Séminaire Lotharingien de Combinatoire 64 (2010), Article B64a

SYMMETRIC AND ANTISYMMETRIC VECTOR-VALUED JACK POLYNOMIALS

CHARLES F. DUNKL

Abstract. Polynomials with values in an irreducible module of the symmetric group can be given the structure of a module for the rational Cherednik algebra, called a standard module. This algebra has one free parameter and is generated by differential-difference (“Dunkl”) operators, multiplication by coordinate functions and the group algebra. By specializing Griffeth’s (arχiv:0707.0251) results for the G(r, p, N) setting, one obtains norm formulae for symmetric and antisymmetric polynomials in the standard module. Such polynomials of minimum degree have norms which involve hook-lengths and generalize the norm of the alternating polynomial.

1. Introduction

Hook-lengths of nodes in Young tableaux appear in a variety of different settings. Griffeth [6] introduced Jack polynomials whose values lie in irreducible modules of the family G (r, p, N) of complex reflection groups. This class of polynomials forms an orthogonal basis for the associated standard module of the rational Cherednik algebra. In this paper we specialize his results to the symmetric group and show how the norms of two special symmetric and antisymmetric polynomials in the standard module depend on the hook-lengths of the partition associated to the representation. These norm formulae prove a necessity condition for aspherical parameter values. This condition was first found by Gordon and Stafford [5].

For N ≥ 2, x = (x_{1}, …, x_{N}) ∈ ℝ^{N} and let ℕ := {0, 1, 2, 3, …}. For a, b ∈ ℕ and a ≤ b let [a, b] = {a, a + 1, …, b} (an interval of integers). The cardinality of a set E is denoted by #E. For α ∈ ℕ^{N} (a composition) let |α| := ∑_{i=1}^{N} α_{i}, x^{α} := ∏_{i=1}^{N} x_{i}^{α_{i}}, a monomial of degree |α|. The spaces of polynomials, respectively homogeneous, polynomials are

2000 Mathematics Subject Classification. Primary 05E05, 20C30; Secondary 33C80, 05E35.

Key words and phrases. Jack polynomials, standard modules, Dunkl operators, hook-lengths.

P := span_{𝔽} {x^{α} : α ∈ ℕ^{N}} ,
P_{n} := span_{𝔽} {x^{α} : α ∈ ℕ^{N}, |α| = n} , n ∈ ℕ,

where 𝔽 is a field ⊃ ℚ. Consider the symmetric group S_{N} as the group of permutations of [1, N]. The group acts on polynomials by linear extension of (xw)_{i} = x_{w(i)}, w ∈ S_{N}, 1 ≤ i ≤ N, that is, wf (x) := f (xw) , f ∈ P. For α ∈ ℕ^{N} let (wα)_{i} = α_{w^{−1}(i)}, then w (x^{α}) = x^{wα}. Also S_{N} is a finite reflection group whose reflections are the transpositions (i, j); x (i, j) = (…, x^{i}_{j}, …, x^{j}_{i}, …). The simple reflections s_{i} := (i, i + 1) , 1 ≤ i < N, generate S_{N}.

Say λ ∈ ℕ^{N} is a partition if λ_{i} ≥ λ_{i+1} for all i. Denote the set of partitions by ℕ^{N,+}. Suppose τ is a partition of N, that is, |τ| = N; then there is an associated Ferrers diagram, namely the set of lattice points {(i, j) ∈ ℕ^{2} : 1 ≤ i ≤ ℓ (τ) , 1 ≤ j ≤ τ_{i}}, also denoted by τ; the length of τ is ℓ (τ) := max {i : τ_{i} > 0}. The conjugate partition τ′ is the partition whose diagram is the transpose of the diagram of τ (that is, τ′_{m} = # {i : τ_{i} ≥ m}). For a node (or point) (i, j) ∈ τ the arm-length is arm (i, j) := τ_{i} − j, the leg-length is leg (i, j) := τ′_{j} − i, and the hooklength is h (i, j) := arm (i, j) + leg (i, j) + 1. We will use arm (i, j; τ) etc. if it is necessary to specify the partition.

To each partition τ of N there is an associated irreducible S_{N}-module V_{τ}. We analyze the space M (τ) of V_{τ}-valued polynomials under the action of differential-difference (“Dunkl”) operators. There is a canonical symmetric bilinear (the contravariant) form ⟨·, ·⟩ on this space. We will construct distinguished polynomials f_{τ}^{s}, f_{τ}^{a} ∈ M (τ), with f_{τ}^{s} being symmetric and f_{τ}^{a} being antisymmetric, such that

⟨f_{τ}^{s}, f_{τ}^{s}⟩ = c_{0} ∏_{(i,j)∈τ} (1 − h (i, j) κ)_{leg(i,j)} ,
⟨f_{τ}^{a}, f_{τ}^{a}⟩ = c_{1} ∏_{(i,j)∈τ} (1 + h (i, j) κ)_{arm(i,j)} ;

and c_{0}, c_{1} ∈ ℚ are constants depending on τ, and the Pochhammer symbol is

(t)_{n} := ∏_{i=1}^{n} (t + i − 1) , n ∈ ℕ.

This result generalizes the situation of the trivial representation of S_{N};

in this case τ = (N) , f_{τ}^{s} = 1, f_{τ}^{a} = ∏_{1≤i<j≤N} (x_{i} − x_{j}) and ⟨f_{τ}^{a}, f_{τ}^{a}⟩ =

c_{1} ∏_{i=2}^{N} (1 + iκ)_{i−1} (see Opdam [10]).

Section 2 collects the needed information about representations of S_{N}. The Dunkl operators and their action on monomials are discussed in Section 3. The (nonsymmetric) Jack polynomials are constructed in Section 4; this material is the specialization of Griffeth’s results for G (r, p, N) to S_{N} = G (1, 1, N) and some of the proofs are in the Appendix. Our main results on symmetric and antisymmetric polynomials are contained in Section 5. There is the description of an orthogonal basis and the detailed exposition of the special polynomials whose norms involve the hook-lengths. In fact these are the symmetric and antisymmetric polynomials of minimum degree. In the standard module M (τ) the orthogonal basis of symmetric (respectively antisymmetric) Jack polynomials is labeled by the column-strict (respectively row-strict) tableaux of shape τ.

2. Representations of S_{N}

Let Y (τ) be the set of reversed standard Young tableaux (RSYT) of shape τ, namely, an assignment of the numbers 1, 2, …, N to each node of τ such that entries decrease in each row and in each column. (Note: we use reversed tableaux because they provide a direct way of expressing eigenvalues of the operators defining the nonsymmetric Jack polynomials, see (4.1).) The node of T containing i is denoted T (i) and the row and column of this node are denoted by rw (i, T) , cm (i, T) respectively, i ∈ [1, N]. The content of T (i) is c (i, T) := cm (i, T) − rw (i, T). Thus c (N, T) = 0 for each T. The well-known hook-length formula asserts that #Y (τ) = N!/ ∏_{(i,j)∈τ} h (i, j). Following Murphy

[8] define an action of S_{N} on the # (Y (τ))-dimensional vector space V_{τ} := span_{𝔽} {v_{T} : T ∈ Y (τ)} as follows:

Proposition 1. Suppose T ∈ Y (τ) and

b_{i} (T) := 1/ (c (i, T) − c (i + 1, T))

for 1 ≤ i < N then:

(1) if b_{i} (T) = 1 (when rw (i, T) = rw (i + 1, T)) then s_{i}v_{T} = v_{T};

(2) if b_{i} (T) = −1 (when cm (i, T) = cm (i + 1, T)) then s_{i}v_{T} = −v_{T};

(3) if 0 < b_{i} (T) ≤ \frac{1}{2} (when rw (i, T) < rw (i + 1, T) and cm (i, T) > cm (i + 1, T)then

s_{i}v_{T} = b_{i} (T) v_{T} + v_{s_{i}T} ,
s_{i}v_{s_{i}T} = (1 − b_{i} (T)^{2}) v_{T} − b_{i} (T) v_{s_{i}T};

(4) if −\frac{1}{2} ≤ b_{i} (T) < 0 (when rw (i, T) > rw (i + 1, T) and cm (i, T) < cm (i + 1, T)) then

s_{i}v_{T} = b_{i} (T) v_{T} + (1 − b_{i} (T)^{2}) v_{s_{i}T} ,

and

s_{i}v_{s_{i}T} = v_{T} − b_{i} (T) v_{s_{i}T} .

In cases (3) and (4) the tableau s_{i}T is obtained by interchanging the entries i, i + 1. Furthermore in case (3) s_{i}f_{0} = f_{0} and s_{i}f_{1} = −f_{1} for

f_{0} = (b_{i} (T) + 1) v_{T} + v_{s_{i}T} ,
f_{1} = (b_{i} (T) − 1) v_{T} + v_{s_{i}T} .

There is an ordering on tableaux such that T > s_{i}T in case (4).

Corollary 1. Let f = ∑_{T ∈Y (τ)} k_{T} v_{T} with the coefficients k_{T} ∈ ℚ and s_{i}f = ±f for some i ∈ [1, N − 1]. Then

(1) T, s_{i}T ∈ Y (τ) implies k_{s_{i}T} = rk_{T} for some r ≠ 0;

(2) s_{i}f = f and cm (i, T) = cm (i + 1, T) implies k_{T} = 0;

(3) s_{i}f = −f and rw (i, T) = rw (i + 1, T) implies k_{T} = 0.

Statement (1) means that k_{s_{i}T} and k_{T} are either both nonzero or both zero.

Definition 1. The Jucys-Murphy elements (in the group algebra ℚS_{N}) are

ω_{i} := ∑_{j=i+1}^{N} (i, j) , i ∈ [1, N] .

There are commutation relations: ω_{i}ω_{j} = ω_{i}ω_{j} for all i, j; ω_{i}s_{j} = s_{j}ω_{i} for j ≠ i−1, i; s_{i}ω_{i}−ω_{i+1}s_{i} = 1 (see Vershik and Okounkov [9, Sec. 4] for the representations of the algebra generated by {ω_{i}, ω_{i+1}, s_{i}}). Murphy proved the following:

Theorem 1. Suppose T ∈ Y (τ) and i ∈ [1, N] then ω_{i}v_{T} = c (i, T) v_{T} .

Let ⟨·, ·⟩_{0} be a S_{N}-invariant positive-definite bilinear form on V_{τ}, (the form is unique up to a multiplicative constant) then each ω_{i} is self-adjoint and hence the vectors v_{T} are pairwise orthogonal, being eigenvectors with different eigenvalues. Denote ∥v∥_{0}^{2} = ⟨v, v⟩_{0}. For given T and i as in case (4) we have ∥v_{T} ∥_{0}^{2} = b (i, T)^{2} ∥v_{T} ∥_{0}^{2} + ∥v_{s_{i}T} ∥_{0}^{2} (since s_{i} is an isometry) and thus ∥v_{s_{i}T} ∥_{0}^{2} = (1 − b (i, T)^{2}) ∥v_{T} ∥_{0}^{2}. There is one formula for ∥v_{T} ∥_{0}^{2} in [8, Thm. 4.1]. The following is based on the content vector of T (that is, (c (1, T) , …, c (N, T))) :

Definition 2. For T ∈ Y (τ) let

∥v_{T} ∥_{c}^{2} = ∏_{1≤i<j≤N, c(i,T)≤c(j,T)−2} \frac{(c (i, T) − c (j, T))^{2} − 1}{(c (i, T) − c (j, T))^{2}}.

For example, let N = 4, τ = (3, 1) and for convenience label tableaux by sequences of rows; then ∥v_{[(4,2,1),(3)]}∥_{c}^{2} = 1, ∥v_{[(4,3,1),(2)]}∥_{c}^{2} = \frac{3}{4}, ∥v_{[(4,3,2),(1)]}∥_{c}^{2} = \frac{2}{3}.

