ the field of complex numbers, ℂ^{×} = ℂ \ {0} and GL(n, ℂ) the group of invertible n × n matrices over ℂ.

A projective representation of degree n of G is a map P : G → GL(n, ℂ) such that for g, h ∈ G

P (g)P (h) = α(g, h)P (gh)

and P (1) = I_{n}, where I_{n} is the identity n×n matrix and α(g, h) ∈ ℂ^{×}. Since multiplication in G and GL(n, ℂ) is associative, it follows that

(2.1)  α(g, h)α(gh, k) = α(g, hk)α(h, k)

for all g, h, k ∈ G. A map α : G × G → ℂ^{×} which satisfies (2.1) is called a 2cocycle(factor set) of G in ℂ and we shall say that P is a projective representation with 2-cocycle α.

Projective representations P and Q of degree n with 2-cocycles α and β respectively are said to be projectively equivalent if there exists a map μ : G → ℂ^{×} and a matrix S ∈ GL(n, ℂ) such that

Q(g) = μ(g)S^{−1}P (g)S

for all g ∈ G. If P and Q are projectively equivalent, it follows that

(2.2)  β(g, h) = \frac{μ(g)μ(h)}{μ(gh)}α(g, h)

for all g, h ∈ G. The corresponding 2-cocycles β and α are then said to be equivalent.

Let H^{2}(G, ℂ^{×}) denote the set of equivalence classes of 2-cocycles; then H^{2}(G, ℂ^{×}) is an abelian group which is called the 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, ℂ^{×}) 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 central extension (H, ϕ) of a group G is a group H together with a homomorphism ϕ : H → G such that kerϕ ⊂ Z(H), where Z(H) is the centre of H, that is,

1 → kerϕ → H →^{ϕ} G → {1}

is exact. Let A = kerϕ, and let {γ(g) | g ∈ G} be a set of coset representatives of A in H which are in 1 − 1 correspondence with the elements of G; thus

H = ⋃_{g∈G} Aγ(g).

Then, for all g, h ∈ G, let a(g, h) be the unique element in A such that

γ(g)γ(h) = a(g, h)γ(gh).

The associative law in H and G implies that

(2.3)  a(g, h)a(gh, k) = a(g, hk)a(h, k)

for all g, h, k ∈ G. Now, let γ be a linear character of the abelian group A and put

α(g, h) = γ(a(g, h))

for all g, h ∈ G, then (2.3) implies that α is a 2-cocycle of G.

Now, let T be an ordinary irreducible representation of H of degree n put P (g) = T (γ(g)) for all g ∈ G, then P is a projective representation of G with 2-cocycle α. A projective representation P of G arising from an irreducible ordinary representation T of H in this way is said to be 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, ℂ^{×}) which linearizes every projective representation of G. Such a group H is called a 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 fixed 2-cocycle α 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 α-covering group of G.

Let α be a 2-cocycle such that {α} has order n and let ω be a primitive n-th root of unity, then α(g, h) = ω^{η(g,h)} for some 0 ≤ η(g, h) < n. Suppose that {ν(g) | g ∈ G} is a set of distinct symbols in one-one correspondence with the elements of G. Let G(α) = {(α^{j}, ν(g)) | 0 ≤ j < n, g ∈ G}, then it is easily verified that G(α) is a group with composition defined by

(α^{j}, ν(g))(α^{k}, ν(h)) = (α^{j+k+η(g,h)}, ν(gh))

for all g, h ∈ G, 0 ≤ j, k < n.

If now P is a projective representation of G of degree n with 2-cocycle α, then define T : G(α) → GL(n, ℂ) by

T (α^{j}, ν(g)) = ω^{j}P (g),

then T is an ordinary representation of G(α). That is, P has been lifted to an ordinary representation of G(α). Such a group G(α) is called an α-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(α) as double covers. As we are basically working with ordinary representations of the G(α), 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 ∈ G(α) is represented faithfully, we refer to these as spin representations of G with 2-cocycle α.

If C denotes a class of conjugate elements in G, let C(α) ∈ G(α) denote the inverse image in G(α). If any g ∈ C(α) is conjugate to −g, then C(α) is a class of conjugate elements in G(α), otherwise C(α) 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.

2.3. Clifford theory for ℤ_{2}^{2}-quotients. Let G be a group with a subgroup H of index 2 and let η be a linear character of G defined by

η(g) = \{ \begin{matrix}1 & if g ∈ H \\ −1 & if g ∉ H. \end{matrix}

If T is an irreducible representation of G with character χ, then η⊗T is also an irreducible representation of G, if these representations are equivalent, then we say that T is selfassociate, but if not, they are said to be η-associate, and are denoted by T_{+} and T_{−}, their characters are denoted by χ_{+} and χ_{−}; clearly χ_{−}(g) = η(g)χ_{+}(g) for all g ∈ G. If T is self-associate, then the unique(up to sign) matrix S such that

T (g)S = η(g)ST (g)

for all g ∈ G, is called the η-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 χ_{1} and χ_{2} respectively, then the difference character Δ^{η}χ, is defined by

Δ^{η}χ(g) = trST (g) = χ_{1}(g) − χ_{2}(g)

for all g ∈ H. Knowledge of the difference character then gives the corresponding characters of H,

\frac{1}{2}(χ ± Δ^{η}χ).

All the above results are classic [12] and date back to A. H. Clifford. Recently, J. R. Stembridge [31] has extended this detailed analysis to the case where G/H ≅ ℤ_{2} × ℤ_{2}; we briefly recall his results. Let L = {1, η, σ, ησ} be the four corresponding linear characters of G. If T is an irreducible representation of G, then ν ⊗ T for all ν ∈ L is also an irreducible representation of G. As before, the question is whether these are equivalent or not. Let L_{T} = {ν ∈ L|ν ⊗ T ∼ T }. Then, the following proposition gives the behavior of T on restriction to H.

Proposition 2.1. 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, ν}, where ν ∈ L, ν ≠ 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 η, σ-associators of T respectively, then

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

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

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}(χ ± Δ^{η}χ ± Δ^{σ}χ ± Δ^{ησ}χ),

where an even number of the − signs occur.

2.4. Clifford algebras and their representations. Let C(n) be the Clifford algebra generated by 1, e_{1}, …, e_{n} subject to the relations

e_{j}^{2} = 1, e_{j}e_{k} = −e_{k}e_{j}, 1 ≤ j, k ≤ n, j ≠ k.

If P in(n) is defined to be the set of invertible elements s of C(n) such that (sα(s^{t}))^{2} = 1, where α is the natural ℤ_{2}-grading on C(n) and ^{t} is the transpose, then we have the short exact sequence

(2.4)  1 ⟶ ℤ_{2} ⟶ P in(n) ⟶^{ρ_{n}} O(n) ⟶ 1 ,

where ρ_{n} is defined by ρ_{n}(s)e_{j} = α(s)e_{j}s^{−1}, for all s ∈ P in(n), 1 ≤ j ≤ n. In fact, the Schur multiplier of O(n) is given by

H^{2}(O_{n}, ℂ^{∗}) = ℤ_{2}. (2.5)

Furthermore, if Spin(n) = ρ_{n}^{−1}(SO(n)), then we also have the classical double covering of the special orthogonal (rotation) group SO(n)

(2.6)  1 ⟶ ℤ_{2} ⟶ Spin(n) ⟶^{ρ_{n}} SO(n) ⟶ 1 ,

Clearly, Spin(n) is of index 2 in P in(n); let η denote the corresponding linear character of P in(n).

We now construct the so-called basic spin representation of Clifford algebras. Let

E = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , I = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , J = ( \begin{matrix}0 & i \\ −i & 0 \end{matrix} ) , K = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )

then

I^{2} = K^{2} = E, J^{2} = E, JI = −IJ = iK,
KI = −IK = iJ, KJ = −JK = I.

Then, if n = 2μ is even, we define an isomorphism P_{n} : C_{n} → ℂ(2^{μ}) by

(2.7)  \{ \begin{matrix}P_{n}(e_{2j−1}) & = & M_{2j−1} := K^{⊗(j−1)} ⊗ I ⊗ E^{⊗(μ−j)} \\ P_{n}(e_{2j}) & = & M_{2j} := K^{⊗(j−1)} ⊗ J ⊗ E^{⊗(μ−j)} \end{matrix}

for 1 ≤ j ≤ μ and if n = 2μ + 1 is odd, we define an isomorphism P_{n,+} : C_{n} → ℂ(2^{μ}) by

(2.8)  \{ \begin{matrix}P_{n,+}(e_{j}) & = & P_{n}(e_{j}) \\ P_{n,+}(e_{2μ+1}) & = & M_{n} = K^{⊗μ} \end{matrix}

for 1 ≤ j ≤ 2μ. Furthermore, for 1 ≤ j ≤ n, put

P_{n,−}(e_{j}) = −P_{n,+}(e_{j})

Then we note that

(2.9)  M_{j}^{2} = I, M_{j}M_{k} = −M_{k}M_{j} for 1 ≤ j, k ≤ n.

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 η-associate representations. From now on, we denote these by P, P_{±}. We shall refer to these as the basic spin representations of the Clifford algebra. It is easily checked that an η−associator of P is K^{⊗μ}. In [18], 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 P in(n). This restricted representation is called the 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 P in(m, n) by taking a product of an irreducible spin representation of P in(m) with an irreducible spin representation of P in(n).

Let P_{1} and P_{2} be irreducible spin representations of P in(m) and P in(n) respectively of degrees d_{1} and d_{2} respectively. Then the twisted product P_{1} ˆ⊗P_{2} is a spin representation of the twisted product P in(m, n) ≅ P in(m) ˆ⊗P in(n) (see [18]) defined as follows; there are 3 cases to be considered.

