Descent Representations and Multivariate Statistics

Ron M. Adin^{∗}^{§}  Francesco Brenti^{†}^{§}  Yuval Roichman^{‡}^{§}

December 6, 2001

Abstract

Combinatorial identities on Weyl groups of types A and B are derived from special bases of the corresponding coinvariant algebras. Using the Garsia-Stanton descent basis of the coinvariant algebra of type A we give a new construction of the Solomon descent representations. An extension of the descent basis to type B, using new multivariate statistics on the group, yields a refinement of the descent representations. These constructions are then applied to refine well-known decomposition rules of the coinvariant algebra and to generalize various identities.

1 Introduction

1.1 Outline

This paper studies the interplay between representations of classical Weyl groups of types A and B and combinatorial identities on these groups. New combinatorial statistics on these groups are introduced, which lead to a new construction of representations. The Hilbert series which emerge give rise to multivariate identities generalizing known ones.

∗Department of Mathematics and Computer Science, Bar-Ilan University, Ramat Gan

52900, Israel. Email: radin@math.biu.ac.il

†Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy. Email: brenti@mat.uniroma2.it

‡Department of Mathematics and Computer Science, Bar-Ilan University, Ramat Gan

52900, Israel. Email: yuvalr@math.biu.ac.il

§Research of all authors was supported in part by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities, and by internal research grants from Bar-Ilan University.

The set of elements in a Coxeter group having a fixed descent set carries a natural representation of the group, called a descent representation. Descent representations of Weyl groups were first introduced by Solomon [31] as alternating sums of permutation representations. This concept was extended to arbitrary Coxeter groups, using a different construction, by Kazhdan and Lusztig [22] [21, §7.15]. For Weyl groups of type A, these representations also appear in the top homology of certain (Cohen-Macaulay) rank-selected posets [34]. Another description (for type A) is by means of zig-zag diagrams [19] [16].

In this paper we give a new construction of descent representations for Weyl groups of type A, using the coinvariant algebra as a representation space. This viewpoint gives rise to a new extension for type B, which refines the one by Solomon.

The construction of a basis for the coinvariant algebra is important for many applications, and has been approached from different viewpoints. A geometric approach identifies the coinvariant algebra with the cohomology ring H^{∗}(G/B) of the flag variety. This leads to the Schubert basis [7] [10], and applies to any Weyl group. This identification also appears in Springer’s construction of irreducible representations [32]; see also [15]. Barcelo [6] found bases for the resulting quotients. An algebraic approach, applying Young symmetrizers, was used by Ariki, Terasoma and Yamada [40] [5] to produce a basis compatible with the decomposition into irreducible representations. This was extended to complex reflection groups in [25].

A combinatorial approach, which produces a basis of monomials, was presented by Garsia and Stanton in [17] (see also [13] and [37]). They actually presented a basis for a finite dimensional quotient of the Stanley-Reisner ring arising from a finite Weyl group. For type A, unlike other types, this quotient is isomorphic to the coinvariant algebra. The Garsia-Stanton descent basis for type A may be constructed from the coinvariant algebra via a straightening algorithm [4]. Using a reformulation of this algorithm we give a natural construction of Solomon’s descent representations as factors of the coinvariant algebra of type A.

An analogue of the descent basis for type B is now given. This analogue (again consisting of monomials) involves extended descent sets and new combinatorial statistics. An extension of the construction of descent representations, using the new basis for type B, gives rise to a family of descent representations, refining Solomon’s. A decomposition of these descent representations into irreducibles, refining theorems of Kraskiewicz-Weyman (for type A) [29, Theorem 8.8] and Stembridge (for type B) [39], is carried out using a multivariate version of Stanley’s formula for the principal specialization of Schur functions.

This algebraic setting is then applied to obtain new multivariate combinatorial identities. Suitable Hilbert series are computed and compared to each other and to generating functions of multivariate statistics. The resulting identities present a far reaching generalization of bivariate identities from [18], [14], and [1].

1.2 Main Results

Let W be a classical Weyl group of type A or B, and let I_{n}^{W} be the ideal of the polynomial ring P_{n} := Q[x_{1}, …, x_{n}] generated by W -invariant polynomials without a constant term. The quotient P_{n}/I_{n}^{W} is called the coinvariant algebra of W. See Subsection 2.5 below.