Lemma 1. Suppose {g_{ij} (T) : 1 ≤ i < j ≤ N} is a collection of functions on Y (τ) such that

(1) g_{ij} (T) = g_{ij} (s_{m}T) for all i, j with {i, j} ∩ {m, m + 1} = ∅,

(2) g_{i,m} (T) = g_{i,m+1} (s_{m}T) and g_{i,m+1} (T) = g_{i,m} (s_{m}T) for i < m,

(3) g_{m,j} (T) = g_{m+1,j} (s_{m}T) and g_{m+1,j} (T) = g_{m,j} (s_{m}T) for j > m + 1, whenever m ∈ [1, N − 1] and {T, s_{m}T } ⊂ Y (τ) , then

\frac{∏_{1≤i<j≤N} g_{ij} (T)}{∏_{1≤i<j≤N} g_{ij} (s_{m}T)} = \frac{g_{m,m+1} (T)}{g_{m,m+1} (s_{m}T)}.

The proof is a straightforward calculation.

Proposition 2. Suppose 0 < b_{i} (T) ≤ \frac{1}{2} for T ∈ Y (τ) and some i ∈ [1, N − 1] then ∥v_{s_{i}T} ∥_{c}^{2} = (1 − b_{i} (T)^{2}) ∥v_{T} ∥_{c}^{2}. Thus ∥·∥_{c} is an S_{N}- invariant norm.

Proof. By hypothesis c (i, T) ≥ c (i + 1, T) + 2 and

c (i, s_{i}T) = c (i + 1, T) , c (i + 1, s_{i}T) = c (i, T) .

All factors in the ratio ∥v_{s_{i}T} ∥_{c}^{2} / ∥v_{T} ∥_{c}^{2} except (1 − (\frac{1}{c(i+1,T)−c(i,T)})^{2})

in the numerator cancel out, by Lemma 1. □

Henceforth we drop the subscript “c” and use “0” for the form. Next we consider invariance properties for certain subgroups of S_{N}, specifically the stabilizer subgroups of the monomials x^{λ}, where λ ∈ ℕ^{N,+}.

Definition 3. For 1 ≤ a < b ≤ N let

S_{[a,b]} := {w ∈ S_{N} : i ∉ [a, b] ⟹ w (i) = i} ,

the subgroup of permutations of [a, b], generated by {s_{i} : a ≤ i < b}.

We look for elements f of V_{τ} which are symmetric or antisymmetric for a group S_{[a,b]}, or the equivalent properties: s_{i}f = f, respectively, s_{i}f = −f, for a ≤ i < b. The idea is to find the expansions of ∑_{w∈S_{[a,b]}} wv_{T}, or ∑_{w∈S_{[a,b]}} sgn (w) wv_{T}, in the basis {v_{S} : S ∈ Y (τ)} for given T ∈ Y (τ).

Definition 4. For T ∈ Y (τ) and a subgroup H of S_{N} let V_{T} (H) = span {wv_{T} : w ∈ H} and let Y (T; H) = {T′ ∈ Y (τ) : v_{T′} ∈ V_{T} (H)}.

In the case H = S_{[a,b]} there are two extremal elements of Y (T; H), namely T_{0} with the property cm (i, T_{0}) ≥ cm (i + 1, T_{0}) for a ≤ i < b, and T_{1} with the property rw (i, T_{0}) ≥ rw (i + 1, T_{0}) (it is possible that T_{0} = T_{1}). To produce T_{0} one applies a sequence of transformations of type (4) (in Proposition 1) (type (3) for T_{1}). If cm (i_{1}, T) = cm (i_{2}, T) for some i_{1}, i_{2} ∈ [a, b] (suppose i_{1} > i_{2} then any entry j in this column of T between i_{1} and i_{2} has to satisfy i_{1} > j > i_{2}) then T_{0} has cm (i, T_{0}) = cm (i + 1, T_{0}) for some i ∈ [a, b − 1]. Similarly if rw (i_{1}, T) = rw (i_{2}, T) for some i_{1}, i_{2} ∈ [a, b] then T_{1} has rw (i, T_{1}) = rw (i + 1, T_{1}) for some i ∈ [a, b − 1].

2.1. Subgroup symmetric vectors. First consider the invariant (symmetric) situation. Corollary 1 and the properties of T_{0} imply the following necessary condition for V_{T} (S_{[a,b]}) to contain a nontrivial S_{[a,b]}-invariant.

Say T satisfies condition [a, b]_{cm} if the entries a, a + 1, …, b are in distinct columns of T, that is, a ≤ i < j ≤ m implies cm (i, T) ≠ cm (j, T). Fix some T satisfying this condition and consider the subspace V_{T} (S_{[a,b]}). Let T_{0} ∈ Y (T; S_{[a,b]}) satisfy cm (i, T) > cm (j, T) for a ≤ i < j ≤ b (equality is ruled out by hypothesis). It is possible that i and i + 1 are in the same row of T_{0} for some i ∈ [a, b] (in which case #Y (T; S_{[a,b]}) < (b − a + 1)! = #S_{[a,b]}). For a ≤ i < b we have rw (i, T_{0}) ≤ rw (i + 1, T_{0}), thus a ≤ i < j ≤ b implies c (j, T_{0}) − c (i, T_{0}) ≤ −2 or j = i + 1 and rw (i + 1, T_{0}) = rw (i, T_{0}); indeed suppose the latter condition does not hold then if j > i + 1

c (j, T_{0}) − c (i, T_{0}) = (cm (j, T_{0}) − cm (i, T_{0})) + (rw (i, T_{0}) − rw (j, T_{0}))
≤ cm (j, T_{0}) − cm (i, T_{0}) ≤ i − j ≤ −2,

or j = i + 1 and

c (i + 1, T_{0}) − c (i, T_{0})
= (cm (i + 1, T_{0}) − cm (i, T_{0})) + (rw (i, T_{0}) − rw (i + 1, T_{0}))
≤ −1 − 1 = −2.

Definition 5. Suppose T ∈ Y (τ) satisfies condition [a, b]_{cm} then let

P_{0} (T; a, b) := ∏_{a≤i<j≤b, cm(i,T)<cm(j,T)} \frac{c (j, T) − c (i, T)}{1 + c (j, T) − c (i, T)}.

The denominator can not vanish, for suppose i < j, cm (i, T) < cm (j, T), and T (i) = T_{0} (i_{1}) , T (j) = T_{0} (i_{2}) with i_{1} < i_{2} (this follows from cm (i_{2}, T) < cm (i_{1}, T)) then c (i, T) − c (j, T) = c (i_{2}, T_{0}) − c (i_{1}, T_{0}) ≤ −2 , and rw (i, T) = rw (i_{2}, T_{0}) ≠ rw (j, T) = rw (i_{1}, T_{0}). For notational convenience we use the fact Y (T; S_{[a,b]}) = Y (T_{0}; S_{[a,b]}) (and let T be variable, henceforth).

Proposition 3. Let f = ∑_{T ∈Y (T_{0};S_{[a,b]})} P_{0} (T; a, b) v_{T} then wf = f for all w ∈ S_{[a,b]}.

Proof. Suppose a ≤ i < b then let

A = {T ∈ Y (T_{0}; S_{[a,b]}) : rw (i, T) = rw (i + 1, T)}

and

B = {T ∈ Y (T_{0}; S_{[a,b]}) : rw (i, T) < rw (i + 1, T)} .

Then

f = ∑_{T ∈A} P_{0} (T; a, b) v_{T} + ∑_{T ∈B} (P_{0} (T; a, b) v_{T} + P_{0} (s_{i}T; a, b) v_{s_{i}T}) .

Fix T ∈ B and compute P_{0} (T; a, b) /P_{0} (s_{i}T; a, b) using Lemma 1; set g_{mn} (T) = 1 if cm (m, T) ≥ cm (n, T) and g_{mn} (T) = \frac{c(n,T)−c(m,T)}{1+c(n,T)−c(m,T)} if cm (m, T) < cm (n, T). Then g_{i,i+1} (T) = 1 and

g_{i,i+1} (s_{i}T) = \frac{c (i + 1, s_{i}T) − c (i, s_{i}T)}{1 + c (i + 1, s_{i}T) − c (i, s_{i}T)} = \frac{1}{1 − b_{i} (s_{i}T)} = \frac{1}{1 + b_{i} (T)}.

Thus P_{0} (T; a, b) /P_{0} (s_{i}T; a, b) = 1 + b_{i} (T) and s_{i}f = f by Proposition 1. □

Corollary 2. Let n_{0} = # {w ∈ S_{[a,b]} : wv_{T_{0}} = v_{T_{0}}}, then

∥f∥_{0}^{2} = \frac{(b − a)!}{n_{0}}P_{0} (T_{1}; a, b) ∥v_{T_{0}}∥_{0}^{2} .

Proof. If T, T′ ∈ Y (τ) and T′ is obtained from T by a sequence of steps of type (3) in Proposition 1 then T′ = wT for some w ∈ S_{N} and v_{T′} = wv_{T} + ∑_{j} b_{j}v_{S_{j}}, where b_{j} ∈ ℚ and [S_{1} = T, S_{2}, …] is the list of intermediate steps. Let f_{1} = ∑_{w∈S_{[a,b]}} wv_{T_{0}} thus f_{1} = cf for some

constant c. In the expansion of f_{1} in the basis {v_{T} : T ∈ Y (T_{0}; S_{[a,b]})} the coefficient of v_{T_{1}} is n_{0}, because T_{0}, T_{1} have the property described above and v_{T_{1}} is extremal in Y (T_{0}; S_{[a,b]}) (heuristically the “bubble sort” is used; first apply (b − 1, b) (b − 2, b − 1) … (a, a + 1) to T_{0}; this moves b to the column with highest possible number; then repeat the process with S_{[a,b−1]}, or S_{[a,b−k]} if b − k + 1, …, b are now in the same row, and so on). The coefficient of v_{T_{1}} in f is P_{0} (T_{1}; a, b). Thus c = \frac{n_{0}}{P_{0}(T_{1};a,b)}in f. Finally

⟨f, f⟩_{0} = \frac{1}{c} ⟨f_{1}, f⟩_{0} = \frac{1}{c} ∑_{w∈S_{[a,b]}} ⟨wv_{T_{0}}, f⟩_{0}

= \frac{(b − a)!}{c} ⟨v_{T_{0}}, f⟩_{0} = \frac{(b − a)!}{c} ⟨v_{T_{0}}, v_{T_{0}}⟩ .

This completes the proof. □

It is straightforward to extend these methods to the case H = S_{[a_{1},b_{1}]} × S_{[a_{2},b_{2}]} × … S_{[a_{n},b_{n}]} where 1 ≤ a_{1} < b_{1} < a_{2} < b_{2} < … < a_{n} < b_{n} ≤ N. This requires a tableau T_{0} ∈ Y (τ) satisfying condition [a_{i}, b_{i}]_{cm} for 1 ≤ i ≤ n. Then

f = ∑_{T ∈Y (T_{0};H)} ∏_{i=1}^{n} P_{0} (T; a_{i}, b_{i}) v_{T}

is the unique H-invariant element of V_{T_{0}} (H).

2.2. Subgroup antisymmetric vectors. We turn to the problem of antisymmetric vectors in V_{T} (H). The previous arguments transfer almost directly by transposing tableaux and inserting minus signs at appropriate places.

Say T satisfies condition [a, b]_{rw} if the entries a, a+1, …, b are in distinct rows of T, that is, a ≤ i < j ≤ m implies rw (i, T) ≠ rw (j, T). Fix some T satisfying this condition and consider the subspace V_{T} (S_{[a,b]}). Let T_{0} ∈ Y (T; S_{[a,b]}) satisfy cm (i, T) ≥ cm (j, T) for a ≤ i < j ≤ b.

Definition 6. Suppose T ∈ Y (τ) satisfies condition [a, b]_{cm} then let

P_{1} (T; a, b) := ∏_{a≤i<j≤b, cm(i,T)<cm(j,T)} \frac{c (j, T) − c (i, T)}{1 − c (j, T) + c (i, T)}.