Case 1: If P_{1} and P_{2} are η-associate spin representations of P in(m) and P in(n) respectively, then put

(P_{1} ˆ⊗P_{2})(τ, σ) = E ⊗ P_{1}(τ) ⊗ P_{2}(σ) if τ ∈ Spin(m), σ ∈ Spin(n),
(P_{1} ˆ⊗P_{2})(τ, 1) = I ⊗ P_{1}(τ) ⊗ I_{d_{2}} if τ ∈ P in(m) \ Spin(m),
(P_{1} ˆ⊗P_{2})(1, σ) = J ⊗ I_{d_{2}} ⊗ P_{2}(σ) if σ ∈ P in(n) \ Spin(n);

the relation IJ = −JI ensures that P_{1} ˆ⊗P_{2} is a spin representation of P in(m) ˆ⊗P in(n) of degree 2d_{1}d_{2}. Furthermore, P_{1} ˆ⊗P_{2} is self-associate, since tr(I) = tr(J) = 0 and so P_{1} ˆ⊗P_{2} and η ⊗ (P_{1} ˆ⊗P_{2}) have equal characters. Case 2: If P_{1} is a self-associate spin representation of P in(m) with η-associator S_{1} and P_{2} is an η-associate spin representation of P in(n), then

S_{1}P_{1}(σ) = \{ \begin{matrix}P_{1}(σ)S_{1} & if σ ∈ Spin(m) \\ −P_{1}(σ)S_{1} & if σ ∈ P in(m) \ Spin(m). \end{matrix}

Now, define

(P_{1} ˆ⊗P_{2})_{±}(τ, σ) = P_{1}(τ) ⊗ P_{2±}(σ) if τ ∈ Spin(m), σ ∈ Spin(n),
(P_{1} ˆ⊗P_{2})_{±}(τ, 1) = P_{1}(τ) ⊗ I_{d_{2}} if τ ∈ P in(m) \ Spin(m),
(P_{1} ˆ⊗P_{2})_{±}(1, σ) = S_{1} ⊗ P_{2±}(σ) if σ ∈ P in(n) \ Spin(n).

Then (P_{1} ˆ⊗P_{2})_{±} are η-associate irreducible spin representations of P in(m) ˆ⊗P in(n) of degree d_{1}d_{2}. Case 3: If P_{1} and P_{2} are both self-associate representations, then define (P_{1} ˆ⊗P_{2})_{±} as in Case 2, but replacing P_{2±} by P_{2}, then (P_{1} ˆ⊗P_{2})_{+} and (P_{1} ˆ⊗P_{2})_{−} are equivalent irreducible spin representations of P in(m) ˆ⊗P in(n), thus (P_{1} ˆ⊗P_{2})_{+} is a self-associate spin representation of degree d_{1}d_{2} in this case.

If we let χ_{P_{1}}, χ_{P_{2}} and χ_{P_{1} ˆ⊗P_{2}} denote the characters of P_{1}, P_{2} and (P_{1} ˆ⊗P_{2}) respectively, and Δ_{P_{1}}, Δ_{P_{2}} and Δ_{P_{1} ˆ⊗P_{2}} denote the difference characters if P_{1}, P_{2} or P_{1} ˆ⊗P_{2} are selfassociate, then as a consequence of the above we have the following proposition.

Proposition 2.2. If P_{1} and P_{2} are spin representations of P in(m) and P in(n) respectively and

(i) if P_{1} and P_{2} are η-associate representations then

χ_{P_{1} ˆ⊗P_{2}}(τ, σ) = \{ \begin{matrix}2χ_{P_{1}}(τ)χ_{P_{2}}(σ) & if τ ∈ Spin(m), σ ∈ Spin(n) \\ 0 & otherwise. \end{matrix}

(ii) if one of P_{1} or P_{2} is self-associate, then

χ_{P_{1} ˆ⊗P_{2}}(τ, σ) = \{ \begin{matrix}χ_{P_{1}}(τ)χ_{P_{2}}(σ) & if τ ∈ Spin(m), σ ∈ Spin(n) \\ Δ_{P_{1}}(τ)χ_{P_{2}}(σ) & if τ ∈ P in(m) \ Spin(m), σ ∈ P in(n) \ Spin(n) \\ 0 & otherwise. \end{matrix}

The above can be generalized, that is, we can define the twisted product P_{1} ˆ⊗ ⋯ ˆ⊗P_{t}, where ˆ⊗ is an associative ’multiplication’.

Let m_{1}, …, m_{t} be positive integers such that m_{1} + ⋯ + m_{t} = l and for 1 ≤ j ≤ t, let P_{j} be an irreducible spin representation of P in(m_{j}) of degree d_{j}. For simplicity, we assume that P_{j}, 1 ≤ j ≤ r ≤ t, are self-associate representations and that the remaining s = t − r representations P_{j} are η-associate representations. Let ±S_{j}, 1 ≤ j ≤ r, be the η-associators of the representations P_{j}, then

(2.10)  P_{j}(σ_{j}) = \{ \begin{matrix}S_{j}P_{j}(σ_{j}) & if σ_{j} ∈ Spin(m_{j}) \\ −S_{j}P_{j}(σ_{j}) & if σ_{j} ∉ Spin(m_{j}). \end{matrix}

Let σ_{j} also denote the element 1⊗⋯⊗1⊗σ_{j} ⊗1⊗⋯⊗1 in P in(m_{1}) ˆ⊗ ⋯ ˆ⊗P in(m_{t}), with σ_{j} in the j-th position, where σ_{j} ∈ P in(m_{j}), 1 ≤ j ≤ t. If σ_{j} ∈ Spin(m_{j}), 1 ≤ j ≤ t, put

(2.11)  P (σ_{j}) = I_{2^{⌊s/2⌋}} ⊗ I_{d_{1}} ⊗ ⋯ ⊗ I_{d_{j−1}} ⊗ P_{j}(σ_{j}) ⊗ I_{d_{j+1}} ⊗ ⋯ ⊗ I_{d_{t}}

and if σ_{j} ∉ Spin(m_{j}), put

(2.12) P (σ_{j}) = \{ \begin{matrix}I_{2^{⌊s/2⌋}} ⊗ S_{1} ⊗ ⋯ ⊗ S_{j−1} ⊗ P_{j}(σ_{j}) ⊗ I_{d_{j+1}} ⊗ ⋯ ⊗ I_{d_{t}} if 1 ≤ j ≤ r, \\ M_{j−r} ⊗ S_{1} ⊗ ⋯ ⊗ S_{r} ⊗ I_{d_{r+1}} ⊗ ⋯ ⊗ I_{d_{j−1}}P_{j}(σ_{j}) ⊗ I_{d_{j+1}} ⊗ ⋯ ⊗ I_{d_{t}} \\ if r + 1 ≤ j ≤ r + s = t. \end{matrix}

The relations (2.3) ensure that P is a spin representation of

P in(m_{1}, …, m_{t}) ≅ P in(m_{1}) ˆ⊗ ⋯ ˆ⊗P in(m_{t}).

The degree of P is 2^{⌊s/2⌋}d_{1} ⋯ d_{t}.

The character of this representation was also calculated in [18] to give the following proposition.

Proposition 2.3. Let ζ be the character of P and ζ_{j}, 1 ≤ j ≤ t, be the characters of P_{j}. (i) If σ_{j} ∈ Spin(m_{j}), 1 ≤ j ≤ t, then

ζ(σ_{1} ⋯ σ_{t}) = 2^{⌊s/2⌋}ζ_{1}(σ_{1}) ⋯ ζ_{t}(σ_{t}).

(ii) If s is odd and Δ_{j} is the difference character of the self-associate representations P_{j}, 1 ≤ j ≤ r, and if σ_{j} ∈ Spin(m_{j}), 1 ≤ j ≤ r, σ_{j} ∉ Spin(m_{j}), r + 1 ≤ j ≤ t, then

ζ(σ_{1} ⋯ σ_{t}) = ±(2i)^{[s/2]}Δ_{1}(σ_{1}) ⋯ Δ_{r}(σ_{r})ζ_{r+1}(σ_{r+1}) ⋯ ζ_{t}(σ_{t}).

(iii) In all other cases

ζ(σ_{1} ⋯ σ_{t}) = 0

The above proposition can be applied in particular to the special case where the P_{i} are the basic spin representations of P in(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 ≤ i ≤ r and that the m_{i} are odd for r+1 ≤ i ≤ t. The degree of the representation P will therefore be 2^{⌊s/2⌋}2^{m_{1}/2} ⋯ 2^{(m_{r+1}−1)/2} ⋯ 2^{(m_{t}−1)/2} = 2^{⌊s/2⌋}2^{(l−s)/2} = 2^{⌊l/2⌋}. 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 σ_{1}, …, σ_{t}. This will not be done at this point, it is postponed for consideration later when these results are applied to certain reflection groups.

3. The Generalized Symmetric Group ℤ_{m}^{n} ⋊ S_{n}

3.1. Presentation. If ℤ_{m} is the cyclic group of order m and S_{n} is the symmetric group of order n!, the generalized symmetric group is the wreath product ℤ_{m} ≀ S_{n} or the semi-direct product ℤ_{m}^{n} ⋊ 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, …, n}, then S_{n} is generated by s_{i}, 1 ≤ i ≤ n − 1 with relations

s_{i}^{2} = 1, (s_{i}s_{i+1})^{3} = 1, 1 ≤ i ≤ n − 2, (s_{i}s_{j})^{2} = 1, |i − j| ≥ 2, 1 ≤ i, j ≤ n − 1,

where s_{i} is the transposition (i, i + 1), 1 ≤ i ≤ n − 1 The group B_{n}^{m} can be considered as the group generated by s_{i}, 1 ≤ i ≤ n − 1, w_{j}, 1 ≤ j ≤ n with relations

s_{i}^{2} = 1, w_{j}^{m} = 1; (s_{i}s_{i+1})^{3} = 1, 1 ≤ i ≤ n − 2, s_{i}w_{i} = w_{i+1}s_{i}, s_{i}w_{j} = w_{j}s_{i}, j ≠ i, i + 1

(s_{i}s_{j})^{2} = 1, |i − j| ≥ 2, 1 ≤ i, j ≤ n − 1, w_{i}w_{j} = w_{j}w_{i}, i ≠ j, 2 ≤ i ≤ 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 ζi, with {1, 2, …, i − 1, i + 1, …, n} fixed, where ζ is a primitive m-th root of unity. It can be verified that w_{j} = s_{j−1}s_{j−2} ⋯ s_{1}w_{1}s_{1} ⋯ s_{j−2}s_{j−1} for 1 ≤ j ≤ n. That is, B_{n}^{m} is the permutation group acting on the set {1, 2, …, n}, but also with the ’sign’ changes w_{i} which are written as

w_{i} = ( \begin{matrix}i \\ ζi \end{matrix} ) .