For any partition λ with (at most) n parts, let P \underline{◁} ; λ be the subspace of the polynomial ring P_{n} = Q[x_{1}, …, x_{n}] spanned by all monomials whose exponent partition is dominated by λ, and let R_{λ} be a distinguished quotient of the image of P \underline{◁} ; λ under the projection of P_{n} onto the coinvariant algebra. For precise definitions see Subsections 3.5 and 5.3. We will show that the homogeneous components of the coinvariant algebra decompose as direct sums of certain R_{λ}-s. This will be done using an explicit construction of a basis for R_{λ}. The construction of this basis involves new statistics on S_{n} and B_{n}.

1.2.1 New Statistics

Let Σ be a linearly ordered alphabet. For any finite sequence σ = σ_{1}, σ_{2}, …, σ_{n} of letters in Σ define

Des(σ) := {i | σ_{i} > σ_{i+1}},

the descent set of σ, and

d_{i}(σ) := |{j ∈ Des(σ) : j ≥ i}|,

the number of descents in σ from position i on.

If Σ consists of integers, let

Neg(σ) := {i | σ_{i} < 0};
n_{i}(σ) := |{j ∈ Neg(σ) : j ≥ i}|;

ε_{i}(σ) := \{ \begin{matrix}1, & if σ_{i} < 0, \\ 0, & otherwise; \end{matrix}

and

f_{i}(σ) := 2d_{i}(σ) + ε_{i}(σ).

The statistics f_{i}(σ) refine the flag-major index fmaj (σ), which was introduced and studied in [2] [3] [1].

For various properties of these statistics see Sections 3, 5, and 6 below.

1.2.2 The Garsia-Stanton Descent Basis and its Extension

To any π ∈ S_{n} Garsia and Stanton [17] associated the monomial

a_{π} := ∏_{i∈Des(π)}(x_{π(1)} ⋯ x_{π(i)}).

It should be noted that in our notation a_{π} = ∏_{i=1}^{n} x_{π(i)}^{d_{i}(π)}. Using Stanley-

Reisner rings Garsia and Stanton showed that the set {a_{π} + I_{n} | π ∈ S_{n}} forms a basis for the coinvariant algebra of type A [17]. This basis will be called the descent basis. The Garsia-Stanton approach is not applicable to the coinvariant algebras of other Weyl groups. In this paper we extend the descent basis to the Weyl groups of type B.

To any σ ∈ B_{n} we associate the monomial

b_{σ} := ∏_{i=1}^{n} x_{|σ(i)|}^{f_{i}(σ)}.

Theorem 1.1 The set

{b_{σ} + I_{n}^{B} | σ ∈ B_{n}}

forms a basis for the coinvariant algebra of type B.

See Corollary 5.3 below.

1.2.3 Descent Representations

For a monomial m in the polynomial ring P_{n} = Q[x_{1}, …, x_{n}], let the exponent partition λ(m) be the partition obtained by rearranging the exponents in a weakly decreasing order. For any partition λ with at most n parts, let P \underline{◁} ; λ be the subspace of P_{n} spanned by all monomials whose exponent partition is dominated by λ :

P \underline{◁} ; λ := span_{Q}\{m | λ(m) \underline{◁} λ\}.

Similarly, define P_{λ}^{◁} by strict dominance :

P_{λ}^{◁} := span_{Q}{m | λ(m) ◁ λ}.

Consider now the canonical projection of P_{n} onto the coinvariant algebra

ψ : P_{n} ⟶ P_{n}/I_{n}.

Define R_{λ} to be a quotient of images under this map :

R_{λ} := ψ(P \underline{◁} ; λ)/ψ(P_{λ}^{◁}).

Then R_{λ} is an S_{n}-module. For any subset S ⊆ {1, …, n} define a partition

λ_{S} := (λ_{1}, …, λ_{n})

by

λ_{i} := | S ∩ {i, …, n} |.

Using a straightening algorithm for the descent basis it is shown that R_{λ} ≠ 0 if and only if λ = λ_{S} for some S ⊆ [n − 1] (Corollary 3.10), and that a basis for R_{λ_{S}} may be indexed by the permutations with descent set equal to S (Corollary 3.11). Let R_{k} be the k-th homogeneous component of the coinvariant algebra P_{n}/I_{n}.