As before we use the basic set Y (T_{0}; S_{[a,b]}) to produce an antisymmetric vector. Note P_{1} (T_{0}; a, b) = 1.

Proposition 4. Let f = ∑_{T ∈Y (T_{0};S_{[a,b]})} P_{1} (T; a, b) v_{T} then s_{i}f = −f for a ≤ i < b and wf = sgn (w) f for all w ∈ S_{[a,b]}. Let n_{0} = # \{w ∈ S_{[a,b]} : wv_{T_{0}} = ±v_{T_{0}}\} then ∥f∥^{2} = \frac{(b−a)!}{n_{0}} |P_{1} (T_{1}; a, b)| ∥v_{T_{0}}∥^{2}.

Proof. Suppose a ≤ i < b then let

A = {T ∈ Y (T_{0}; S_{[a,b]}) : cm (i, T) = cm (i + 1, T)}

and

B = {T ∈ Y (T_{0}; S_{[a,b]}) : cm (i, T) > cm (i + 1, T)};

T ∈ A implies s_{i}v_{T} = −v_{T}. Then

f = ∑_{T ∈A} P_{1} (T; a, b) v_{T} + ∑_{T ∈B} (P_{1} (T; a, b) v_{T} + P_{1} (s_{i}T; a, b) v_{s_{i}T}) .

Fix T ∈ B and compute P_{1} (T; a, b) /P_{1} (s_{i}T; a, b) using Lemma 1; set g_{mn} (T) = 1 if cm (m, T) ≥ cm (n, T) and g_{mn} (T) = \frac{c(n,T)−c(m,T)}{1−c(n,T)+c(m,T)} if cm (m, T) < cm (n, T). Then g_{i,i+1} (T) = 1 and

g_{i,i+1} (s_{i}T) = \frac{c (i + 1, s_{i}T) − c (i, s_{i}T)}{1 − c (i + 1, s_{i}T) + c (i, s_{i}T)}

= \frac{1}{−1 − b_{i} (s_{i}T)} = \frac{1}{b_{i} (T) − 1}.

Thus P_{0} (T; a, b) /P_{0} (s_{i}T; a, b) = b_{i} (T) − 1 and s_{i}f = −f by Proposition 1. The norm formula follows from the proof of Corollary 2 with some small modifications to take care of sign-changes. □

There are corresponding statements for H = S_{[a_{1},b_{1}]} × … × S_{[a_{n},b_{n}]}, using disjoint intervals. The branching theorem for the restriction of irreducible representations of S_{N} to those of the parabolic subgroups (like H) implicitly appears in the previous discussion, in connection with the conditions [a, b]_{cm} and [a, b]_{rw}.

3. Dunkl operators

Let κ be a transcendental (formal parameter) and set 𝔽 = ℚ (κ). Consider the space P ⊗ V_{τ} = span_{𝔽} {x^{α}v_{T} : α ∈ ℕ^{N}, T ∈ Y (τ)}, polynomials p (x) on ℝ^{N} with values in V_{τ}. The space is an S_{N}-module with the action w (x^{α}v_{T}) = x^{wα} (wv_{T}) for w ∈ S_{N}, extended to all of P ⊗ V_{τ} by linearity. For p ∈ P and u ∈ V_{τ} and 1 ≤ i ≤ N let

(3.1) D_{i} (p (x) u) := \frac{∂}{∂x_{i}} p (x) u + κ ∑_{j=1,j≠i}^{N} \frac{p (x) − p (x (i, j))}{x_{i} − x_{j}} (i, j) u.

The definition is extended to P⊗V_{τ} by linearity. Then D_{i}D_{j} = D_{j}D_{i} for 1 ≤ i, j ≤ N. The proof is a straightforward adaptation of the original proof for scalar polynomials p (x) (see [3, Ch. 4]). There are important commutators (appearing in the definition of the rational Cherednik algebra, the algebra generated by 𝔽S_{N} and {x_{i}, D_{i} : i ∈ [1, N]}):

(3.2)  D_{i}x_{j} − x_{j}D_{i} = −κ (i, j) , i ≠ j
D_{i}x_{i} − x_{i}D_{i} = 1 + κ ∑_{j≠i} (i, j) .

Definition 7. The space P ⊗ V_{τ} equipped with the action of 𝔽S_{N} and {x_{i}, D_{i} : i ∈ [1, N]} is a standard module of the rational Cherednik algebra and is denoted by M (τ). For n ∈ ℕ the linear subspace P_{n} ⊗ V_{τ} is denoted by M_{n} (τ).

The representation theory of rational Cherednik algebras is described in the survey [11] by Rouquier. For p (x) ∈ P ⊗ V_{τ} set

U_{i}p (x) = D_{i} (x_{i}p (x)) − κ ∑_{j=1}^{i−1} (i, j) p (x) , 1 ≤ i ≤ N.

The operators U_{i} also commute pairwise. They have a triangularity property (a special case of a result of Griffeth [6] for the complex reflection groups G (r, p, N)). There is an important function on compositions:

Definition 8. For α ∈ ℕ^{N} and 1 ≤ i ≤ N let

r (α, i) := # {j : α_{j} > α_{i}} + # {j : 1 ≤ j ≤ i, α_{j} = α_{i}}

be the rank function.

A consequence of the definition is that r (α, i) < r (α, j) is equivalent to α_{i} > α_{j}, or α_{i} = α_{j} and i < j. For any α the function i ↦ r (α, i) is one-to-one on {1, 2, …, N}. Let w_{α} denote the inverse function, thus r (α, w_{α} (i)) = i. Further α is a partition if and only if r (α, i) = i for all i. In general (w_{α}^{−1}α)_{i} = α_{w_{α}(i)} for 1 ≤ i ≤ N, and thus w_{α}^{−1}α is a partition, denoted by α^{+}. For example, let α = (1, 0, 4, 2, 4) then

[r (α, i)]_{i=1}^{5} = [4, 5, 1, 3, 2] ,
[w_{α} (i)]_{i=1}^{5} = [3, 5, 4, 1, 2] ,
α^{+} = (4, 4, 2, 1, 0) .

The order on compositions is derived from the dominance order.

Definition 9. For α, β ∈ ℕ^{N} the partial order α ≻ β (α dominates β) means that α ≠ β and ∑_{i=1}^{j} α_{i} ≥ ∑_{i=1}^{j} β_{i} for 1 ≤ j ≤ N; and α ▷ β means that |α| = |β| and either α^{+} ≻ β^{+} or α^{+} = β^{+} and α ≻ β.

For example (5, 1, 4) ▷ (1, 5, 4) ▷ (4, 3, 3), while (1, 5, 4) and (6, 2, 2) are not comparable in ▷. There are some results useful in analyzing U_{i}x^{α}u. Let ε (i) be the ith standard basis vector in ℕ^{N}, for 1 ≤ i ≤ N. By [3, Lemma 8.2.3] the following hold for α ∈ ℕ^{N}:

(1) if α_{i} > α_{j} and i < j then (i, j) α ◁ α;

(2) α^{+} ⊵ α;

(3) if 1 ≤ s < α_{i} − α_{j} then α^{+} ▷ (α − s (ε (i) − ε (j)))^{+}.

The following is a consequence of these relations and an easy computation.

Lemma 2. For α ∈ ℕ^{N} and i ≠ j let B_{ij}x^{α} := \frac{x_{i}x^{α} − x_{j} (i, j) x^{α}}{x_{i} − x_{j}},

then (1) if α_{i} = α_{j} then B_{ij}x^{α} = x^{α};

(2) if α_{i} > α_{j} then

B_{ij}x^{α} = x^{α} + (i, j) x^{α} + ∑_{s=1}^{α_{i}−α_{j}−1} x^{α−s(ε(i)−ε(j))}

and α^{+} ▷ (α − s (ε (i) − ε (j)))^{+} for 1 ≤ s ≤ α_{i} − α_{j} − 1;

(3) if α_{i} < α_{j} then

B_{ij}x^{α} = − ∑_{s=1}^{α_{j}−α_{i}−1} x^{α−s(ε(j)−ε(i))}

and α^{+} ▷ (α − s (ε (j) − ε (i)))^{+} for 1 ≤ s ≤ α_{j} − α_{i} − 1.

The following proposition can be elegantly stated in terms of conjugates of Jucys-Murphy elements. Recall the conjugation relation w (i, j) w^{−1} = (w (i) , w (j)).

Definition 10. For α ∈ ℕ^{N} and 1 ≤ i ≤ N let ω_{i}^{α} := w_{α}ω_{r(α,i)}w_{α}^{−1}, where w_{α} is the inverse of r (α, ·). Equivalently ω_{i}^{α} = ∑_{r(α,j)>r(α,i)} (i, j).

To justify the second equation observe that

w_{α}ω_{r(α,i)}w_{α}^{−1} = ∑_{r(α,i)<j} (w_{α} (r (α, i)) , w_{α} (j)) = ∑_{r(α,i)<j} (i, w_{α} (j))

and r (α, w_{α} (j)) = j.

Proposition 5. Suppose α ∈ ℕ^{N}, u ∈ V_{τ} and 1 ≤ i ≤ N then

U_{i}x^{α}u = x^{α} [(α_{i} + 1) u + κω_{i}^{α}u] + κ ∑_{β◁α}x^{β}u_{β},

where each u_{β} = 0 or ± (i, j) u for some j.

Proof. Let q_{α} denote elements of span {x^{β} : β ◁ α}. In the case 1 ≤ j < i the coefficient of κ (i, j) u is B_{ij}x^{α} − (i, j) x^{α} which equals (1) 0 if α_{i} = α_{j}, (2) x^{α} + q_{α} if α_{i} > α_{j}, (3) − ((i, j) x^{α} + q_{α}) if α_{j} > α_{i}, so that (i, j) α ◁ α. In the case i < j ≤ N the coefficient of κ (i, j) u is B_{ij}x^{α} which equals (1) x^{α} if α_{i} = α_{j}, (2) x^{α} + (i, j) x^{α} + q_{α} if α_{i} > α_{j}, so that (i, j) α ◁ α, (3) q_{α} if α_{i} < α_{j}. Thus κx^{α} (i, j) u appears in U_{i}x^{α}u exactly when α_{i} > α_{j} or α_{i} = α_{j} and j > i, that is, r (α, j) > r (α, i). □

Following Griffeth we define an order on the pairs {(α, u) : α ∈ ℕ^{N}}: (α, u_{1}) ▷ (β, u_{2}) means that α ▷ β. For this order the leading term of U_{i}x^{α}u is x^{α} (α_{i} + 1 + κω_{i}^{α}) u.

4. Nonsymmetric Jack polynomials

This section presents the structure of the simultaneous eigenvectors of {U_{i} : 1 ≤ i ≤ N} in M (τ). These are vector-valued generalizations of the nonsymmetric Jack polynomials (see [3, Ch. 8]). The operators U_{i} are self-adjoint with respect to the contravariant form, which is described as follows:

The contravariant form ⟨·, ·⟩ on M (τ) is the canonical symmetric S_{N}-invariant bilinear form, extending the form ⟨·, ·⟩_{0} on V_{τ}, : such that

⟨x_{i}f, g⟩ = ⟨f, D_{i}g⟩ , i ∈ [1, N] , f, g ∈ M (τ) .

An existence proof can be based on the operator

∑_{i=1}^{N} x_{i}D_{i} + κ ∑_{1≤i<j≤N} (i, j)

and induction. The important properties of the form are:

(1) if f ∈ P_{m} ⊗ V_{τ}, g ∈ P_{n} ⊗ V_{τ} and m ≠ n then ⟨f, g⟩ = 0;

(2) if w ∈ S_{N} then ⟨wf, wg⟩ = ⟨f, g⟩ for all f, g ∈ M (τ), if 1 ≤ i < j ≤ N then ⟨(i, j) f, g⟩ = ⟨f, (i, j) g⟩;

(3) if i ∈ [1, N] and f, g ∈ M (τ) then ⟨D_{i}x_{i}f, g⟩ = ⟨f, D_{i}x_{i}g⟩.