3.2. Classes of conjugate elements. The classes of conjugate elements of S_{n} are parameterized by the partitions (1^{n_{1}}2^{n_{2}} … n^{n_{n}}) of n, where n_{i} ≥ 0, 1 ≤ i ≤ n.

The classes of B_{n}^{m} are defined similarly in terms of m-partitions (see, for example, [12]). The elements of B_{n}^{m} permute the set {1, 2, …, n} and multiply each of the elements of this set by a power of ζ. Thus the elements of B_{n}^{m} are of the form

x = ( \begin{matrix}1 & 2 & … & n \\ ζ^{k_{1}}b_{1} & ζ^{k_{2}}b_{2} & … & ζ^{k_{n}}b_{n} \end{matrix} ) ,

where {b_{1}, b_{2}, …, b_{n}} is a permutation of the set {1, 2, …, n} and 1 ≤ k_{i} ≤ m, 1 ≤ i ≤ n. Any element of B_{n}^{m} can be uniquely expressed as a product of disjoint cycles x = ∏_{i=1}^{t} θ_{i}. where

θ_{i} = ( \begin{matrix}b_{i_{1}} & b_{i_{2}} & … & b_{i_{l_{i}}} \\ ζ^{k_{i_{1}}}b_{i_{2}} & ζ^{k_{i_{2}}}b_{i_{3}} & … & ζ^{k_{i_{l_{i}}}} b_{i_{1}} \end{matrix} ) ,

where ∑_{i=1}^{t} l_{i} = n; put f(θ_{i}) = ∑_{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}} … n^{a_{1n}}; 1^{a_{21}}2^{a_{22}} … n^{a_{2n}}; …; 1^{a_{m1}}2^{a_{m2}} … n^{a_{mn}}),

where ∑_{i=1}^{n} a_{ij} = n_{j} 1 ≤ j ≤ m, where a_{pq} denotes the number of cycles θ_{i} in the above decomposition of σ of length q such that f(θ_{i}) ≡ p − 1 (mod m). The order of this class is

(3.1)  \frac{m^{n}n!}{∏_{p,q} a_{pq}!(qm)^{a_{pq}}}

We have, by definition, the short exact sequence

(3.2)  1 ⟶ ℤ_{m}^{n} ⟶ B_{n}^{m} ⟶^{υ_{n}} S_{n} ⟶ 1 ,

where υ_{n} is defined by υ_{n}(s_{i}) = s_{i}, υ_{n}(w_{i}) = 1 for all 1 ≤ i ≤ n, where ℤ_{m}^{n} = ℤ_{m}⊗…⊗ℤ_{m}, (n copies), where the i-th copy of 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

(3.3)  1 ⟶ ℤ_{m/2}^{n} ⟶ B_{n}^{m} ⟶^{τ_{n}} B_{n}^{2} ⟶ 1 ,

where τ_{n} is defined by τ_{n}(s_{i}) = s_{i}, τ_{n}(w_{i}) = w_{i} for all 1 ≤ j ≤ n, where now the i-th copy of ℤ_{m/2} should be regarded as the cyclic group generated by w_{i}^{2}.

Under the homomorphism υ_{n} the class

(1^{a_{11}}2^{a_{12}} … n^{a_{1n}}; 1^{a_{21}}2^{a_{22}} … n^{a_{2n}}; …; 1^{a_{m1}}2^{a_{m2}} … n^{a_{mn}}),

of B_{n}^{m} fuses to the class (1^{∑_{i=1}^{m} a_{i1}}2^{∑_{i=1}^{m} a_{i2}} … n^{∑_{i=1}^{m} a_{in}}) of S_{n} and under the homomorphism τ_{n} this class fuses to the class

(1^{∑m ; {i=1 \atop i odd} a_{i1}}2^{∑m ; {i=1 \atop i odd} a_{i2}} … n^{∑m ; {i=1 \atop i odd} a_{in}}; 1^{∑m ; {i=1 \atop i even} a_{i1}}2^{∑m ; {i=1 \atop i even} a_{i2}} … n^{∑m ; {i=1 \atop i even} 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

(3.4)  \{ \begin{matrix}σ_{k}(s_{i}) = 1, 1 ≤ i ≤ n − 1 & σ_{k}(w_{j}) = ζ^{k}, 1 ≤ j ≤ n \\ η(s_{i}) = −1, 1 ≤ i ≤ n − 1 & η(w_{j}) = 1, 1 ≤ j ≤ n \\ ϵ_{k}(s_{i}) = −1, 1 ≤ i ≤ n − 1 & ϵ_{k}(w_{j}) = ζ^{k}, 1 ≤ j ≤ n, \end{matrix}

where 1 ≤ k ≤ m − 1, together with the identity character. In the special case k = m/2, we write ϵ for ϵ_{m/2} and σ for σ_{m/2}. The values of these characters for an element in the class (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) are as follows

η(λ_{(1)}; λ_{(2)}; …; λ_{(m)}) = (−1)^{∑_{i=1}^{m} l(λ_{(i)})},
σ(λ_{(1)}; λ_{(2)}; …; λ_{(m)}) = (−1)^{n−∑_{i=2,i even}^{m} l(λ_{(i)})},
ϵ(λ_{(1)}; λ_{(2)}; …; λ_{(m)}) = (−1)^{n−∑_{i=1,i odd}^{m−1} l(λ_{(i)})}.

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}.

Lemma 3.1. (i) ker η = {x ∈ (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) | (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is even}, (ii) ker σ = {x ∈ (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) | ∑_{i=2 i even}^{m} ∑_{j=1}^{n} a_{ij} is even }, (iii) ker ϵ = ησ = {x ∈ (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) | (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is even and

∑_{i=2 i even}^{m} ∑_{j=1}^{n} a_{ij} is even , or (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is odd and ∑_{i=2 i even}^{m} ∑_{j=1}^{n} a_{ij} is odd }.

Proof. (i) Since η(w_{j}) = 1 for 1 ≤ j ≤ n and η(s_{j}) = −1 for 1 ≤ j ≤ n, then for x ∈ ker η, 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 [λ_{(1)}; λ_{(2)}; …; λ_{(m)}] is even.

(ii) Since σ(w_{j}) = −1 for 1 ≤ j ≤ n and σ(s_{j}) = 1 for 1 ≤ j ≤ n, then for x ∈ ker σ, the total number of the generators w_{i} in any expression for x must be even. In an expression x = ∏_{i=1}^{t} θ_{i} of x as a product of cycles, the cycles θ_{i} for which f(θ_{i} is even (odd) give rise to an even (odd) number of w_{j}. Thus, for σ(x) = 1, we require an even number of cycles θ_{i} with f(θ_{i}) odd. This can only occur if ∑_{i=2 i even}^{m} ∑_{j=1}^{n} a_{ij} is even.

(iii) Since ϵ(w_{j}) = −1 for 1 ≤ j ≤ n and ϵ(s_{j}) = −1 for 1 ≤ j ≤ n, then for x ∈ ker σ, 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 (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) and ∑_{i=2 i even}^{m} ∑_{j=1}^{n} a_{ij} are even or both are odd which results in the required conclusion.

If we now let M = ker η ⋂ ker σ ⋂ ker ϵ, then M = {x ∈ (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) | (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is even and ∑_{i=2 i even}^{m} ∑_{j=1}^{n} a_{ij} is even }. Then, the following lemma can be proved.

Lemma 3.2. If m is even, the following is a short exact sequence

1 ⟶ M ⟶ B_{n}^{m} ⟶ ℤ_{2} × ℤ_{2} ⟶ 1 .

Proof. Define ϕ : B_{n}^{m} ⟶ ℤ_{2} × ℤ_{2} by

ϕ(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 ϕ is well-defined. Clearly, the map ϕ is surjective and it only remains to determine kerϕ.

For x ∈ kerϕ, 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 kerϕ is indeed the subgroup M.

4. A Covering Group B̃_{n}^{m} of B_{n}^{m} and its Basic Spin Representations

The Schur multiplier of B_{n}^{m} was obtained in [8]

(4.1) H^{2}(B_{n}^{m}, ℂ^{∗}) = \{ \begin{matrix}ℤ_{2} = \{γ\} & if m is odd, n ≥ 4, \\ ℤ_{2} × ℤ_{2} × ℤ_{2} = \{(γ, λ, μ)\} & if m is even, n ≥ 4, \\ ℤ_{2} × ℤ_{2} = \{(λ, μ)\} & if m is even, n = 3, \\ ℤ_{2} = \{μ\} & if m is even, n = 2, \\ \{1\} & otherwise, \end{matrix}

where γ = λ = μ = ±1.

This means that if m is even B_{n}^{m} has eight 2-cocycles {(γ, λ, μ)|γ^{2} = λ^{2} = μ^{2} = 1} and two 2-cocycles if m is odd, {(γ)|γ^{2} = 1}. A corresponding representation group is denoted by B̃_{n}^{m} which has a presentation

B̃_{n}^{m} = < t_{i}, 1 ≤ i ≤ n − 1, u_{j}, 1 ≤ j ≤ n | t_{i}^{2} = 1, u_{j}^{m} = 1
(t_{i}t_{i+1})^{3} = 1, 1 ≤ i ≤ n − 2, t_{i}u_{i} = u_{i+1}t_{i}, t_{i}u_{j} = λu_{j}t_{i}, j ≠ i, i + 1

(t_{i}t_{j})^{2} = γ1, |i − j| ≥ 2, 1 ≤ i, j ≤ n − 1, (4.2)

u_{i}u_{j} = μu_{j}u_{i}, i ≠ j, 2 ≤ i ≤ n − 1⟩,

where

γ^{2} = λ^{(2,m)} = μ^{(2,m)} = 1

and γ, λ, μ commute with each other and with the t_{i}, u_{j}.