Theorem 1.2 For every 0 ≤ k ≤ \binom{n}{2},

R_{k} ≅ ⨁_{S} R_{λ_{S}}

as S_{n}-modules, where the sum is over all subsets S ⊆ [n − 1] such that

∑_{i∈S} i = k.

See Theorem 3.12 below.

Let

R_{λ}^{B} := ψ^{B}(P \underline{◁} ; λ) / ψ^{B}(P_{λ}^{◁}),

where ψ^{B} : P_{n} ⟶ P_{n}/I_{n}^{B} is the canonical map from P_{n} onto the coinvariant algebra of type B. For subsets S_{1} ⊆ [n − 1] and S_{2} ⊆ [n], let λ_{S_{1},S_{2}} be the vector

λ_{S_{1},S_{2}} := 2λ_{S_{1}} + 1_{S_{2}},

where λ_{S_{1}} is as above and 1_{S_{2}} ∈ {0, 1}^{n} is the characteristic vector of S_{2}. Again, R_{λ}^{B} ≠ 0 if and only if λ = λ_{S_{1},S_{2}} for some S_{1} ⊆ [n − 1], S_{2} ⊆ [n] (Corollary 5.6). In this case, a basis for R_{λ_{S_{1},S_{2}}} may be indexed by the signed permutations σ ∈ B_{n} with Des(σ) = S_{1} and Neg(σ) = S_{2} (Corollary 5.7).

Let R_{k}^{B} for the k-th homogeneous component of the coinvariant algebra of type B. The following theorem is a B-analogue of Theorem 1.2.

Theorem 1.3 For every 0 ≤ k ≤ n^{2},

R_{k}^{B} ≅ ⨁_{S_{1},S_{2}} R_{λ_{S_{1},S_{2}}}

as B_{n}-modules, where the sum is over all subsets S_{1} ⊆ [n − 1] and S_{2} ⊆ [n] such that λ_{S_{1},S_{2}} is a partition and

2 · ∑_{i∈S_{1}} i + |S_{2}| = k.

See Theorem 5.8 below.

1.2.4 Decomposition into Irreducibles

Theorem 1.4 For any subset S ⊆ [n − 1] and partition μ ⊢ n, the multiplicity in R_{λ_{S}} of the irreducible S_{n}-representation corresponding to μ is

m_{S,μ} := | { T ∈ SY T (μ) | Des(T) = S } |,

the number of standard Young tableaux of shape μ and descent set S.

See Theorem 4.1 below. This theorem implies the well known Kraskiewicz-Weyman theorem for decomposing the homogeneous components of the coinvariant algebra into irreducibles. See Subsection 2.5.

For type B we have Theorem 1.5 For any pair of subsets S_{1} ⊆ [n − 1], S_{2} ⊆ [n], and a bipartition (μ^{1}, μ^{2}) of n, the multiplicity of the irreducible B_{n}-representation corresponding to (μ^{1}, μ^{2}) in R_{λ_{S_{1},S_{2}}} is

m_{S_{1},S_{2},μ^{1},μ^{2}} := | { T ∈ SY T (μ^{1}, μ^{2}) | Des(T) = S_{1} and Neg(T) = S_{2} } |,

the number of pairs of standard Young tableaux of shapes μ^{1} and μ^{2} with descent set S_{1} and sets of entries [n] \ S_{2} and S_{2}, respectively.

For definitions and more details see Subsection 5.4 and Theorem 5.9 below.

The proofs apply multivariate extensions of Stanley’s formula for the principal specialization of a Schur function. See Lemmas 4.3 and 5.10 below.

1.2.5 Combinatorial Identities

For any partition λ = (λ_{1}, …, λ_{n}) with at most n parts define

m_{j}(λ) := | {1 ≤ i ≤ n | λ_{i} = j} | (∀j ≥ 0).