We use ∥f∥^{2} to denote ⟨f, f⟩ although the form may not be positivedefinite . For a specific value κ ∈ ℚ the kernel of the form, that is, {f : ⟨g, f⟩ = 0, ∀g ∈ M (τ)}, is called the radical of M (τ) and denoted J_{κ} (τ), and the quotient module M (τ) /J_{κ} (τ) is denoted L_{κ} (τ). Values of κ such that J_{κ} (τ) ≠ (0) are called singular values.

If λ ∈ ℕ^{N,+} then the leading term in U_{i}x^{λ}u is x^{λ} (λ_{i} + 1 + κω_{i}) u; this suggests that eigenvectors of ω_{i} have good properties under the action of U_{i}. For compositions the coordinates have to be appropriately permuted. From (5) we see that for T ∈ Y (τ) and α ∈ ℕ^{N} the leading term in U_{i}x^{α}w_{α}v_{T} is (α_{i} + 1 + κc (r (α, i) , T)) x^{α}w_{α}v_{T}, because ω_{i}^{α}w_{α}v_{T} = w_{α}ω_{r(α,i)}v_{T} = c (r (α, i) , T) w_{α}v_{T}. For any n ∈ ℕ the set {x^{α}w_{α}v_{T} : α ∈ ℕ^{N}, |α| = n, T ∈ Y (τ)} is a basis of M_{n} (τ) on which the operators U_{i} act in a triangular manner (with respect to ▷). For α ∈ ℕ^{N}, T ∈ Y (τ), let

(4.1)  ξ_{i} (α, T) = α_{i} + 1 + κc (r (α, i) , T) , i ∈ [1, N] .

For any β ≠ α and |β| = |α| there is at least one i such that α_{i} ≠ 0 and α_{i} ≠ β_{i} thus ξ_{i} (α, T) ≠ ξ_{i} (β, T′) for any T, T′ ∈ Y (τ) (and generic κ). (The restriction to α_{i} ≠ 0 is needed in the next section; if |α| = |β| and α_{i} ≠ 0 implies α_{i} = β_{i} then α = β.) Thus there exists a basis of simultaneous eigenvectors of {U_{i} : i ∈ [1, N]}. The following is the specialization to S_{N} of Griffeth’s construction [6, Thm. 5.2] of nonsymmetric Jack polynomials.

Proposition 6. For α ∈ ℕ^{N}, T ∈ Y (τ) there exists a unique element ζ_{α,T} of M (τ) such that U_{i}ζ_{α,T} = ξ_{i} (α, T) ζ_{α,T} for 1 ≤ i ≤ N and

ζ_{α,T} (x) = x^{α}w_{α}v_{T} + ∑_{β◁α}x^{β}u_{βα},

where u_{βα} ∈ V_{τ}.

Existence of this set of simultaneous eigenvectors of {U_{i} : i ∈ [1, N]} follows from the triangular property, the commutativity, and the separation properties of the eigenvalues (α, T) ↦ [ξ_{i} (α, T)]_{i=1}^{N}.

Because each U_{i} is self-adjoint for ⟨·, ·⟩ we have ⟨ζ_{α,T} , ζ_{β,T′}⟩ = 0 when α ≠ β or T ≠ T′.

We consider the action of S_{N} on the polynomials ζ_{α,T}. As usual there are explicit formulae for the action of s_{i} = (i, i + 1) based on the commutations U_{j}s_{i} = s_{i}U_{j} for j ≠ i, i + 1 and s_{i}U_{i}s_{i} = U_{i+1} + κ. These are special cases of [6, Thm. 5.3], however we use the nonnormalized basis for V_{τ} rather than the orthonormal one used there (so coefficients in ℚ (κ) suffice). As in [6] let σ_{i} denote the formal operator s_{i} + \frac{κ}{U_{i+1}−U_{i}}; suppose f ∈ M (τ) and U_{j}f = λ_{j}f for 1 ≤ j ≤ N (with λ_{j} ∈ ℚ (κ) and λ_{i} ≠ λ_{i+1}) then U_{j}σ_{i}f = λ_{j}σ_{i}f for j ≠ i, i + 1 and U_{i}σ_{i}f = λ_{i+1}σ_{i}f, U_{i+1}σ_{i}f = λ_{i}σ_{i}f (where σ_{i}f = s_{i}f + \frac{κ}{λ_{i+1}−λ_{i}} f. Specifically there are two main cases α_{i} ≠ α_{i+1} and α_{i} = α_{i+1}. For α ∈ ℕ^{N} and T ∈ Y (τ) let

b_{i} (α, T) = \frac{κ}{ξ_{i} (α, T) − ξ_{i+1} (α, T)}

= \frac{κ}{α_{i} − α_{i+1} + κ (c (r (α, i) , T) − c (r (α, i + 1) , T))}.

The proof of the following is in the Appendix.

Proposition 7. Suppose α ∈ ℕ^{N} and α_{i} > α_{i+1} for some i < N. Then

s_{i}ζ_{α,T} = b_{i} (α, T) ζ_{α,T} + (1 − b_{i} (α, T)^{2}) ζ_{s_{i}α,T} ,
s_{i}ζ_{s_{i}α,T} = ζ_{α,T} − b_{i} (α, T) ζ_{s_{i}α,T};
∥ζ_{α,T} ∥^{2} = (1 − b_{i} (α, T)^{2}) ∥ζ_{s_{i}α,T} ∥^{2} .

Remark 1. A necessary condition for the form ⟨·, ·⟩ to be positivedefinite now becomes apparent: b_{i} (α, T)^{2} < 1 for all i, α, T. The

14 CHARLES F. DUNKL

“trivial” cases are τ = (N) and τ = (1, …, 1) for which κ > −\frac{1}{N} and κ < \frac{1}{N} are necessary and sufficient, respectively. Otherwise let h_{τ} := τ_{1} + ℓ (τ) − 1, the maximum hook-length of τ, then −\frac{1}{h_{τ}} < κ < \frac{1}{h_{τ}} implies b_{i} (α, T)^{2} < 1 for all i, α, T. Note that 1 ≤ i, j ≤ N, T ∈ Y (τ) implies |c (i, T) − c (j, T)| ≤ h_{τ} − 1.

Etingof, Stoica and Griffeth [4, Thm. 5.5] found the complete description of the set of values of κ for which L_{κ} (τ) provides a unitary representation of the rational Cherednik algebra. We can find an expression for ∥ζ_{α,T} ∥^{2} in terms of ∥ζ_{α^{+},T} ∥^{2}, following the approach used in [3, Thm. 8.5.8]. Definition 11. For α ∈ ℕ^{N}, T ∈ Y (τ) and ε = ± let

E_{ε} (α, T) = ∏_{\begin{matrix}1≤i<j≤N \\ α_{i}<α_{j} \end{matrix}} (1 + \frac{εκ}{α_{j} − α_{i} + κ (c (r (α, j) , T) − c (r (α, i) , T))}) ,

and let E_{2} (α, T) = E_{+} (α, T) E_{−} (α, T) . Definition 12. For α ∈ ℕ^{N} let

inv (α) := # {(i, j) : 1 ≤ i < j ≤ N, α_{i} < α_{j}} ,

the number of inversions in α. Proposition 8. Suppose α ∈ ℕ^{N}, T ∈ Y (τ) , ε = ± and α_{i+1} > α_{i} for some i ∈ [1, N − 1] then E_{ε} (s_{i}α, T) /E_{ε} (α, T) = 1 + εb_{i} (α, T). Proof. Using an argument similar to that of Lemma 1 we have

\frac{E_{ε} (s_{i}α, T)}{E_{ε} (α, T)}

= 1 + \frac{εκ}{(s_{i}α)_{i+1} − (s_{i}α)_{i} + κ (c (r (s_{i}α, i + 1) , T) − c (r (s_{i}α, i) , T))}

= 1 + εb_{i} (α, T) , because r (s_{i}α, i + 1) = r (α, i) and r (s_{i}α, i) = r (α, i + 1) . □ Corollary 3. Suppose α ∈ ℕ^{N}, T ∈ Y (τ) then

∥ζ_{α,T} ∥^{2} = E_{2} (α, T)^{−1} ∥ζ_{α^{+},T} ∥^{2} .

Proof. Argue by induction on inv (α). If the formula is valid for some α with α_{i} > α_{i+1} then by Proposition 7

∥ζ_{s_{i}α,T} ∥^{2} = (1 − b_{i} (α, T)^{2})^{−1} ∥ζ_{α,T} ∥^{2}
= (1 − b_{i} (α, T)^{2})^{−1} E_{2} (α, T)^{−1} ∥ζ_{α^{+},T} ∥^{2}
= E_{2} (s_{i}α, T)^{−1} ∥ζ_{α^{+},T} ∥^{2} .

This completes the induction. □

Consider the case α_{i} = α_{i+1} and let I = r (α, i) so that r (α, i + 1) = I + 1 and b_{i} (α, T) = (c (I, T) − c (I + 1, T))^{−1} = b_{I} (T) (see Proposition 1). Furthermore

s_{i}w_{α} = w_{α} (w_{α}^{−1} (i) , w_{α}^{−1} (i + 1)) = w_{α} (I, I + 1) = w_{α}s_{I}.

The transformation properties depend on the positions of I and I + 1 in T.

Proposition 9. Suppose α ∈ ℕ^{N}, T ∈ Y (τ) and α_{i} = α_{i+1} for some i < N. For I = r (α, T) the following hold:

(1) if b_{I} (T) = 1 then s_{i}ζ_{α,T} = ζ_{α,T},

(2) if b_{I} (T) = −1 then s_{i}ζ_{α,T} = −ζ_{α,T},

(3) if −\frac{1}{2} ≤ b_{I} (T) < 0 then

s_{i}ζ_{α,T} = b_{I} (T) ζ_{α,T} + (1 − b_{I} (T)^{2}) ζ_{α,s_{I}T} ,

(4) if 0 < b_{I} (T) ≤ \frac{1}{2} then s_{i}ζ_{α,T} = b_{I} (T) ζ_{α,T} + ζ_{α,s_{I}T}.

Proof. It suffices to consider the action of s_{i} on the leading term of ζ_{α,T}. Indeed s_{i}x^{α}w_{α}v_{T} = x^{α}w_{α} (s_{I}v_{T}) and we use the equations from Proposition 1. □

Note that in case (3) ∥ζ_{α,T} ∥^{2} = (1 − b_{I} (T)^{2}) ∥ζ_{α,s_{I}T} ∥^{2} (and the reciprocal in case (4)). The closed formula for the norm is proven by means of induction and a raising operator involving a cyclic shift and multiplication by x_{N}. The details are in the Appendix.

The following result is due to Griffeth [6, Thm. 6.1] for the general setting of G (r, 1, N). There is a slight change due to our use of nonnormalized vectors v_{T} and reversed standard tableaux.

Theorem 2. Suppose λ ∈ ℕ^{N,+} and T ∈ Y (τ) then

∥ζ_{λ,T} ∥^{2} = ∥v_{T} ∥_{0}^{2} ∏_{i=1}^{N} (1 + κc (i, T))_{λ_{i}}

× ∏_{1≤i<j≤N} ∏_{l=1}^{λ_{i}−λ_{j}} (1 − \frac{κ^{2}}{(l + κ (c (i, T) − c (j, T)))^{2}} ) .