For simplicity, from now on, we will fix a 2-cocycle [γ, λ, μ] ∈ (γ, λ, μ), with γ^{2} = λ^{(2,m)} = μ^{(2,m)} = 1 and with the convention that λ = μ = 1 if m is odd; γ = 1 if m is even and n = 3; γ = λ = 1 if m is even and n = 2; and γ = λ = μ = 1 if n = 1. Thus, the 2-cocycles will be denoted by [±1, ±1, ±1]; we note that only the 2-cocycles [±1, 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 [23] (who in [24] was the first to determine all the irreducible spin representations of B_{n}^{m} for all 2-cocycles). Later, Stembridge [31] 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 η and σ. 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 (λ, μ) of B_{n} for which η = −1, σ = −1 are of the form (DOP; DEP), that is, λ has distinct odd parts and μ has distinct even parts.

\begin{matrix}2-cocycle & η = 1, σ = 1 & η = −1, σ = 1 & η = 1, σ = −1 & η = −1, σ = −1 \\ [1, −1, 1] & (P; P) & (EP; ∅) & (DOP; DOP) & (∅; EP) \\ [−1, 1, 1] & (OP; OP) & (DP; DP) & (OP; OP) & (DP; DP) \\ [−1, −1, 1] & (OP; OP) & (DEP; ∅) & (DP; DP) & (∅; DEP) \\ [1, 1, −1] & (OP; ∅) & ∅ & (∅; DP) & (∅; DP) \\ [1, −1, −1] & (OP; ∅) & (∅; DP) & (∅; OP) & (DOP; DEP) \\ [−1, 1, −1] & (OP; EP) & ∅ & (∅; DOP) & (∅; P) \\ [−1, −1, −1] & (OP; EP) & (∅; P) & (∅; P) & (OP; EP) \end{matrix}

Table 1. Splitting classes for B_{n}^{2}

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} ⟶^{τ_{n}} 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 [23] 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{matrix}2-cocycle & η = 1, σ = 1 & η = −1, σ = 1 & η = 1, σ = −1 & η = −1, σ = −1 \\ [1, −1, 1] & (P; …; P) & (EP; ∅; …; EP; ∅) & (DOP; …; DOP) & (∅; EP; …; ∅; EP) \\ [−1, 1, 1] & (OP; …; OP) & (DP; …; DP) & (OP; …; OP) & (DP; …; DP) \\ [−1, −1, 1] & (OP; …; OP) & (DEP; ∅; …; DEP; ∅) & (DP; …; DP) & (∅; DEP; …; ∅; DEP) \\ [1, 1, −1] & (OP; ∅; …; OP; ∅) & ∅ & (∅; DP; …; ∅; DP) & (∅; DP; …; ∅; DP) \\ [1, −1, −1] & (OP; ∅; …; OP; ∅) & (∅; DP; …; ∅; DP) & (∅; OP; …; ∅; OP) & (DOP; DEP; …; DOP; DEP) \\ [−1, 1, −1] & (OP; EP; …; OP; EP) & ∅ & (∅; DOP; …; ∅; DOP) & (∅; P; …; ∅; P) \\ [−1, −1, −1] & (OP; EP; …; OP; EP) & (∅; P; …; ∅; P) & (∅; P; …; ∅; P) & (OP; EP; …; OP; EP) \end{matrix}

Table 2. Splitting classes for B_{n}^{m}

For example, for the 2-cocycle [−1, 1, 1], the splitting classes of B_{n}^{m} (or of B̃_{n}^{m}) in the notation of this paper are classes of the m-partition form (OP, OP, …, OP) and (DP, DP, …, DP).

4.1. Basic spin representations of generalized symmetric groups. Let W (Φ) be the irreducible finite reflection group of rank l with root system Φ and simple system Π = {α_{1}, …, α_{l}} and let τ_{j} = τ_{α_{j}} be the reflection corresponding to α_{j} ∈ Π. Then the group W (Φ) is generated by the simple reflections τ_{j}, 1 ≤ j ≤ l subject to the relations

τ_{j}^{2} = 1, 1 ≤ j ≤ l, (τ_{j}τ_{k})^{m_{jk}} = 1, 1 ≤ j, k ≤ l, j ≠ k,

where m_{jk} are positive integers such that m_{kj} = m_{jk}.

If the group W (Φ) of rank l is embedded in the orthogonal group O(n); say ϕ : W (Φ) ↪ O(n) is an embedding of W (Φ) into an orthogonal group O(n), for some n, then let M_{ϕ}(Φ) = ρ_{n}^{−1}(W (Φ)). Then we have the following (see [2] and [18] for the details including notation)

(4.3)  \begin{aligned}1 & ⟶ & ℤ_{2} & ⟶ & P in(n) & ⟶^{ρ_{n}} & O(n) & ⟶ & 1 \\ & & & & & & ↑ϕ \\ 1 & ⟶ & ℤ_{2} & ⟶ & M_{ϕ}(Φ) & ⟶^{ρ_{n}} & W (Φ) & ⟶ & 1. \end{aligned}

It is clear that the lower sequence in (4.3) is also an exact sequence, but to show that M_{ϕ}(Φ) is a covering group of W (Φ), it is necessary to show that M_{ϕ}(Φ) is a stem extension of W (Φ), that is, to verify that

ℤ_{2} ⊂ Z(M_{ϕ}(Φ)) ∩ (M_{ϕ}(Φ))′.

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 (Φ). Furthermore, it was shown in [18], that if n = l the basic spin representations P, P_{±} remain irreducible on restriction to the finite irreducible reflection groups W (Φ), where rank(Φ) = 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 η : W (Φ) ↪ O(l), where rankΦ = l. In this case, put

M(Φ) = ϕ_{l}^{−1}(W (Φ)).

Then, a presentation of M(Φ) is obtained. We have that

ρ_{l}(α_{j}) = τ_{j}, 1 ≤ j ≤ l.

and if we let r_{j} = α_{j}/∥ α_{j} ∥, 1 ≤ j ≤ l, then we also have

ρ_{l}(r_{j}) = τ_{j} = τ_{r_{j}}, 1 ≤ j ≤ l.

If, in addition, z ∈ P in(l), is such that ρ_{l}(z) = I_{l}, then z ∈ ℤ_{2}, that is, z^{2} = 1. Then, we have that the group M(Φ) is generated by r_{j}, 1 ≤ j ≤ l, z subject to the relations

(r_{j}r_{k})^{m_{jk}} = z^{m_{jk}−1}, 1 ≤ j, k ≤ l, z^{2} = 1, zr_{j} = r_{j}z, 1 ≤ j ≤ 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). Type A_{n−1}. In order to apply the above, we use an embedding of the root system A_{n−1} in ℝ^{n−1} where the simple system is given by

\{α_{j} = \sqrt{j − 1}e_{j−1} − \sqrt{j + 1}e_{j}, 1 ≤ j ≤ 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 ≤ j ≤ n − 1

is the irreducible basic spin representation of S_{n} if n is odd and P_{±} are the two associate basic spin representation of W (A_{n−1}) if n is even. In the above, the generators τ_{j} have been replaced by the corresponding ones in this setting. In fact, we obtain the presentation

Ã_{n−1} = ⟨t_{i}, 1 ≤ i ≤ n − 1, z | t_{i}^{2} = 1, z^{2} = 1, (t_{i}t_{i+1})^{3} = 1, 1 ≤ i ≤ n − 2,
(t_{i}t_{j})^{2} = z, t_{i}z = zt_{i}, |i − j| ≥ 2, 1 ≤ i, j ≤ n − 1⟩

and thus, this representation, as was shown in [18] 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.

Proposition 4.1. Let ψ, (ψ_{±}) be the character of the basic spin representation P, (P_{±}). (i) If x ∈ (ρ), ρ ∈ OP (n),then

ψ(x) = 2^{⌊ \frac{1}{2} (l(ρ)−1)⌋}.

(ii) If x ∈ (n), then

ψ_{±}(x) = ±i^{\frac{1}{2} (n−2)}\sqrt{n/2} if n is even.

(iii)

ψ(x) = 0 otherwise.

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 υ_{n} defined in (3.2). This results in the following proposition.

Proposition 4.2. Let ψ, (ψ_{±}) be the character of the basic spin representation P, (P_{±}). (i) If x ∈ (λ_{(1)}; λ_{(2)}; …; λ_{(m)}), λ_{(i)} ∈ (OP (|λ_{(i)}|), 1 ≤ i ≤ m, then

ψ(x) = 2^{⌊ \frac{1}{2} (l(λ_{(1)};λ_{(2)};...;λ_{(m)})−1)⌋}.

(ii) If x ∈ (∅; …; ∅; n; ∅; …; ∅), then

ψ_{±}(x) = ±i^{\frac{1}{2} (n−2)}\sqrt{n/2} if n is even,

where n can be in any one of m possible positions.

(iii)

ψ(x) = 0 otherwise.

It was I. Schur [30] who first showed that the irreducible representations for this 2cocycle correspond to partitions λ ∈ DP (n). These were constructed in a remarkable way by M. L. Nazarov [21] which is a generalization of the above construction which corresponds to the partition (n). We briefly recall his results.

Let λ ∈ DP (n), the shifted diagram for λ is

D_{λ} = {(i, j) ∈ ℤ^{2} | 1 ≤ i ≤ l(λ); i ≤ j ≤ λ_{i} + i − 1}.

This is represented graphically where a point (i, j) ∈ ℤ^{2} is represented by the unit square in the plane ℝ^{2} with centre (i, j), the coordinates i and j increasing from top to bottom and from left to right respectively. A shifted tableau of shape λ is a bijection Δ : D_{λ} → {1, 2, …, n}; a bijection is represented as a filling of the squares of D_{λ} with the numbers 1, 2, …, n, each of these numbers being used once only. A shifted tableau Δ is standard if the numbers increase down its columns and across its rows. Now, let S_{λ} denote the set of all standard shifted tableaux of shape λ. Let Δ ∈ S_{λ} and let k ∈ {1, 2, …, n} be fixed. Let k and k+1 have the coordinates (i, j) and (i′, j′) in Δ. Put a = j −i+1, b = j′−i′+1. Consider Δ and s_{k}Δ, then s_{k}Δ ∈ S_{λ} or s_{k}Δ ∉ S_{λ}. Assume a < b, otherwise work with s_{k}Δ, even if s_{k}Δ ∉ S_{λ}. 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 = ( \begin{matrix}x & z \\ z & y \end{matrix} ) , B = ( \begin{matrix}−y & z \\ z & −x \end{matrix} ) and C = ( \begin{matrix}x & u \\ u & −x \end{matrix} ) .