By considering Hilbert series of the polynomial ring with respect to rearranged multi-degree and applying the Straightening Lemma for the coinvariant algebra of type A we obtain

Theorem 1.6 For any positive integer n

∑_{ℓ(λ)≤n} \binom{n}{m_{0}(λ), m_{1}(λ), …} ∏_{i=1}^{n} q_{i}^{λ_{i}} = \frac{∑_{π∈S_{n}} ∏_{i=1}^{n} q_{i}^{d_{i}(π)}}{∏_{i=1}^{n}(1 − q_{1} ⋯ q_{i})}

in Z[[q_{1}, …, q_{n}]], where the sum on the left-hand side is taken over all partitions with at most n parts.

See Theorem 6.2 below. This theorem generalizes Gessel’s Theorem for the bivariate distribution of descent number and major index [18].

The main combinatorial result for type B asserts :

Theorem 1.7 For any positive integer n

∑_{σ∈B_{n}} ∏_{i=1}^{n} q_{i}^{d_{i}(σ)+n_{i}(σ^{−1})} = ∑_{σ∈B_{n}} ∏_{i=1}^{n} q_{i}^{2d_{i}(σ)+ε_{i}(σ)}.

See Theorem 6.1 below.

For further identities see Section 6. In particular, it is shown that central results from [1] follow from Theorem 1.7.

2 Preliminaries

2.1 Notations

Let P := {1, 2, 3, …}, N := P ∪ {0}, Z be the ring of integers, and Q be the field of rational numbers; for a ∈ N let [a] := {1, 2, …, a} (where [0] := ∅). Given n, m ∈ Z, n ≤ m, let [n, m] := {n, n + 1, …, m}. For S ⊂ N write S = {a_{1}, …, a_{r}}_{<} to mean that S = {a_{1}, …, a_{r}} and a_{1} < … < a_{r}. The cardinality of a set A will be denoted by |A|. More generally, given a multiset M = {1^{a_{1}}, 2^{a_{2}}, …, r^{a_{r}}} denote by |M| its cardinality, so |M| = ∑_{i=1}^{r} a_{i}.

Given a variable q and a commutative ring R, denote by R[q] (respectively, R[[q]]) the ring of polynomials (respectively, formal power series) in q with coefficients in R. For i ∈ N let, as customary, [i]_{q} := 1+q+q^{2}+…+q^{i−1} (so [0]_{q} = 0). Given a vector v = (v_{1}, …, v_{n}) ∈ P^{n} and a sequence of variables x_{1}, …, x_{n} denote by x̄^{v} the monomial ∏_{i=1}^{n} x_{i}^{v_{i}} ∈ R[x_{1}, …, x_{n}].

2.2 Sequences and Permutations

Let Σ be a linearly ordered alphabet. Given a sequence σ = (σ_{1}, …, σ_{n}) ∈ Σ^{n} we say that a pair (i, j) ∈ [n] × [n] is an inversion of σ if i < j and σ_{i} > σ_{j}. We say that i ∈ [n − 1] is a descent of σ if σ_{i} > σ_{i+1} ;

Des(σ) := {1 ≤ i ≤ n − 1 | σ_{i} > σ_{i+1}}

is the descent set of σ. Denote by inv(σ) (respectively, des(σ)) the number of inversions (respectively, descents) of σ. We also let

maj(σ) := ∑_{i∈Des(σ)} i

and call it the major index of σ.

Given a set T let S(T) be the set of all bijections π : T → T, and S_{n} := S([n]). For π ∈ S_{n} write π = π_{1} … π_{n} to mean that π(i) = π_{i}, for i = 1, …, n. Any π ∈ S_{n} may also be written in disjoint cycle form (see, e.g., [35, p.17]), usually omitting the 1-cycles of π. For example, π = 365492187 may also be written as π = (9, 7, 1, 3, 5)(2, 6). Given π, τ ∈ S_{n} let πτ := π∘τ (composition of functions) so that, for example, (1, 2)(2, 3) = (1, 2, 3).

Denote by B_{n} the group of all bijections σ of the set [−n, n] \ {0} onto itself such that

σ(−a) = −σ(a)

for all a ∈ [−n, n] \ {0}, with composition as the group operation. This group is usually known as the group of “signed permutations” on [n], or as the hyperoctahedral group of rank n. We identify S_{n} as a subgroup of B_{n}, and B_{n} as a subgroup of S_{2n}, in the natural ways.