5. Symmetric and Antisymmetric Polynomials

We consider symmetric and antisymmetric linear combinations of {ζ_{α,T} }. Recall

b_{i} (α, T) = \frac{κ}{α_{i} − α_{i+1} + κ (c (r (α, i) , T) − c (r (α, i + 1) , T))},
b_{i} (T) = \frac{1}{c (i, T) − c (i + 1, T)},

for α ∈ ℕ^{N}, T ∈ Y (τ) , i ∈ [1, N − 1]. Here is a description of s_{i}- invariant polynomials for a given i:

(1) ζ_{α,T} + (1 − b_{i} (α, T)) ζ_{s_{i}α,T}, for α_{i} > α_{i+1};

(2) (b_{I} (T) + 1) ζ_{α,T} + ζ_{α,s_{I}T}, for α_{i} = α_{i+1}, I = r (α, i) and 0 < b_{I} (T) ≤ \frac{1}{2};

(3) ζ_{α,T}, for α_{i} = α_{i+1}, I = r (α, i) and b_{I} (T) = 1 (rw (I, T) = rw (I + 1, T)).

The antisymmetric polynomials for s_{i} (s_{i}f = −f) are

(1) ζ_{α,T} − (1 + b_{i} (α, T)) ζ_{s_{i}α,T}, for α_{i} > α_{i+1};

(2) (b_{I} (T) − 1) ζ_{α,T} + ζ_{α,s_{I}T}, for α_{i} = α_{i+1}, I = r (α, i) and 0 < b_{I} (T) ≤ \frac{1}{2};

(3) ζ_{α,T}, for α_{i} = α_{i+1}, I = r (α, i) and b_{I} (T) = −1 (cm (I, T) = cm (I + 1, T)).

Now we construct invariants. In any orbit span {wζ_{α,T} : w ∈ S_{N}} there must be a polynomial with leading term x^{α^{+}} so it suffices to consider the situation ζ_{λ,T} for partitions λ. We collect concepts for use in the sequel.

Notation 1. For λ ∈ ℕ^{N,+} let W_{λ} = {w ∈ S_{N} : wλ = λ}, the stabilizer subgroup of λ. Thus

W_{λ} = S_{[a_{1},b_{1}]} × S_{[a_{2},b_{2}]} × … S_{[a_{n},b_{n}]},

where 1 ≤ a_{1} < b_{1} < a_{2} < b_{2} < … < a_{n} < b_{n} ≤ N (this means λ_{a_{1}} = λ_{b_{1}} > λ_{b_{1}+1} and so forth). These intervals depend on λ but we will not incorporate this into the notation. Let λ^{R} = (λ_{N}, λ_{N−1}, …, λ_{1}) ∈ ℕ^{N}, the reverse of λ. The permutation w_{λ^{R}} is defined by (w_{λ^{R}})^{−1} (i) = r (λ^{R}, i) , i ∈ [1, N] (Definition 10).

Generally w_{λ^{R}} ≠ w_{0} where w_{0} is the longest permutation given by w_{0} (i) = N + 1 − i (example: λ = (3, 2, 2, 1) then [w_{λ^{R}} (i)]_{i=1}^{4} = [4, 2, 3, 1]). The composition λ^{R} is the unique minimum for the order “≻” on {α : α^{+} = λ}. For α^{+} = λ and T ∈ Y (τ) the leading term of ζ_{α,T} is x^{α}w_{α}v_{T} (where w_{α}^{−1} (i) = r (α, i)) and the minimality of λ^{R} implies that the expansion of ζ_{λ^{R},T} has no term of the form x^{α}u with u ∈ V_{τ} when α ≠ λ^{R}, α^{+} = λ. From the expressions in (2) above we see that the subgroup W_{λ} is an important part of the analysis. The formulae developed in Section 2 will be used. The method is an analog of the one for scalar nonsymmetric Jack polynomials, introduced by Baker and Forrester [1].

Definition 13. For λ ∈ ℕ^{N,+} and T ∈ Y (τ) define the tableau ⌊λ, T ⌋ to be the assignment of λ_{1}, λ_{2}, …, λ_{N} to the nodes of the Ferrers diagram of τ so that the entry at T (i) is λ_{i}, i ∈ [1, N]. Thus the entries of ⌊λ, T ⌋ are weakly increasing (≤) in each row and in each column. The set of T′ satisfying ⌊λ, T′⌋ = ⌊λ, T ⌋ is exactly Y (T; W_{λ}).

5.1. Case: #Y (T; W_{λ}) = 1. We begin with the situation of symmetrizing ζ_{λ,T} when #Y (T; W_{λ}) = 1, that is, v_{T} is W_{λ}-invariant so that each interval [a_{i}, b_{i}] is contained in a row of T,( 1 ≤ i ≤ n (rw (j, T) = rw (b_{i}, T) for 1≤ i ≤ n and a_{i} ≤ j ≤ b_{i}). Then ∑_{w∈S_{N}} wζ_{λ,T} = ∑_{α^{+}=λ} A_{α}ζ_{α,T} with coefficients to be determined.

Theorem 3. Suppose λ ∈ ℕ^{N,+} and T ∈ Y (τ) such that w ∈ W_{λ} implies wv_{T} = v_{T} then the polynomial f_{λ,T} defined by

f_{λ,T}^{s} = ∑_{α^{+}=λ} E_{−} (α, T) ζ_{α,T} ,

is S_{N}-invariant and

∥f_{λ,T}^{s} ∥^{2} = \frac{N!}{#W_{λ}} \frac{1}{E_{+} (λ^{R}, T)} ∥ζ_{λ,T} ∥^{2} .

Proof. Fix i ∈ [1, N − 1] and let

A = {α : α^{+} = λ, α_{i} = α_{i+1}} ,
B = {α : α^{+} = λ, α_{i} > α_{i+1}} .

Write

f_{λ,T}^{s} = ∑_{α∈A} E_{−} (α, T) ζ_{α,T} + ∑_{α∈B} (E_{−} (α, T) ζ_{α,T} + E_{−} (s_{i}α, T) ζ_{s_{i}α,T}) .

Suppose α ∈ A then r (i + 1, α) = r (i, α) + 1 thus the values r (i, α) and r (i + 1, α) belong to some interval [a_{j}, b_{j}] (where S_{[a_{j},b_{j}]} is a factor of W_{λ}) and are adjacent entries in some row of T, hence s_{i}ζ_{α,T} = ζ_{α,T}. Next let α ∈ B then the corresponding term in the sum is E_{−} (α, T) (ζ_{α,T} + \frac{E_{−}(s_{i}α,T)}{E_{−}(α,T)}ζ_{s_{i}α,T} ). Using the techniques of Lemma 1 we find that

\frac{E_{−} (s_{i}α, T)}{E_{−} (α, T)}

= 1 − \frac{κ}{(s_{i}α)_{i+1} − (s_{i}α)_{i} + κ (c (r (s_{i}α, i + 1) , T) − c (r (s_{i}α, i) , T))}

= 1 − \frac{κ}{α_{i} − α_{i+1} + κ (c (r (α, i) , T) − c (r (α, i + 1) , T))}

= 1 − b_{i} (α, T) ,

and thus the term for α in the sum over B is s_{i}-invariant. Consider g = ∑_{w∈S_{N}} wζ_{λ^{R},T}; since g is S_{N}-invariant it must equal a constant multiple γ of f_{λ,T}^{s}. To find γ consider the coefficients of x^{λ}v_{T} in f_{λ,T}^{s} and g. The leading term of ζ_{λ^{R},T} is x^{λ^{R}}w_{λ^{R}} (v_{T}) . The coefficient in f_{λ,T}^{s} is 1 (by definition of ζ_{λ,T}). The term x^{λ}v_{T} appears in wζ_{λ^{R},T} with coefficient 1 exactly when w = w_{1}w_{λ^{R}}^{−1} for w_{1} ∈ W_{λ}. Thus g = (#W_{λ}) f_{λ,T}^{s} and

∥f_{λ,T}^{s} ∥^{2} = \frac{1}{#W_{λ}} ⟨g, f_{λ,T}^{s} ⟩ = \frac{1}{#W_{λ}} ∑_{w∈S_{N}} ⟨wζ_{λ^{R},T} , f_{λ,T}^{s} ⟩

= \frac{N!}{#W_{λ}} ⟨ζ_{λ^{R},T} , f_{λ,T}^{s} ⟩ = \frac{N!}{#W_{λ}} E_{−} (λ^{R}, T ) ∥ζ_{λ^{R},T} ∥^{2}

= \frac{N!E_{−} (λ^{R}, T )}{(#W_{λ}) E_{2} (λ^{R}, T)} ∥ζ_{λ,T} ∥^{2} .

This completes the proof. □

Continuing with the case #Y (T; W_{λ}) = 1 we turn to the corresponding antisymmetric function involving ζ_{λ,T} such that v_{T} is antisymmetric for W_{λ}. That is each interval [a_{i}, b_{i}] (appearing in W_{λ}) is contained in a column of T, a_{i} ≤ j ≤ b_{i} implies cm (j, T) = cm (b_{i}, T)). The number of inversions inv (α) takes the place of the sign of a permutation in order to allow λ to have some repeated values.

Theorem 4. Suppose λ ∈ ℕ^{N,+} and T ∈ Y (τ) such that s_{i} ∈ W_{λ} implies s_{i}v_{T} = −v_{T} then the polynomial f_{λ,T}^{a} defined by

f_{λ,T}^{a} = ∑_{α^{+}=λ} (−1)^{inv(α)} E_{+} (α, T) ζ_{α,T} ,

is S_{N}-alternating, and

∥f_{λ,T}^{a} ∥^{2} = \frac{N!}{#W_{λ}} \frac{1}{E_{−} (λ^{R}, T)} ∥ζ_{λ,T} ∥^{2} .

Proof. Fix i ∈ [1, N − 1] and let

A = {α : α^{+} = λ, α_{i} = α_{i+1}} ,
B = {α : α^{+} = λ, α_{i} > α_{i+1}} .

Note α ∈ B implies inv (s_{i}α) = inv (α) + 1. Write

f_{λ,T}^{a} = ∑_{α∈A} (−1)^{inv(α)} E_{+} (α, T) ζ_{α,T}

+ ∑_{α∈B} (−1)^{inv(α)} (E_{+} (α, T) ζ_{α,T} − E_{+} (s_{i}α, T) ζ_{s_{i}α,T}) .

Suppose α ∈ A then r (i + 1, α) = r (i, α) + 1 thus the values r (i, α) and r (i + 1, α) belong to some interval [a_{j}, b_{j}] (where S_{[a_{j},b_{j}]} is a factor of W_{λ}) and are adjacent entries in some column of T, hence s_{i}ζ_{α,T} = −ζ_{α,T}. Next let α ∈ B then the corresponding term in the sum is (−1)^{inv(α)} E_{+} (α, T) (ζ_{α,T} − \frac{E_{+}(s_{i}α,T)}{E_{+}(α,T)}ζ_{s_{i}α,T} ), a scalar multiple of ζ_{α,T} − (1 + b_{i} (α, T)) ζ_{s_{i}α,T}, by an argument similar to the previous theorem. This term satisfies s_{i}f = −f. Thus s_{i}f_{λ,T}^{a} = −f_{λ,T}^{a}. Consider g = ∑_{w∈S_{N}} sgn (w) wζ_{λ^{R},T}; since g is S_{N}-alternating it must equal a constant multiple γ of f_{λ,T}^{a} . To find γ consider the coefficients of x^{λ}v_{T} in f_{λ,T}^{a} and g. The coefficient in f_{λ,T}^{a} is 1 (by definition of ζ_{λ,T}). The term x^{λ}v_{T} appears in wζ_{λ^{R},T} exactly when w = w_{1}w_{λ^{R}}^{−1} for w_{1} ∈ W_{λ}. Let

ε = sgn (w_{λ^{R}}) = (−1)^{inv(λ^{R})}, because the length of w_{λ^{R}} is inv (λ^{R})). Furthermore

sgn (w_{1}w_{λ^{R}}^{−1}) w_{1}w_{λ^{R}}^{−1}ζ_{λ^{R},T} = sgn (w_{1}w_{λ^{R}}^{−1}) w_{1}w_{λ^{R}}^{−1} (x^{λ^{R}}w_{λ^{R}}v_{T} ) + h_{1}
= ε sgn (w_{1}) w_{1} (x^{λ}v_{T} ) + h_{2}
= εx^{λ}v_{T} + h_{2},

where h_{1} and h_{2} are terms of lower order, that is, of the form ∑_{β◁λ} x^{β}u_{β} with u_{β} ∈ V_{τ}. Thus g = ε (#W_{λ}) f_{λ,T} and

∥f_{λ,T}^{a} ∥^{2} = \frac{ε}{#W_{λ}} ⟨g, f_{λ,T}^{a} ⟩ = \frac{ε}{#W_{λ}} ∑_{w∈S_{N}} sgn (w) ⟨wζ_{λ^{R},T} , f_{λ,T}^{a} ⟩

= \frac{ε}{#W_{λ}} ∑_{w∈S_{N}} sgn (w) ⟨ζ_{λ^{R},T} , w^{−1}f_{λ,T}^{a} ⟩ = \frac{εN!}{#W_{λ}} ⟨ζ_{λ^{R},T} , f_{λ,T}^{a} ⟩

= \frac{εN!}{#W_{λ}} (−1)^{inv(λ^{R})} E_{+} (λ^{R}, T ) ∥ζ_{λ^{R},T} ∥^{2}

= \frac{N!E_{+} (λ^{R}, T )}{(#W_{λ}) E_{2} (λ^{R}, T)} ∥ζ_{λ,T} ∥^{2} .