Let h be the number of rows in Δ occupied by 1, 2, …, k + 1.Then, if Δ, Δ′ ∈ S_{λ}, put

X_{±}^{⟨λ⟩}(s_{k}) = \{ \begin{matrix}A ⊗ M_{k−h+1} + B ⊗ M_{k−h} & if 1 < a < b \\ C ⊗ M_{k−h+1} & if 1 = a < b \end{matrix}

and if Δ′ ∉ S_{λ}, replace the matrices A, B and C by the element which appears in the (2, 2)-position. This is repeated for all the tableaux Δ ∈ S_{λ} and we obtain the following proposition.

Proposition 4.3. The X^{⟨λ⟩}, λ ∈ DP^{+}(n), X_{±}^{⟨λ⟩}, λ ∈ DP^{−}(n) form a complete set of irreducible spin representations of S_{n} of degree 2^{⌊ \frac{n−l(λ)}{2} ⌋}g_{λ}, where g_{λ} is the number of shifted standard tableaux of shape λ.

We note that the η-associator of X^{⟨λ⟩} is id ⊗ K^{⊗μ}, where μ = ⌊n/2⌋. Type B_{n}. In order to apply the above, we use an embedding of the root system B_{n} in ℝ^{n} where the simple system is given by

{α_{j} = e_{j−1} − e_{j}, 1 ≤ j ≤ n − 1, α_{n} = e_{n}},

then

Q(s_{j}) = \frac{1}{\sqrt{2}}(M_{j−1} − M_{j}), 1 ≤ j ≤ n − 1, Q(w_{1}) = M_{n}

is the irreducible basic spin representation of W (B_{n}) if n is even and Q_{±} are the two associate basic spin representation of W (B_{n}) if n is odd. Here, we have replaced the notation P (P_{±}) by Q (Q_{±}) for obvious reasons.

In this case, we obtain the presentation

B̃_{n} = ⟨t_{i}, 1 ≤ i ≤ n − 1, u_{j}, 1 ≤ j ≤ n | t_{i}^{2} = 1, u_{j}^{2} = 1, (t_{i}t_{i+1})^{3} = 1,
1 ≤ i ≤ n − 2, t_{i}u_{i} = u_{i+1}t_{i}, t_{i}u_{j} = λu_{j}t_{i}, j ≠ i, i + 1, (t_{i}t_{j})^{2} = γ,
|i − j| ≥ 2, 1 ≤ i, j ≤ n − 1, u_{i}u_{j} = μu_{j}u_{i}, i ≠ j, 2 ≤ i ≤ n − 1⟩,

where γ^{2} = λ^{(2,m)} = μ^{(2,m)} = 1 and γ, λ, μ 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 ≤ i ≤ 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 [18] as given in the following proposition.

Proposition 4.4. Let χ, (χ_{±}) be the character of the basic spin representation Q, (Q_{±}). (i) If x ∈ (ρ; ϱ), (ρ; ϱ) ∈ (OP (|ρ|); EP (|ϱ|)), then

χ(x) = \{ \begin{matrix}2^{\frac{1}{2} (l(ρ;ϱ))} & if n is even \\ 2^{\frac{1}{2} (l(ρ;ϱ)−1)} & if n is odd \end{matrix}

(ii) If x ∈ (∅; ϱ), ϱ ∈ P (n), then

χ_{±}(x) = ±i^{\frac{1}{2} (n−1)}2^{\frac{1}{2} (l(ϱ)−1)} if n is odd,

(iii)

χ(x) = 0 otherwise.

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

Proposition 4.5. Let χ, (χ_{±}) be the character of the basic spin representation Q, (Q_{±}). (i) If x ∈ (ρ_{1}; ϱ_{1}; …; ρ_{m/2}; ϱ_{m/2}), ρ_{i} ∈ OP; ϱ_{i} ∈ EP, 1 ≤ i ≤ m/2, then

χ(x) = \{ \begin{matrix}2^{\frac{1}{2} (∑_{i=1}^{m/2} l(ρ_{i};ϱ_{i}))} & if n is even \\ 2^{\frac{1}{2} (∑_{i=1}^{m/2} l(ρ_{i};ϱ_{i})−1)} & if n is odd \end{matrix}

(ii) If x ∈ (∅, ϱ_{1}, …, ∅, ϱ_{m/2}), ϱ_{i} ∈ P, 1 ≤ i ≤ m/2, then

χ_{±}(x) = ±i^{\frac{1}{2} (n−1)}2^{\frac{1}{2} (∑_{i=1}^{m/2} l(ϱ_{i})−1)} if n is odd,

(iii)

χ(x) = 0 otherwise.

In [19] 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} ↪ O(2) given by

(σ ⊕ η)(s_{i}) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , 1 ≤ i ≤ l − 1, (σ ⊕ η)(w_{l}) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) .

Now, if we use the exact sequence

1 ⟶ ℤ_{2} ⟶ P in(2) ⟶^{ρ_{2}} O(2) ⟶ 1
↑σ ⊕ η
B_{n}^{m}/M

we ultimately, by putting

R(s_{i}) = M_{2} = ( \begin{matrix}0 & i \\ −i & 0 \end{matrix} ) , 1 ≤ i ≤ n − 1, R(w_{l}) = M_{1} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ,

obtain an irreducible spin representation R of degree 2 of B_{n}^{m} corresponding to the 2cocycle [1, −1, 1]. The σ−, η− and ϵ−associators of this representation are J, I and K respectively.

The character of this representation is given by the following proposition.

Proposition 4.6. If ξ denotes the character of R, then

ξ(x) = \{ \begin{matrix}2 & if x ∈ M \\ 0 & otherwise. \end{matrix}

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^{⌊ \frac{1}{2} (n−1)⌋}, 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 [31], 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 α = [1, −1, 1], let P be any projective representation of B_{n}^{m} with 2-cocycle β, then R ⊗ P is a projective representation of B_{n}^{m} with 2-cocycle αβ. Then we have the following proposition.

Proposition 4.7. Let P be an irreducible projective representation of degree d of B_{n}^{m} with 2-cocycle β.

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

(ii) If L_{P} = {1, ν}, where ν ∈ L, then R ⊗ P is the direct sum of two inequivalent irreducible projective representation of degree d of B_{n}^{m} with 2-cocycle αβ.

(iii) If L_{P} = L, and U, V are the η, σ-associators of P respectively, then

(a) if UV = −V U, then R⊗P is the direct sum of four inequivalent irreducible projective representation of degree d/2 of B_{n}^{m} with 2-cocycle αβ.

(b) if UV = V U, then R ⊗ P is the direct sum of two equivalent irreducible projective representation of degree d of B_{n}^{m} with 2-cocycle αβ.

5. Irreducible spin representations of generalized symmetric groups.

5.1. 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 [12], but also for a treatment which is more in line with our requirements, see the work of M. Saeed-ul Islam [27]. As this work is not easily available a review of his presentation is given below. Furthermore, H. Can [6] has given a description of the construction of the corresponding Specht modules and also in [7], 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 [1].

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

If we let p_{0} = 0 and p_{i} = ∑_{j=1}^{i} k_{j}, 1 ≤ i ≤ 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, …, p_{i}}, 1 ≤ i ≤ m, where ∑_{i=1}^{m} k_{i} = n. Then let B_{(k_{1},...,k_{m})}^{m} = B_{k_{1}}^{m}×⋯×B_{k_{m}}^{m} be the corresponding generalized Young subgroup. Recall that we have defined earlier the linear characters σ_{k}, 1 ≤ k ≤ m − 1 by

σ_{k}(s_{i}) = 1, 1 ≤ i ≤ n − 1; σ_{k}(w_{j}) = ζ^{k}, 1 ≤ j ≤ n.

The representation X^{[λ;∅;...;∅]} is obtained by lifting X^{[λ]} from S_{n} to B_{n}^{m}, we define X^{[∅;...;∅;λ;∅;...;∅]}, where λ is in the (k + 1)-th position, 1 ≤ k ≤ m − 1 to be σ_{k} ⊗ X^{[λ;∅;...;∅]}. If |λ_{i}| = k_{i} 1 ≤ i ≤ m, define

X^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} = (X^{[λ_{(1)};∅;...;∅]} ⊗ X^{[∅;λ_{(2)};...;∅]} ⊗ ⋯ ⊗ X^{[∅;∅;...;λ_{(m)}]}) ↑ B_{n}^{m},

inducing from B_{(k_{1},...,k_{m})}^{m} to B_{n}^{m}. If we let

χ_{(k_{1},...,k_{m})} = 1 ⊗ σ_{1} ⊗ ⋯ ⊗ σ_{m−1},

then we have

X^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} = (χ_{(k_{1},...,k_{m})} ⊗ X^{[λ_{(1)};∅;...;∅]} ⊗ X^{[λ_{(2)};∅;...;∅]} ⊗ ⋯ ⊗ X^{[λ_{(m)};∅;...;∅]}) ↑ B_{n}^{m}.

We can now prove the following lemma and theorem.

Lemma 5.1. χ_{(k_{1},...,k_{m})} is a linear character of B_{(k_{1},...,k_{m})}^{m} such that χ_{(k_{1},...,k_{m})}^{g}(x) ≠ χ_{(k̃_{1},...,k̃_{m})}(x) for some x ∈ ker υ_{n} and for all g ∈ B_{n}^{m} unless (k_{1}, …, k_{m}) = (k̃_{1}, …, k̃_{m}) in which case this holds for all g ∈ B_{n}^{m}\B_{(k_{1},...,k_{m})}^{m}.

Proof. If (k̃_{1}, …, k̃_{m}) ≠ (k_{1}, …, k_{m}), assume without loss of generality that k̃_{i} > k_{i} for some i, that is, P_{i} ⊂ P̃_{i}. If g ∈ B_{n}^{m} is such that υ_{n}(g)P_{i} = P_{i}, if j ∈ P̃_{i}\P_{i}, then

υ_{n}(j) ∈ P_{l}, l ≠ i and we put x = ( \begin{matrix}1 \\ 1 \end{matrix} ) ⋯ ( \begin{matrix}j \\ ζj \end{matrix} ) ⋯ ( \begin{matrix}n \\ n \end{matrix} ).

If υ_{n}(g)P_{i} ≠ P_{i}, then there exists j ∈ P_{i} ⊂ P̃_{i} such that υ_{n}(j) ∈ P_{l}, 1 ≤ l ≤ m, l ≠ i and for this j, we define x as above.

In each case, χ_{(k̃_{1},...,k̃_{m})}(x) = ζ^{i}, but

χ_{(k_{1},...,k_{m})}^{g}(x) = χ_{(k_{1},...,k_{m})}(υ_{n}(g)xυ_{n}(g)^{−1}) = ζ^{l}, l ≠ i.

If (k̃_{1}, …, k̃_{m}) = (k_{1}, …, k_{m}) and g ∈ B_{n}^{m}\B_{(k_{1},...,k_{m})}^{m}, then there exists at least one index i, 1 ≤ i ≤ m and an integer j ∈ P_{j} such that υ_{n}(j) ∈ P_{l}, 1 ≤ l ≤ m, l ≠ i. Once again we define x ∈ ker υ_{n} as above for this particular j. Clearly,

χ_{(k_{1},...,k_{m})}^{g}(x) = ζ^{l} ≠ ζ^{i} = χ_{(k̃_{1},...,k̃_{m})}(x),

which completes the proof of the lemma. Theorem 5.2. A complete set of inequivalent irreducible (ordinary) representations of B_{n}^{m} is given by the X^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]}, where (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is any m-partition of n.

Proof. Let (k_{1}, …, k_{m}) and (k̃_{1}, …, k̃_{m}) be two arbitrary m-tuples of n and let X^{[λ_{(1)},λ_{(2)},...,λ_{(m)}]} = X^{[λ_{(1)};∅;...;∅]} ⊗ X^{[∅;λ_{(2)};...;∅]} ⊗ ⋯ ⊗ X^{[∅;∅;...;λ_{(m)}]} and X^{[λ̃_{(1)},λ̃_{(2)},...,λ̃_{(m)}]} = X^{[λ̃_{(1)};∅;...;∅]}⊗X^{[∅;λ̃_{(2)};...;∅]}⊗⋯⊗X^{[∅;∅;...;λ̃_{(m)}]} be two corresponding representations of B_{(k_{1},...,k_{m})}^{m} and B_{(k̃_{1},...,k̃_{m})}^{m} with characters χ^{[λ_{(1)},λ_{(2)},...,λ_{(m)}]} and χ^{[λ̃_{(1)},λ̃_{(2)},...,λ̃_{(m)}]} respectively as defined above. We will prove that

(χ^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]}, χ^{[λ̃_{(1)};λ̃_{(2)};...;λ̃_{(m)}]})_{B_{n}^{m}} = 0

unless k_{i} = k̃_{i}, 1 ≤ i ≤ 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

(χ^{[λ_{(1)},λ_{(2)},...,λ_{(m)}]}, (χ^{[λ̃_{(1)};λ̃_{(2)};...;λ̃_{(m)}]}) ↓ B_{(k_{1},...,k_{m})}^{m})_{B_{(k_{1},...,km)}^{m}}
= ∑_{x}(χ^{[λ_{(1)},λ_{(2)},...,λ_{(m)}]}, (χ^{[λ̃_{(1)},λ̃_{(2)},...,λ̃_{(m)}]} ↓ H_{x}) ↑ B_{(k_{1},...,k_{m})}^{m})_{B_{(k_{1},...,km)}^{m}}
= ∑_{x}(χ^{[λ_{(1)},λ_{(2)},...,λ_{(m)}]} ↓ H_{x}, (χ^{[λ̃_{(1)},λ̃_{(2)},...,λ̃_{(m)}]})^{x} ↓ H_{x})_{H_{x}},

where H_{x} = B_{(k_{1},...,k_{m})}^{m} ⋂ x^{−1}B_{(k̃_{1},...,k̃_{m})}^{m}x, (χ^{[λ̃_{(1)},λ̃_{(2)},...,λ̃_{(m)}]})^{x}(x^{−1}gx) =

(χ^{[λ̃_{(1)},λ̃_{(2)},...,λ̃_{(m)}]})(g) for all g ∈ B_{(k̃_{1},...,k̃_{m})}^{m} and x ranges over the double coset representatives of the generalized Young subgroups B_{(k_{1},...,k_{m})}^{m} and B_{(k̃_{1},...,k̃_{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,

∑_{x} χ^{[λ_{(1)},λ_{(2)},...,λ_{(m)}]} ↓ H_{x} = ∑_{x}(χ^{[λ̃_{(1)},λ̃_{(2)},...,λ̃_{(m)}]})^{x} ↓ H_{x}

have an irreducible component in common, then so do

∑_{x} χ^{[λ_{(1)},λ_{(2)},...,λ_{(m)}]} ↓ ker υ_{n} = ∑_{x}(χ^{[λ̃_{(1)},λ̃_{(2)},...,λ̃_{(m)}]})^{x} ↓ ker υ_{n},

since ker υ_{n} ⊂ H_{x}. Then, using the alternative form for X^{[λ_{(1)},λ_{(2)},...,λ_{(m)}]} given above, we see that these respectively coincide with χ_{(k_{1},...,k_{m})} ↓ ker υ_{n} and χ_{(k̃_{1},...,k̃_{m})}^{x} ↓ ker υ_{n}. Since both of these are irreducible, they are equal on ker υ_{n} and so, by Lemma 5.1, we have (k_{1}, …, k_{m}) = (k̃_{1}, …, k̃_{m}) and x ∈ B_{(k_{1},...,k_{m})}^{m}. Now, using this information, an elementary inner product calculation shows that

(χ^{[λ_{(1)},λ_{(2)},...,λ_{(m)}]}, χ^{[λ̃_{(1)},λ̃_{(2)},...,λ̃_{(m)}]})_{B_{(k_{1},...,km)}^{m}} = (χ^{λ_{1}} ⋯ χ^{λ_{m}}, χλ̃_{1} ⋯ χλ̃_{m})_{S_{(k_{1},...,km)}},

which is non-zero only if these two characters are equal and we have the desired result.

5.2. 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 λ ∈ DP (n)^{+}, X^{⟨λ⟩} are the irreducible spin representation of S_{n} and if λ ∈ DP (n)^{−}, X_{±}^{⟨λ⟩} are the two η-associate irreducible spin representations. The corresponding spin characters are denoted by χ^{⟨λ⟩}, χ_{±}^{⟨λ⟩}.

We show that the irreducible representations of B_{n}^{m} for the 2-cocycle [-1,1,1] are indexed by m-partitions (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) of n, where λ_{(i)} ∈ DP (|λ_{(i)}|), 1 ≤ i ≤ m, the corresponding representations and characters will be denoted by X^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} (X_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩})and χ^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} (χ_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩})(if they are η-associate) respectively.

The representation X^{⟨λ;∅;...;∅⟩} is obtained by lifting X^{⟨λ⟩} from S_{n} to B_{n}^{m}, we define X^{⟨∅;...;∅;λ;∅;...;∅⟩}, where λ is in the (k + 1)-th position, 1 ≤ k ≤ m − 1 to be σ_{k} ⊗ X^{⟨λ;∅;...;∅⟩}. If |λ_{i}| = k_{i} 1 ≤ i ≤ m, define

X^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = (X^{⟨λ_{(1)};∅;...;∅⟩} ˆ⊗X^{⟨∅;λ_{(2)};...;∅⟩} ˆ⊗ ⋯ ˆ⊗X^{⟨∅;∅;...;λ_{(m)}⟩}) ↑ B_{n}^{m},

inducing from B_{(k_{1},...,k_{m})}^{m} to B_{n}^{m}, where ˆ⊗ is the twisted tensor product [18],[9]. Then we have

X^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = (χ_{(k_{1},...,k_{m})} ⊗ X^{⟨λ_{(1)};∅;...;∅⟩} ˆ⊗X^{⟨λ_{(2)};∅;...;∅⟩} ˆ⊗ ⋯ ˆ⊗X^{⟨λ_{(m)};∅;...;∅⟩}) ↑ B_{n}^{m}.

There are similar statements for the η-associate representations and characters.

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

Theorem 5.3. A complete set of inequivalent irreducible spin representations of B_{n}^{m} for the 2-cocycle [−1, 1, 1] is given by the

X^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} if n − l(λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is even

and

X_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} if n − l(λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is odd,

where (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is an m-partition of n, with λ_{(i)} ∈ DP (|λ_{(i)}|), 1 ≤ i ≤ m.

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.

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

5.3. The 2-cocycle [-1,-1,-1]. Let Q, Q_{±} 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 (λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅), put

QX^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} = Q ⊗ X^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} (n even)

and

Q_{±}X^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} = Q_{±} ⊗ X^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} (n odd).

Then, we prove the following theorem.

Theorem 5.4. A complete set of inequivalent irreducible spin representations of B_{n}^{m} for the 2-cocycle [−1, −1, −1] is given by the QX^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} (n even) and Q_{±}X^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} (n odd), where [λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅] is an m-partition of n.