This completes the proof. □

5.2. Case: #Y (T; W_{λ}) > 1. Let T_{0} ∈ Y (τ) such that T_{0} satisfies condition [a_{i}, b_{i}]_{cm} for each factor S_{[a_{i},b_{i}]} of W_{λ} and a_{i} ≤ j_{1} < j_{2} ≤ b_{i} implies cm (j_{1}, T_{0}) > cm (j_{2}, T_{0}). This condition is equivalent to the tableau ⌊λ, T_{0}⌋ being column-strict (the entries strictly increase in each column, see [7, p. 5], such tableaux are also called semistandard Young tableaux) and T_{0} has a certain extremal property among all T ∈ Y (T_{0}; W_{λ}). Let

f_{λ,T_{0}}^{s} = ∑_{α^{+}=λ} ∑_{T ∈Y (T_{0};W_{λ})} ∏_{j=1}^{n} P_{0} (T; a_{j}, b_{j}) E_{−} (α, T) ζ_{α,T} ,

u_{λ,T_{0}} = ∑_{T ∈Y (T_{0};W_{λ})} ∏_{j=1}^{n} P_{0} (T; a_{j}, b_{j}) v_{T} ∈ V_{τ}.

The term involving x^{λ} is h_{0} = ∑_{T ∈Y (T_{0};W_{λ})} ∏_{j=1}^{n} P_{0} (T; a_{j}, b_{j}) ζ_{λ,T}, thus

the leading term in f_{λ,T_{0}}^{s} is x^{λ}u_{λ,T_{0}}. From the transformation rules in Proposition 9 it follows that ∥h_{0}∥^{2} = ∥ζ_{λ,T_{0}}∥^{2} ∥u_{λ,T_{0}}∥_{0}^{2} / ∥v_{T_{0}}∥_{0}^{2} (see Corollary 2). Also h_{0} is W_{λ}-invariant. In the symbol f_{λ,T_{0}}^{a} one could replace T_{0} by any T ∈ Y (T_{0}; W_{λ}); then ∏_{j=1}^{n} P_{0} (T; a_{j}, b_{j}) = 1 and T ∈ Y (T_{0}; W_{λ}) implies T = T_{0}.

Theorem 5. wf_{λ,T_{0}}^{s} = f_{λ,T_{0}}^{s} for all w ∈ S_{N} and

∥f_{λ,T_{0}}^{s}∥^{2} = \frac{N!}{#W_{λ}} \frac{∥u_{λ,T_{0}}∥_{0}^{2}}{E_{+} (λ^{R}, T_{0}) ∥v_{T_{0}}∥_{0}^{2}} ∥ζ_{λ,T_{0}}∥^{2} .

Proof. Let F (α, T) = ∏_{j=1}^{n} P_{0} (T; a_{j}, b_{j}) E_{−} (α, T). Fix i ∈ [1, N − 1] and collect the terms of f_{λ,T_{0}}^{s} into three parts. Let

L = {(α, T) : α^{+} = λ, T ∈ Y (T_{0}; W_{λ})}
A = {(α, T) ∈ L : α_{i} = α_{i+1}, rw (r (α, i) , T) = rw (r (α, i) + 1, T)} ,
B = {(α, T) ∈ L : α_{i} > α_{i+1}} ,
C = {(α, T) ∈ L : α_{i} = α_{i+1}, rw (r (α, i) , T) < rw (r (α, i) + 1, T)} .

The first part is ∑_{(α,T)∈A} F (α, T) ζ_{α,T} and in this case s_{i}ζ_{α,T} = ζ_{α,T}. The second part is

∑_{(α,T)∈B} (F (α, T) ζ_{α,T} + F (s_{i}α, T) ζ_{s_{i}α,T})

= ∑_{(α,T)∈B} F (α, T) (ζ_{α,T} + \frac{F (s_{i}α, T)}{F (α, T)}ζ_{s_{i}α,T} ) .

Just as in Proposition 1 \frac{F(s_{i}α,T)}{F(α,T)} = 1 − b_{i} (α, T), and hence this sum is s_{i}-invariant. For use in C let I (α) = rw (α, i). Then the third part is

∑_{(α,T)∈C} (F (α, T) ζ_{α,T} + F (α, s_{I(α)}T ) ζ_{α,s_{I(α)}T} )

= ∑_{(α,T)∈C} F (α, s_{I(α)}T ) (\frac{F (α, T)}{F (α, s_{I(α)}T )}ζ_{α,T} + ζ_{α,s_{I(α)}T} ) .

To show that each term is s_{i}-invariant we must show

\frac{F (α, T)}{F (α, s_{I(α)}T )} = b_{I(α)} + 1.

Fix such a term. The equality α_{i} = α_{i+1} implies [I (α) , I (α) + 1] ⊂ [a_{i}, b_{i}] for some i. Thus

\frac{∏_{j=1}^{n} P_{0} (T; a_{j}, b_{j})}{∏_{j=1}^{n} P_{0} (s_{I(α)}T; a_{j}, b_{j})} = \frac{P_{0} (T; a_{i}, b_{i})}{P_{0} (s_{I(α)}T; a_{i}, b_{i})} = 1 + b_{I(α)} (T) .

Finally consider E_{−} (α, T) /E_{−} (α, s_{I(α)}T ); let

g_{lj} (T) = 1 − \frac{κ}{α_{j} − α_{l} + κ (c (r (α, j) , T) − c (r (α, l) , T))}

if l < j and α_{l} < α_{j}, and g_{lj} (T) = 1 otherwise. Then r (α, j) ∉ {I (α) , I (α) + 1} implies that

c (r (α, j) , T) = c (r (α, j) , s_{I(α)}T ) ,

also

c (r (α, i) , T) = c (r (α, i + 1) , s_{I(α)}T )

and

c (r (α, i + 1) , T) = c (r (α, i) , s_{I(α)}T ) .

Thus g_{l,i} (T) = g_{l,i+1} (s_{I(α)}T ) and g_{l,i} (s_{I(α)}T ) = g_{l,i+1} (T) for 1 ≤ l < i with similar relations for g_{ij} and g_{i+1,j} when i + 1 < j ≤ N. Also

g_{i,i+1} (T) = 1 = g_{i,i+1} (s_{I(α)}T )

thus

E_{−} (α, T) = ∏_{1≤l<j≤N} g_{lj} (T) = E_{−} (α, s_{I(α)}T ) .

Hence s_{i}f_{λ,T_{0}}^{s} = f_{λ,T_{0}}^{s}.

To compute ∥f_{λ,T_{0}}^{s}∥^{2} consider ∑_{w∈S_{N}} wh_{R} where

h_{R} := ∑_{T ∈Y (T_{0};W_{λ})} ∏_{j=1}^{n} P_{0} (T; a_{j}, b_{j}) ζ_{λ^{R},T} .

By the argument used above for type (4), we have

E_{−} (λ^{R}, T ) = E_{−} (λ^{R}, T_{0})

for all T ∈ Y (T_{0}; W_{λ}). Thus the term for α = λ^{R} in f_{λ,T_{0}}^{s} is E_{−} (λ^{R}, T_{0}) h_{R} and the leading term in h_{R} is x^{λ^{R}}w_{λ^{R}}u_{λ,T_{0}}. Similarly to the proof of Theorem 3 we conclude ∑_{w∈S_{N}} wh_{R} = (#W_{λ}) f_{λ,T_{0}}^{s} and

(#W_{λ}) ∥f_{λ,T_{0}}^{s}∥^{2} = N! ⟨h_{R}, f_{λ,T_{0}}^{s}⟩
= N!E_{−} (λ^{R}, T_{0}) ∥g∥^{2} ;

thus

∥g∥^{2} = \begin{matrix}∥ζ_{λ^{R},T_{0}} ∥ & 2 & ∥u_{λ,T_{0}}∥ & {2 \atop 0} \\ ∥v_{T_{0}}∥ & {2 \atop 0} \end{matrix} = \frac{∥ζ_{λ,T_{0}}∥^{2} ∥u_{λ,T_{0}}∥_{0}^{2}}{E_{2} (λ^{R}, T_{0}) ∥v_{T_{0}}∥_{0}^{2}} .

□

Corollary 4. Suppose λ, μ ∈ ℕ^{N,+} and T_{1}, T_{2} ∈ Y (τ) such that ⌊λ, T_{1}⌋ and ⌊μ, T_{2}⌋ are column-strict. If λ ≠ μ or T_{2} ∉ Y (T_{1}; W_{λ}) then ⟨f_{λ,T_{1}}^{s}, f_{μ,T_{2}}^{s}⟩ = 0.

Let T_{0} ∈ Y (τ) such that T_{0} satisfies condition [a_{i}, b_{i}]_{rw} for each factor S_{[a_{i},b_{i}]} of W_{λ} and a_{i} ≤ j_{1} < j_{2} ≤ b_{i} implies cm (j_{1}, T_{0}) ≤ cm (j_{2}, T_{0}). This condition is equivalent to the tableau ⌊λ, T ⌋ being row-strict (the entries strictly increase in each row), and T_{0} having a certain extremal property. Let

f_{λ,T_{0}}^{a} = ∑_{α^{+}=λ} (−1)^{inv(α)} ∑_{T ∈Y (T_{0};W_{λ})} ∏_{j=1}^{n} P_{1} (T; a_{j}, b_{j}) E_{+} (α, T) ζ_{α,T} ,

u_{λ,T_{0}} = ∑_{T ∈Y (T_{0};W_{λ})} ∏_{j=1}^{n} P_{1} (T; a_{j}, b_{j}) v_{T} ∈ V_{τ}.

The term involving x^{λ} is h_{0} = ∑_{T ∈Y (T_{0};W_{λ})} ∏_{j=1}^{n} P_{1} (T; a_{j}, b_{j}) ζ_{λ,T}, thus the

leading term in f_{λ,T_{0}}^{a} is x^{λ}u_{λ,T_{0}}. From the transformation rules in Proposition 9 it follows that ∥h_{0}∥^{2} = ∥ζ_{λ,T_{0}}∥^{2} ∥u_{λ,T_{0}}∥^{2} / ∥v_{T_{0}}∥^{2} (see Proposition 4). Also h_{0} is W_{λ}-antisymmetric. Theorem 6. wf_{λ,T_{0}}^{a} = sgn (w) f_{λ,T_{0}}^{a} for all w ∈ S_{N} and

∥f_{λ,T_{0}}^{a}∥^{2} = \frac{N!}{#W_{λ}} \frac{∥u_{λ,T_{0}}∥_{0}^{2}}{E_{−} (λ^{R}, T_{0}) ∥v_{T_{0}}∥_{0}^{2}} ∥ζ_{λ,T_{0}}∥^{2} .