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^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}, for any element of cycle type

(P (|λ_{(1)}|), 0, P (|λ_{(3)}|), 0, …, P (|λ_{(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 (|ρ_{1}|), EP (|ϱ_{1}|), …, OP (|ρ_{m/2}|), EP (|ϱ_{m/2}|)), where |ρ_{i}| + |ϱ_{i}| = |λ_{(2i−1)}|, 1 ≤ i(odd) ≤ m − 1. If ζ is the character of Q, then by Proposition 4.4, we have that ζ(ρ_{1}, ϱ_{1}, …, ρ_{m/2}, ϱ_{m/2}) are nonzero on the classes of type (OP (|ρ_{1}|), EP (|ϱ_{1}|), …, OP (|ρ_{m/2}|), EP (|ϱ_{m/2}|)), thus it follows that the characters (ζχ^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}) 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, …, OP, EP) and (P, ∅, P, ∅, …, P, ∅); 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

||ζχ^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}||^{2}

= ∑_{1≤i≤m/2^{ρ_{i}∈OP,ϱ_{i}∈EP}} \frac{1}{z_{ρ_{1},ϱ_{1},...,ρ_{m/2},ϱ_{m/2}}} |ζχ^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}(ρ_{1}, ϱ_{1}, …, ρ_{m/2}, ϱ_{m/2})|^{2}

= ∑_{1≤i≤m/2^{ρ_{i}∈OP,ϱ_{i}∈EP}} \frac{1}{z_{ρ_{1},ϱ_{1},...,ρ_{m/2},ϱ_{m/2}}} 2^{l(ρ,ϱ))}|χ^{λ_{(1)}}(ρ_{1} ∪ ϱ_{1}) ⋯ χ^{λ_{m−1}}(ρ_{m/2} ∪ ϱ_{m/2})|^{2}

= ∑_{1≤i≤m/2^{ρ_{i}∈OP,ϱ_{i}∈EP}} \frac{1}{z_{ρ_{1},ϱ_{1}}, ⋯, z_{ρ_{m/2},ϱ_{m/2}}} ∏_{i=1}^{m/2} 2^{l(ρ_{i},ϱ_{i}))}| ∏_{i(odd)}^{m/2} χ^{λ_{(2i−1)}}(ρ_{i} ∪ ϱ_{i})|^{2} = 1

using the corresponding result, Theorem 9.2, in [31] and where z_{ρ_{1},ϱ_{1},...,ρ_{m/2},ϱ_{m/2}} is the order of the centralizer of that class in B_{n}^{m}.

5.4. The 2-cocycle [1,-1,-1]. In this case, a similar process to that used in the previous section is applied to the representations

QX^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} = Q ⊗ X^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩},
QX_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} = Q ⊗ X_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩},
Q_{±}X^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} = Q_{±} ⊗ X^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩}

or

Q_{±}X_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} = Q_{±} ⊗ X_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩},

where (λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is an m-partition of n, with λ_{(i)} ∈ DP (|λ_{(i)}|), 1 ≤ i ≤ m − 1 as the case may be. These representations have 2-cocycle [1, −1, −1].

We prove the following theorem.

Theorem 5.5. 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^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} if n − l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is even

and

QX_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} if n − l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is odd,

and (ii) if n is odd

Q_{±}X^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} if n − l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is even

and

Q_{±}X_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} if n − l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is odd,

where (λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is an m-partition of n, with λ_{(i)} ∈ DP (|λ_{(i)}|), 1 ≤ i ≤ m − 1.

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

(OP; ∅; …; OP; ∅), (DOP; DEP; …; DOP; DEP),
(∅; OP; …; ∅; OP) and (∅; DP; …; ∅; DP),

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.

5.5. The 2-cocycle [1,-1,1]. The following lemma is required.

Lemma 5.6. If X^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} is an ordinary representation of B_{n}^{m} with character χ^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} corresponding to the m-partition (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) of n, then

(i) ηχ^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} = χ^{[λ′_{(1)};λ′_{(2)};...;λ′_{(m)}]},
(ii) σχ^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} = χ^{[λ_{( \frac{m}{2} +1)};...;λ_{(m)};λ_{(1)};...;λ_{( \frac{m}{2} )}]},
(iii) εχ^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} = χ^{[λ′_{( \frac{m}{2} +1)};...;λ′_{(m)};λ′_{(1)};...;λ′_{( \frac{m}{2} )}]},

where η, σ, ε = ζσ are linear characters of B_{n}^{m}.

Proof Using the well-known fact [12] that ηχ^{[λ]} = χ^{[λ′]}, where χ^{[λ′]} is the character of S_{n} η−associate to χ^{[λ]} and noting that

χ^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} = χ^{[λ_{(1)};∅;...;∅]} ⊗ (σ_{1} ⊗ X^{[λ_{(2)};∅;...;∅]}) ⊗ ⋯ ⊗ (σ_{m−1} ⊗ X^{[λ_{(m)};∅;...;∅]})

then (i) follows. Furthermore, since by definition

σχ^{[λ;∅;...;∅]} = σ_{m/2} ⊗ χ^{[λ;∅;...;∅]} = χ^{[∅;...;∅;λ;∅;...;∅]},

(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 η ⋂ ker σ ⋂ ker ε, thus we have the following corollary.

Corollary 5.7.

χ_{M}^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} = χ_{M}^{[λ′_{(1)};λ′_{(2)};...;λ′_{(m)}]} = χ_{M}^{[λ_{( \frac{m}{2} +1)};...;λ_{(m)};λ_{(1)};...;λ_{( \frac{m}{2} )}]}
= χ_{M}^{[λ′_{( \frac{m}{2} +1)};...;λ′_{(m)};λ′_{(1)};...;λ′_{( \frac{m}{2} )}]}.

If we now define ξ^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} = ξ ⊗ χ^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]}, where ξ 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

ξ^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} = ξ^{[λ′_{(1)};λ′_{(2)};...;λ′_{(m)}]} = ξ^{[λ_{( \frac{m}{2} +1)};...;λ_{(m)};λ_{(1)};...;λ_{( \frac{m}{2} )}]}
= ξ^{[λ′_{( \frac{m}{2} +1)};...;λ′_{(m)};λ′_{(1)};...;λ′_{( \frac{m}{2} )}]}.

If we now put RX^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} = R⊗X^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} for each m-partition (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) of n, then the RX^{[λ_{(1)};λ_{(2)};...;λ_{(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.

Theorem 5.8. The representation RX^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]} is

(i) is a sum of two equivalent irreducible representations if λ_{(i)} = λ_{( \frac{m}{2} +i)} ∈ SCP, 1 ≤ i ≤ \frac{m}{2} and n ≡ 0 (mod 4),

(ii) is a sum of four inequivalent irreducible spin representations of equal degrees if λ_{(i)} = λ_{( \frac{m}{2} +i)} ∈ SCP, 1 ≤ i ≤ \frac{m}{2} and n ≡ 2 (mod 4),

(iii) is a sum of two inequivalent representations of equal degrees if λ_{(i)} ∈ SCP, 1 ≤ i ≤ m or λ_{(i)} = λ_{( \frac{m}{2} +i)} or λ_{(i)} = λ′_{( \frac{m}{2} +i)}, 1 ≤ i ≤ \frac{m}{2} but not λ_{(i)} = λ_{( \frac{m}{2} +i)} ∈ SCP, 1 ≤ i ≤ \frac{m}{2}.

In all other cases, the four representations RX^{[λ_{(1)};λ_{(2)};...;λ_{(m)}]}, RX^{[λ′_{(1)};λ′_{(2)};...;λ′_{(m)}]}, RX^{[λ_{( \frac{m}{2} +1)};...;λ_{(m)};λ_{(1)};...;λ_{( \frac{m}{2} )}]}, RX^{[λ′_{( \frac{m}{2} +1)};...;λ′_{(m)};λ′_{(1)};...;λ′_{( \frac{m}{2} )}]} are equivalent irreducible spin representations of B_{n}^{m}.

5.6. The 2-cocycle [-1,-1,1]. The procedure in this case follows that of the previous section, but now we consider the representations RX^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = R ⊗ X^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} (RX_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = R ⊗ X_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩}) if n − l(λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is odd (even), where (λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is an m-partition of n, with λ_{(i)} ∈ DP (|λ_{(i)}|), 1 ≤ i ≤ m. Then the RX^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩}, RX_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} are spin representations of B_{n}^{m} with 2-cocycle [−1, −1, 1].

In this case the following lemma is required.

Lemma 5.9. If χ^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩}, χ_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} are the spin character of B_{n}^{m} then

(i) ηχ_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = χ_{∓}^{⟨λ′_{(1)};λ′_{(2)};...;λ′_{(m)}⟩},
ηχ^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = χ^{⟨λ′_{(1)};λ′_{(2)};...;λ′_{(m)}⟩},
(ii) σχ^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = χ^{⟨λ_{( \frac{m}{2} +1)};...;λ_{(m)};λ_{(1)};...;λ_{( \frac{m}{2} )}⟩},
σχ_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = χ_{±}^{⟨λ_{( \frac{m}{2} +1)};...;λ_{(m)};λ_{(1)};...;λ_{( \frac{m}{2} )}⟩},
(iii) εχ_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = χ_{∓}^{⟨λ′_{( \frac{m}{2} +1)};...;λ′_{(m)};λ′_{(1)};...;λ′_{( \frac{m}{2} )}⟩},
εχ^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = χ^{⟨λ′_{( \frac{m}{2} +1)};...;λ′_{(m)};λ′_{(1)};...;λ′_{( \frac{m}{2} )}⟩},

where ζ, σ, ε = ζσ are linear characters of B_{n}^{m}.

Proof From [30] we have that ηχ_{±}^{⟨λ⟩} = χ_{∓}^{⟨λ⟩}, and noting that

χ^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = χ^{⟨λ_{(1)};∅;...;∅⟩} ⊗ (σ_{1} ⊗ X^{⟨λ_{(2)};∅;...;∅⟩}) ⊗ ⋯ ⊗ (σ_{m−1} ⊗ X^{⟨λ_{(m)};∅;...;∅⟩})

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).

Corollary 5.10.

χ_{±M}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = χ_{∓M}^{⟨λ′_{(1)};λ′_{(2)};...;λ′_{(m)}⟩} = χ_{±M}^{⟨λ_{( \frac{m}{2} +1)};...;λ_{(m)};λ_{(1)};...;λ_{( \frac{m}{2} )}⟩}
= χ_{∓M}^{⟨λ′_{( \frac{m}{2} +1)};...;λ′_{(m)};λ′_{(1)};...;λ′_{( \frac{m}{2} )}⟩}.

If we now define ξ^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = ξ ⊗ χ^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩}, where ξ 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

ξ_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} = ξ_{∓}^{⟨λ′_{(1)};λ′_{(2)};...;λ′_{(m)}⟩} = ξ_{±}^{⟨λ_{( \frac{m}{2} +1)};...;λ_{(m)};λ_{(1)};...;λ_{( \frac{m}{2} )}⟩}
= ξ_{∓}^{⟨λ′_{( \frac{m}{2} +1)};...;λ′_{(m)};λ′_{(1)};...;λ′_{( \frac{m}{2} )}⟩}.

Theorem 5.11. The representation RX^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩} is

(i) is a sum of two irreducible equivalent representations if λ_{(i)} = λ_{( \frac{m}{2} +i)}, 1 ≤ i ≤ \frac{m}{2} and l(λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is even ,

(ii) is a sum of four inequivalent irreducible representations of equal degrees if λ_{(i)} = λ_{( \frac{m}{2} +i)}, 1 ≤ i ≤ \frac{m}{2} and l(λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is odd,

(iii) is a sum of two inequivalent representations of equal degrees if

n − l(λ_{(1)}; λ_{(2)}; …; λ_{(m)})

is even, but λ_{(i)} ≠ λ_{( \frac{m}{2} +i)}, for some 1 ≤ i ≤ \frac{m}{2}.

If n − l(λ_{(1)}; λ_{(2)}; …; λ_{(m)}) is odd, the four representations

RX_{±}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩}, RX_{∓}^{⟨λ_{(1)};λ_{(2)};...;λ_{(m)}⟩},
RX_{±}^{⟨λ_{( \frac{m}{2} +1)};...;λ_{(m)};λ_{(1)};...;λ_{( \frac{m}{2} )}⟩}, RX_{∓}^{⟨λ_{( \frac{m}{2} +1)};...;λ_{(m)};λ_{(1)};...;λ_{( \frac{m}{2} )}⟩}

are equivalent irreducible spin representations of B_{n}^{m}.

5.7. The 2-cocycle [-1,1,-1]. In this case, for the m-partition of n (λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅), if n is even, put

RQX^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} = R ⊗ QX^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}

and if n is odd, put

RQ_{±}X^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} = R ⊗ Q_{±}X^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}.

Then, these representations have factor set [−1, 1, −1] and we prove the following theorem.

Theorem 5.12. The representation RQX^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} is

(i) is a sum of two equivalent irreducible representations if λ_{(i)} ∈ SCP, 1 ≤ i(odd) ≤ m and n is even (ii) is a sum of two inequivalent representations of equal degrees if λ_{(i)} ∈ SCP, 1 ≤ i(odd) ≤ m and n is odd or λ_{(i)} ∉ SCP, for some 1 ≤ i(odd) ≤ m and n is even. If λ_{(i)} ∉ SCP, 1 ≤ i(odd) ≤ m and n is odd, the representations RQ_{±}X^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}, RQ_{±}X^{[λ′_{(1)};∅;λ′_{(3)};∅;...;λ′_{(m−1)};∅]}, RQ_{±}X^{[λ_{( \frac{m}{2} )};∅;...;λ_{(m−1)};∅;...;λ_{(1)};∅;...;λ_{( \frac{m}{2} −1)};∅]}, RQ_{±}X^{[λ′_{( \frac{m}{2} )};∅;...;λ′_{(m−1)};∅;...;λ′_{(1)};∅;...;λ′_{( \frac{m}{2} −1)};∅]} are equivalent irreducible spin representations of B_{n}^{m}.

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^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} by ζχ^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}.

Lemma 5.13. If n is even, then

\begin{aligned}(i) η(ζχ^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}) & = & ζχ^{[λ′_{(1)};∅;λ′_{(3)};∅;...;λ′_{(m−1)};∅]}, \\ (ii) σ(ζχ^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}) & = & ζχ^{[λ′_{(1)};∅;λ′_{(3)};∅;...;λ′_{(m−1)};∅]}, \\ (iii) ε(ζχ^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}) & = & ζχ^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}. \end{aligned}

If n is odd, then

(i) η(ζ_{±}χ^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]})=ζ_{∓}χ^{[λ′_{(1)};∅;λ′_{(3)};∅;...;λ′_{(m−1)};∅]},
(ii) σ(ζ_{±}χ^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]})=\begin{cases}ζ_{±}χ^{[λ_{( \frac{m}{2} +1)};∅;...;λ_{(m−1)};∅;λ_{1};∅;...;λ_{( \frac{m}{2} −1)};∅]} & if m ≡ 0 (mod 4) \\ ζ_{±}χ^{[∅;λ_{( \frac{m}{2} +2)};...;λ_{(m−1)};∅;λ_{1};∅;...;∅;λ_{( \frac{m}{2} )}]} & if m ≡ 2 (mod 4) \end{cases}

(iii) ε(ζ_{±}χ^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]})=\begin{cases}ζ_{±}χ^{[λ′_{( \frac{m}{2} +1)};∅;...;λ′_{(m−1)};∅;λ′_{1};∅;...;λ′_{( \frac{m}{2} −1)};∅]} & if m ≡ 0 (mod 4) \\ ζ_{±}χ^{[∅;λ′_{( \frac{m}{2} +2)};...;λ′_{(m−1)};∅;λ′_{1};∅;...;∅;λ′_{( \frac{m}{2} )}]} & if m ≡ 2 (mod 4) \end{cases}

where ζ, σ, ε = ζσ are linear characters of B_{n}^{m}.

From this lemma, we obtain, as before in Section 5.5, the following corollary.

Corollary 5.14. If n is even, then

ζχ_{M}^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} = ζχ_{M}^{[λ′_{(1)};∅;λ′_{(3)};∅;...;λ′_{(m−1)};∅]}.

If n is odd, then

ζχ_{M}^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} = ζχ_{M}^{[λ′_{(1)};∅;λ′_{(3)};∅;...;λ′_{(m−1)};∅]}

= \begin{cases}ζ_{±}χ^{[λ_{( \frac{m}{2} +1)};∅;...;λ_{(m−1)};∅;λ_{1};∅;...;λ_{( \frac{m}{2} −1)};∅]} & if m ≡ 0 (mod 4) \\ ζ_{±}χ^{[∅;λ_{( \frac{m}{2} +2)};...;λ_{(m−1)};∅;λ_{1};∅;...;∅;λ_{( \frac{m}{2} )}]} & if m ≡ 2 (mod 4) \end{cases}

= \begin{cases}ζ_{±}χ^{[λ′_{( \frac{m}{2} +1)};∅;...;λ′_{(m−1)};∅;λ′_{1};∅;...;λ′_{( \frac{m}{2} −1)};∅]} & if m ≡ 0 (mod 4) \\ ζ_{±}χ^{[∅;λ′_{( \frac{m}{2} +2)};...;λ′_{(m−1)};∅;λ′_{1};∅;...;∅;λ′_{( \frac{m}{2} )}]} & if m ≡ 2 (mod 4) \end{cases} .

In turn, this leads to the corresponding results for the characters ξζχ_{M}^{[λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]} and in turn to the proof of Theorem 5.12.

5.8. The 2-cocycle [1,1,-1]. In this case, let (λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) be the mpartition of n with λ_{(i)} ∈ DP (|λ_{(i)}|), 1 ≤ i ≤ m. If n is even and n − l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is even, put

RQX^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} = R ⊗ QX^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩}

and if n − l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is odd, put

RQX_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} = R ⊗ QX_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩};

and if n is odd and n − l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is even, put

RQ_{±}X^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} = R ⊗ Q_{±}X^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩}

and if n − l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is odd, put

RQ_{±}X_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} = R ⊗ Q_{±}X_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩}.

These representations have 2-cocycle [1, 1, −1].

Theorem 5.15. The representation RQX^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} is

(i) is a sum of two equivalent irreducible representations if n−l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is even and n is even ,

(ii) is a sum of two inequivalent representations of equal degrees if n−l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is even and n is odd or if n−l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is odd and n is even.

If n − l(λ_{(1)}; ∅; λ_{(3)}; ∅; …; λ_{(m−1)}; ∅) is odd and n is odd, the representations

RQ_{±}X_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩} and RQ_{±}X_{±}^{⟨λ_{( \frac{m}{2} )};∅;...;λ_{(m−1)};∅;...;λ_{(1)};∅;...;λ_{( \frac{m}{2} −1)};∅⟩} are equivalent irreducible spin representations of B_{n}^{m}.

In this case, we merely state the corresponding lemma to Lemma 5.13.

Lemma 5.16. If n is even, then

\begin{aligned}(i) η(ζχ_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩}) & = & ζχ_{∓}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅]}, \\ (ii) σ(ζχ_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩}) & = & ζχ_{∓}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩}, \\ (iii) ε(ζχ_{±}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩}) & = & ζχ_{∓}^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩}. \end{aligned}

If n is odd, then

(i) η(ζ_{±}χ^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩})=ζ_{∓}χ^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩},
(ii) σ(ζ_{±}χ^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩})=\begin{cases}ζ_{∓}χ^{⟨λ_{( \frac{m}{2} +1)};∅;...;λ_{(m−1)};∅;λ_{1};∅;...;λ_{( \frac{m}{2} −1)};∅⟩} & if m ≡ 0 (mod 4) \\ ζ_{∓}χ^{⟨∅;λ_{( \frac{m}{2} +2)};...;λ_{(m−1)};∅;λ_{1};∅;...;∅;λ_{( \frac{m}{2} )}⟩} & if m ≡ 2 (mod 4) \end{cases}

(iii) ε(ζ_{±}χ^{⟨λ_{(1)};∅;λ_{(3)};∅;...;λ_{(m−1)};∅⟩})=\begin{cases}ζ_{∓}χ^{⟨λ_{( \frac{m}{2} +1)};∅;...;λ_{(m−1)};∅;λ_{1};∅;...;λ_{( \frac{m}{2} −1)};∅⟩} & if m ≡ 0 (mod 4) \\ ζ_{∓}χ^{⟨∅;λ_{( \frac{m}{2} +2)};...;λ′_{(m−1)};∅;λ_{1};∅;...;∅;λ_{( \frac{m}{2} )}⟩} & if m ≡ 2 (mod 4) \end{cases}

where ζ, σ, ε = ζσ are linear characters of B_{n}^{m}.

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Institute of Mathematical and Physical Sciences, University of Wales, Aberystwyth, Ceredigion SY23 3BZ, Wales, U.K.

E-mail address: alun@morus25.fsnet.co.uk