Proof. Let F (α, T) = ∏_{j=1}^{n} P_{1} (T; a_{j}, b_{j}) E_{+} (α, T). Fix i ∈ [1, N − 1] and collect the terms of f_{λ,T_{0}}^{a} into three parts. Let

L = {(α, T) : α^{+} = λ, T ∈ Y (T_{0}; W_{λ})}
A = {(α, T) ∈ L : α_{i} = α_{i+1}, cm (r (α, i) , T) = cm (r (α, i) + 1, T)} ,
B = {(α, T) ∈ L : α_{i} > α_{i+1}} ,
C = {(α, T) ∈ L : α_{i} = α_{i+1}, cm (r (α, i) , T) < cm (r (α, i) + 1, T)} .
□

The proof that each of the following satisfies s_{i}f = −f is analogous to the proof of the previous theorem:

∑_{(α,T)∈A} (−1)^{inv(α)} F (α, T) ζ_{α,T} ,

∑_{(α,T)∈B} (−1)^{inv(α)} (F (α, T) ζ_{α,T} − F (s_{i}α, T) ζ_{s_{i}α,T})

= ∑_{(α,T)∈B} (−1)^{inv(α)} F (α, T) (ζ_{α,T} − \frac{F (s_{i}α, T)}{F (α, T)}ζ_{s_{i}α,T} ) ,

∑_{(α,T)∈C} (−1)^{inv(α)} (F (α, T) ζ_{α,T} + F (α, s_{I(α)}T ) ζ_{α,s_{I(α)}T} )

= ∑_{(α,T)∈C} (−1)^{inv(α)} F (α, s_{I(α)}T ) (\frac{F (α, T)}{F (α, s_{I(α)}T )}ζ_{α,T} + ζ_{α,s_{I(α)}T} ) .

In the second equation\frac{F(s_{i}α,T)}{F(α,T)} = 1 + b_{i} (α, T). In the third equation

I = r (α, i) and \frac{F(α,T)}{F(α,s_{I(α)}T)} = b_{I(α)} − 1. The proof for the norm formula

is also analogous, based on ∑_{w∈S_{N}} wh_{R} where

h_{R} = ∑_{T ∈Y (T_{0};W_{λ})} ∏_{j=1}^{n} P_{1} (T; a_{j}, b_{j}) ζ_{λ^{R},T} .

Note sgn (w_{λ^{R}}) = (−1)^{inv(λ^{R})}.

Remark 2. The polynomials in Theorems 5 and 6 form orthogonal bases for the symmetric and antisymmetric polynomials, respectively, in M (τ).

5.3. Minimum degree polynomials. For a given partition τ of N there are unique symmetric and antisymmetric polynomials of minimum degree in the standard module M (τ). We now establish the key results concerning the norms of these polynomials. It is obvious that the column-strict tableau ⌊λ, T ⌋ with minimum |λ| has the entries 0 in row #1, 1 in row #2 and so on (consider the minimum entries in each column). Denote this partition by δ^{s} (τ) and the unique possible T by T^{s} (the entries N, N − 1, …, 2, 1 are entered row-by-row in the Ferrers diagram of τ). Example: let τ = (5, 3, 2) then

T^{s} = 10 9 8 7 6 5 4 3 2 1 , ⌊δ^{s} (τ) , T^{s}⌋ = 0 0 0 0 0 1 1 1 2 2 ,

and δ^{s} (τ) = (2, 2, 1, 1, 1, 0, 0, 0, 0, 0).

Similarly the row-strict tableau ⌊λ, T ⌋ with minimum |λ| has the entries 0 in column #1, 1 in column #2 and so on (consider the minimum entries in each row). Denote this partition by δ^{a} (τ) and the unique possible T by T^{a} (the entries N, N − 1, …, 2, 1 are entered columnby-column in the Ferrers diagram of τ). Example: let τ = (5, 3, 2) then

T^{a} = 10 7 4 2 1 9 6 3 8 5 , ⌊δ^{a} (τ) , T^{a}⌋ = 0 1 2 3 4 0 1 2 0 1 ,

and δ^{a} (τ) = (4, 3, 2, 2, 1, 1, 1, 0, 0, 0). The sum of the hook-lengths of τ equals |δ^{s} (τ)| + |δ^{a} (τ)| + N (see [7, Ex. 2, p. 11]).

Let f_{τ}^{s} = f_{δ^{s}(τ),T^{s}}^{s} and f_{τ}^{a} = f_{δ^{a}(τ),T^{a}}^{a}. These polynomials are actually independent of κ; there is no composition α such that α ◁ δ^{s} (τ) and α^{+} ≠ δ^{s} (τ) which can occur in a symmetric polynomial, due to the minimality of δ^{s} (τ). A similar argument applies to δ^{a} (τ). To compute the norms ∥f_{τ}^{s}∥^{2} and ∥f_{τ}^{a}∥^{2} we use the special properties of δ^{s} (τ) to write simplified formulae. To use the formulae in Theorems 2 and 3 note that δ_{a} (τ)_{j} = i − 1 when j appears in row #i of T^{s}, and the corresponding contents of T^{s} are 1 − i, …, τ_{i} − i. Let L = ℓ (τ) and

P_{1} (τ) = ∏_{i=2}^{L} ∏_{j=1}^{τ_{i}} (1 + κ (j − i))_{i−1} ,

P_{2} (τ) = ∏_{1≤i<j≤L} ∏_{j_{1}=1}^{τ_{i}} ∏_{j_{2}=1}^{τ_{j}} ∏_{r=1}^{j−i} (1 − \frac{κ^{2}}{(r + κ (j_{2} − j_{1} − j + i))^{2}} ) ,

P_{3} (τ) = ∏_{1≤i<j≤L} ∏_{j_{1}=1}^{τ_{i}} ∏_{j_{2}=1}^{τ_{j}} (1 + \frac{κ}{j − i + κ (j_{2} − j_{1} − j + i)});
H^{s} (τ) = ∏_{(i,j)∈τ} (1 − κh (i, j))_{leg(i,j)} .

Then ∥ζ_{δ^{s}(τ),T^{s}} ∥^{2} = ∥v_{T^{s}}∥^{2} P_{1} (τ) P_{2} (τ) and E_{+} (δ^{s} (τ)^{R} , T^{s}) = P_{3} (τ).

Theorem 7. Suppose τ is a partition then \frac{P_{1} (τ) P_{2} (τ)}{P_{3} (τ)} = H^{s} (τ).

Proof. We use induction on the last part τ_{L}. The induction begins with τ = (N) where each product equals 1. Let σ = (τ_{1}, τ_{2}, …, τ_{L} − 1) and assume the formula is valid for σ. (It is possible that τ_{L} = 1 and ℓ (σ) = L−1). The nodes in σ and τ have the same hook-lengths except for the nodes (i, L) with 1 ≤ i < L and (L, j) with 1 ≤ j ≤ τ_{L}. The latter have zero leg-length and do not contribute to H^{s} (σ) or H^{s} (τ). Then for 1 ≤ i < L

arm (i, L; σ) = arm (i, L; τ) = τ_{i} − τ_{L}
leg (i, L; σ) + 1 = leg (i, L; τ) = L − i
h (i, L; σ) + 1 = h (i, L; τ) = 1 + τ_{i} − τ_{L} + L − i,

and thus

\frac{H^{s} (τ)}{H^{s} (σ)} = ∏_{i=1}^{L−1} \begin{matrix}(1 − κ (τ_{i} − τ_{L} + L − i + 1)) & L−i \\ (1 − κ (τ_{i} − τ_{L} + L − i)) & L−i−1 \end{matrix} .

Firstly,

\frac{P_{1} (τ)}{P_{1} (σ)} = (1 + κ (τ_{L} − L))_{L−1} ;

secondly

\frac{P_{2} (τ)}{P_{2} (σ)} = ∏_{i=1}^{L−1} p_{i} (τ) ,

where

p_{i} (τ)

= ∏_{r=1}^{L−i} ∏_{j=1}^{τ_{i}} (\frac{(r + κ (τ_{L} − L + i) − (j − 1) κ)}{(r + κ (τ_{L} − L + i) − jκ)}

· \frac{(r + κ (τ_{L} − L + i) − (j + 1) κ)}{(r + κ (τ_{L} − L + i) − jκ)})

= ∏_{r=1}^{L−i} \begin{matrix}(r + κ (τ_{L} − L + i)) & (r + κ (τ_{L} − L + i) − (τ_{i} + 1) κ) \\ (r + κ (τ_{L} − L + i) − κ) & (r + κ (τ_{L} − L + i) − τ_{i}κ) \end{matrix}

= \begin{matrix}(1 + κ (τ_{L} − L + i)) & L−i & (1 + κ (τ_{L} − τ_{i} − L + i − 1)) & L−i \\ (1 + κ (τ_{L} − L + i − 1)) & L−i & (1 + κ (τ_{L} − τ_{i} − L + i)) & L−i \end{matrix};

a telescoping product argument is used to produce the third line from the second.

Thirdly,

\frac{P_{3} (σ)}{P_{3} (τ)} = ∏_{i=1}^{L−1} ∏_{j_{1}=1}^{τ_{i}} (\frac{L − i + κ (τ_{L} − j_{1} − L + i)}{L − i + κ (τ_{L} + 1 − j_{1} − L + i)})

= ∏_{i=1}^{L−1} \frac{L − i + κ (τ_{L} − τ_{i} − L + i)}{L − i + κ (τ_{L} − L + i)}.

Combining these products and by use of

\frac{L − i + κ (τ_{L} − τ_{i} − L + i)}{(1 + κ (τ_{L} − τ_{i} − L + i))_{L−i}} = \frac{1}{(1 + κ (τ_{L} − τ_{i} − L + i))_{L−i}} ,

\frac{(1 + κ (τ_{L} − L + i))_{L−i}}{L − i + κ (τ_{L} − L + i)} = (1 + κ (τ_{L} − L + i))_{L−i−1} ,

we obtain

\frac{P_{1} (τ) P_{2} (τ) P_{3} (σ) H^{s} (σ)}{P_{1} (σ) P_{2} (σ) P_{3} (τ) H^{s} (τ)}

= (1 + κ (τ_{L} − L))_{L−1} ∏_{i=1}^{L−1} \begin{matrix}(1 + κ (τ_{L} − L + i)) & L−i−1 \\ (1 + κ (τ_{L} − L + i − 1)) & L−i \end{matrix}

= 1.

The last step is easy: replace i by i − 1 in the numerator (and now 2 ≤ i ≤ L) and cancel. This completes the induction. □

Theorem 8. Suppose τ is a partition of N then

∥f_{τ}^{s}∥^{2} = \frac{N!}{∏_{i=1}^{ℓ(τ)} τ_{i}!} ∥v_{T^{s}}∥_{0}^{2} ∏_{(i,j)∈τ} (1 − κh (i, j))_{leg(i,j)} .

Proof. The formulae of Theorem 5 and the previous theorem imply this result. The stabilizer subgroup W_{δ^{s}(τ)} acts on the rows of τ and has order τ_{1}!τ_{2}! . . .. □

Theorem 9. Suppose τ is a partition of N then

∥f_{τ}^{a}∥^{2} = \frac{N!}{∏_{i=1}^{τ_{1}} τ′_{i}!} ∥v_{T^{a}}∥_{0}^{2} ∏_{(i,j)∈τ} (1 + κh (i, j))_{arm(i,j)} .

Proof. Apply Theorems 3, 6 and the formula in Theorem 8 to the conjugate τ′ of τ and with κ replaced by −κ. Then leg (j, i; τ′) = arm (i, j; τ) for (i, j) ∈ τ. Note however that ∥v_{T^{s}}∥_{0}^{2} / ∥v_{T^{a}}∥_{0}^{2} is computed by use of Proposition 4. □

As example we use τ = (5, 3, 2) again. The hook-lengths and norms are

7 6 4 2 1 4 3 1 2 1 ,
∥f_{τ}^{s}∥^{2} = c_{0} (1 − 7κ)_{2} (1 − 6κ)_{2} (1 − 4κ)^{2} (1 − 3κ) ,
∥f_{τ}^{a}∥^{2} = c_{1} (1 + 7κ)_{4} (1 + 6κ)_{3} (1 + 4κ)_{2}^{2} (1 + 3κ) (1 + 2κ)^{2} .

Analogously to the M ((N)) (trivial representation) result, each hooklength m appears in m − 1 factors (mκ + r) involving each nonzero residue class mod m. In the example for m = 6 we obtain 6κ − 2, 6κ − 1, 6κ + 1, 6κ + 2, 6κ + 3. We conjecture that the singular values for M (τ) form a subset of \{\frac{n}{m} : m = h (i, j) , (i, j) ∈ τ, \frac{n}{m} ∉ ℤ\} (κ_{0} ∈ ℚ is a singular value if there exists nonzero f ∈ M (τ) such that D_{i} (κ_{0}) f = 0 for all i ∈ [1, N]; that is, the generic κ is specialized to κ_{0}; the condition is equivalent to J_{κ_{0}} (τ) ≠ (0)). As yet there is insufficient evidence for speculation about any further restrictions.

5.4. Aspherical values. S. Griffeth (personal communication, gratefully acknowledged) points out that Theorem 8 provides a new proof for one of the parts of the Gordon-Stafford Theorem [5, Cor. 3.13]; another proof was found by Bezrukavnikov and Etingof [2, Cor. 4.2]; note that these papers use c = −κ as parameter. An aspherical module of the rational Cherednik algebra is one containing no nonzero S_{N}- invariant. If some quotient module of a standard module is aspherical for a numerical value κ_{0} of κ then κ_{0} is called an aspherical value. Theorem 8 shows that any aspherical value is in \{\frac{m}{n} : 1 ≤ m < n ≤ N\} (this is one component of the Gordon-Stafford theorem, which deals with the problem of Morita equivalence of rational Cherednik algebras for parameters κ and κ − 1). Suppose M_{0} is a proper submodule of M (τ) for κ = κ_{0} ∈ ℚ (that is, a specific numerical value). This means that M_{0} is closed under multiplication by x_{i} and the action of D_{i} for i ∈ [1, N] and under the action of S_{N}. Then f ∈ M_{0} implies ⟨g, f⟩ = 0 for all g ∈ M (τ) (M_{0} is a submodule of the radical J_{κ_{0}} (τ), the maximal submodule.). Indeed, by the definition of the contravariant form, ⟨x^{α}u, f⟩ = ⟨u, D^{α}f (x) |_{x=0}⟩_{0} for α ∈ ℕ^{N}, u ∈ V_{τ} (and D^{α} = ∏_{i=1}^{N} D_{i}^{α_{i}}). If f ∈ P_{n} ⊗ V_{τ} and |α| = n then D^{α}f (x) ∈ V_{τ}. If also f ∈ M_{0} then D^{α}f (x) = 0, or else M_{0} = M (τ). If M (τ) /M_{0} is aspherical then f_{τ}^{s} ∈ M_{0} and κ = κ_{0} is a zero of ∏_{(i,j)∈τ} (1 − κh (i, j))_{leg(i,j)}.

6. Appendix

This contains proofs of the specializations of Griffeth’s results in the G (r, 1, N)-context, which were stated in Section 3. Here is the restatement and proof of Proposition 7

Proposition 10. Suppose α ∈ ℕ^{N} and α_{i} > α_{i+1} for some i < N. Then s_{i}ζ_{α,T} = b_{i} (α, T) ζ_{α,T} + (1 − b_{i} (α, T)^{2}) ζ_{s_{i}α,T} , s_{i}ζ_{s_{i}α,T} = ζ_{α,T} − b_{i} (α, T) ζ_{s_{i}α,T}; and ∥ζ_{α,T} ∥^{2} = (1 − b_{i} (α, T)^{2}) ∥ζ_{s_{i}α,T} ∥^{2}.

Proof. The condition α_{i} ≠ α_{i+1} implies r (s_{i}α, i) = r (α, i + 1) and r (s_{i}α, i + 1) = r (α, i), thus ξ_{i} (s_{i}α, T) = ξ_{i+1} (α, T) and ξ_{i+1} (s_{i}α, T) = ξ_{i} (α, T) (and ξ_{j} (s_{i}α, T) = ξ_{j} (α, T) for j ≠ i, i + 1). Since the eigenvalues determine the eigenvectors uniquely we have that

s_{i}ζ_{α,T} − b_{i} (α, T) ζ_{α,T} = aζ_{s_{i}α,T} ,
s_{i}ζ_{s_{i}α,T} + b_{i} (α, T) ζ_{s_{i}α,T} = a′ζ_{α,T} ,

for some scalars a, a′. The fact that s_{i}^{2} = 1 implies aa′ = 1 − b_{i} (α, T)^{2}. We show that a′ = 1 by finding the leading term in s_{i}ζ_{s_{i}α,T}, namely x^{α}s_{i}w_{s_{i}α}v_{T}. It remains to show that w_{s_{i}α} = s_{i}w_{α}, that is, r (s_{i}α, s_{i}w_{α} (j)) = j for all j. If w_{α}^{−1} (j) ≠ i, i+1 then r (s_{i}α, s_{i}w_{α} (j)) = r (α, w_{α} (j)) = j. If w_{α}^{−1} (j) = i then r (s_{i}α, s_{i}w_{α} (j)) = r (s_{i}α, i + 1) = r (α, i) = j. The case w_{α}^{−1} (j) = i + 1 follows similarly. The second displayed equation shows that ∥s_{i}ζ_{s_{i}α,T} ∥^{2} = ∥ζ_{s_{i}α,T} ∥^{2} = ∥a′ζ_{α,T} ∥^{2} − b_{i} (α, T)^{2} ∥ζ_{s_{i}α,T} ∥^{2}. □

The proof of the norm formula of Theorem 2 uses a raising operator. From the commutators 3.2 we obtain:

U_{i}x_{N}f = x_{N} (U_{i} − κ (i, N)) f, 1 ≤ i < N,
U_{N}x_{N}f = x_{N} (1 + D_{N}x_{N}) f.

Let θ_{N} = s_{1}s_{2} … s_{N−1} thus θ_{N} (N) = 1 and θ_{N} (i) = i+1 for 1 ≤ i < N (a cyclic shift). Then

U_{i}x_{N}f = x_{N} (θ_{N}^{−1}U_{i+1}θ_{N}) f, 1 ≤ i < N,
U_{N}x_{N}f = x_{N} (1 + θ_{N}^{−1}U_{1}θ_{N}) f.

If f satisfies U_{i}f = λ_{i}f for 1 ≤ i ≤ N then

U_{i} (x_{N}θ_{N}^{−1}f) = λ_{i+1} (x_{N}θ_{N}^{−1}f)

for 1 ≤ i < N and

U_{N} (x_{N}θ_{N}^{−1}f) = (λ_{1} + 1) (x_{N}θ_{N}^{−1}f) .

For α ∈ ℕ^{N} let ϕ (α) := (α_{2}, α_{3}, …, α_{N}, α_{1} + 1), then x_{N}θ_{N}^{−1}x^{α} = x^{ϕ(α)}.

Proposition 11. Suppose α ∈ ℕ^{N}, T ∈ Y (τ), then

ζ_{ϕ(α),T} = x_{N}θ_{N}^{−1}ζ_{α,T} .

Proof. By straightforward arguments it follows that r (ϕ (α) , i) = r (α, i + 1) for 1 ≤ i < N and r (ϕ (α) , N) = r (α, 1), that is, r (ϕ (α) , i) = r (α, θ_{N} (i)) for all i. This is equivalent to r (ϕ (α) , θ_{N}^{−1} (w_{α} (j))) = r (α, w_{α} (j)) = j for all j, or w_{ϕ(α)} = θ_{N}^{−1}w_{α}. The leading term x^{α}w_{α}v_{T} of ζ_{α,T} is mapped to x^{ϕ(α)}w_{ϕ(α)}v_{T} by f ↦ x_{N}θ_{N}^{−1}f. Note that U_{i}ζ_{ϕ(α),T} = (α_{i+1} + 1 + κc (r (ϕ (α) , i) , T)) ζ_{ϕ(α),T} for 1 ≤ i < N and U_{N}ζ_{ϕ(α),T} = (α_{1} + 2 + κc (r (ϕ (α) , N) , T)) ζ_{ϕ(α),T}. Thus x_{N}θ_{N}^{−1}ζ_{α,T} and ζ_{ϕ(α),T} have the same eigenvalues for {U_{i}} and the same coefficient of x^{ϕ(α)}.Hence x_{N}θ_{N}^{−1}ζ_{α,T} = ζ_{ϕ(α),T}. □

Corollary 5. ∥ζ_{ϕ(α),T} ∥^{2} = (α_{1} + 1 + κc (r (α, 1) , T)) ∥ζ_{α,T} ∥^{2}.

Proof. Indeed

∥ζ_{ϕ(α),T} ∥^{2} = ⟨θ_{N}^{−1}ζ_{α,T} , D_{N}x_{N}θ_{N}^{−1}ζ_{α,T} ⟩
= ⟨θ_{N}^{−1}ζ_{α,T} , θ_{N}^{−1}D_{1}x_{1}ζ_{α,T} ⟩ = ⟨ζ_{α,T} , U_{1}ζ_{α,T} ⟩
= ξ_{1} (α, T) ∥ζ_{α,T} ∥^{2} .
□

The following is a restatement and proof of Theorem (2) (Griffeth [6, Thm. 6.1]).

Theorem 10. Suppose λ ∈ ℕ^{N,+} and T ∈ Y (τ) then

∥ζ_{λ,T} ∥^{2} = ∥v_{T} ∥_{0}^{2} ∏_{i=1}^{N} (1 + κc (i, T))_{λ_{i}}

× ∏_{1≤i<j≤N} ∏_{l=1}^{λ_{i}−λ_{j}} (1 − \frac{κ^{2}}{(l + κ (c (i, T) − c (j, T)))^{2}} ) .

Proof. Argue by induction. Suppose λ_{1} = λ_{2} = … = λ_{m} > λ_{m+1}. Let

β = (λ_{1}, …, λ_{m−1}, λ_{m+1}, …, λ_{N}, λ_{1}) ,
α = (λ_{1} − 1, λ_{1}, …, λ_{m−1}, λ_{m+1}, …, λ_{N}) ,
μ = (λ_{1}, …, λ_{m−1}, λ_{1} − 1, λ_{m+1}, …, λ_{N}) .

Thus β = ϕ (α) and

∥ζ_{β,T} ∥^{2} = (λ_{1} + κc (m, T)) ∥ζ_{α,T} ∥^{2}
= (λ_{1} + κc (m, T)) E_{2} (α, T)^{−1} ∥ζ_{μ,T} ∥^{2} ,
∥ζ_{λ,T} ∥^{2} = E_{2} (β, T) ∥ζ_{β,T} ∥^{2} .

We have

E_{ε} (α, T) = ∏_{j=2}^{m} (1 + \frac{εκ}{1 + κ (c (j − 1, T) − c (m, T))}) ,

E_{ε} (β, T) = ∏_{j=m+1}^{N} (1 + \frac{εκ}{λ_{1} − λ_{j} + κ (c (m, T) − c (j, T))}) .

The validity of the formula for ∥ζ_{μ,T} ∥^{2} thus implies the validity for ∥ζ_{λ,T} ∥^{2} (that is, the value of ∥ζ_{λ,T} ∥^{2} / ∥ζ_{μ,T} ∥^{2} from the formula agrees

with (λ_{1} + κc (m, T)) \frac{E_{2} (β, T)}{E_{2} (α, T)}). □

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Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, US

E-mail address: cfd5z@virginia.edu
URL: http://people.virginia.edu/~cfd5z