For σ ∈ B_{n} write σ = [a_{1}, …, a_{n}] to mean that σ(i) = a_{i} for i = 1, …, n, and (using the natural linear order on [−n, n] \ {0}) let

inv(σ) := inv(a_{1}, …, a_{n}),
des(σ) := des(a_{1}, …, a_{n}),
maj (σ) := maj (a_{1}, …, a_{n}),
Neg(σ) := {i ∈ [n] : a_{i} < 0},
neg(σ) := |Neg(σ)|,

and fmaj (σ) := 2 · maj (σ) + neg(σ).

The statistic fmaj was introduced in [2] [3] and further studied in [1].

2.3 Partitions and Tableaux

Let n be a nonnegative integer. A partition of n is an infinite sequence of nonnegative integers with finitely many nonzero terms λ = (λ_{1}, λ_{2}, …),

where λ_{1} ≥ λ_{2} ≥ … and ∑_{i=1}^{∞} λ_{i} = n. The sum ∑ λ_{i} = n is called the

size of λ, denoted |λ|); write also λ ⊢ n. The number of parts of λ, ℓ(λ), is the maximal j for which λ_{j} > 0. The unique partition of n = 0 is the empty partition ∅ = (0, 0, …), which has length ℓ(∅) := 0. For a partition λ = (λ_{1}, …, λ_{k}, …) define the conjugate partition λ′ = (λ′_{1}, …, λ′_{i}, …) by letting λ′_{i} be the number of parts of λ that are ≥ i (∀i ≥ 1).

The dominance partial order on partitions is defined as follows : For any two partitions μ and λ of the same integer, μ dominates λ (denoted μ \underline{▷} λ) if and only if ∑_{j=1}^{i} μ_{j} ≥ ∑_{j=1}^{i} λ_{j} for all i (and, by assumption, ∑_{j=1}^{∞} μ_{j} = ∑_{j=1}^{∞} λ_{j}).

The subset {(i, j) | i, j ∈ P, j ≤ λ_{i}} of P^{2} is called the Young diagram of shape λ. (i, j) is the cell in row i and column j. The diagram of the conjugate shape λ′ may be obtained from the diagram of shape λ by interchanging rows and columns.

A Young tableau of shape λ is obtained by inserting the integers 1, 2, …, n (where n = |λ|) as entries in the cells of the Young diagram of shape λ, allowing no repetitions. A standard Young tableau of shape λ is a Young tableau whose entries increase along rows and columns.

We shall draw Young tableaux as in the following example.

Example 1.

1 3 4 6 9
2 7 8 12
5 11

10

A descent in a standard Young tableau T is an entry i such that i + 1 is strictly south (and hence weakly west) of i. Denote the set of all descents in T by Des(T). The descent number and the major index (for tableaux) are defined as follows :

des(T) := ∑_{i∈Des(T)} 1 ; maj (T) := ∑_{i∈Des(T)} i.

Example 1. (Cont.) Let T be the standard Young tableau drawn in Example 1. Then Des(T) = {1, 4, 6, 9}, des(T) = 4, and maj (T) = 1 + 4 + 6 + 9 = 20.

A semistandard Young tableau of shape λ is obtained by inserting nonnegative integers as entries in the cells of the Young diagram of shape λ, so that the entries weakly increase along rows and strictly increase down columns. A reverse semistandard Young tableau is obtained by inserting nonnegative integers, so that the entries weakly decrease along rows and strictly decrease down columns (note that zero entries are allowed).

A bipartition of n is a pair (λ^{1}, λ^{2}) of partitions of total size |λ^{1}|+|λ^{2}| = n. A skew diagram of shape (λ^{1}, λ^{2}) is a disjoint union of a diagram of shape λ^{1} and a diagram of shape λ^{2}, where the second diagram lies southwest of the first. A standard Young tableau T = (T^{1}, T^{2}) of shape (λ^{1}, λ^{2}) is obtained by inserting the integers 1, 2, …, n as entries in the cells, such that the entries increase along rows and columns. The descent set Des(T), the descent number des(T), and the major index maj (T) of T are defined as above. The negative set, Neg(T), of such a tableau T is the set of entries in the cells of λ^{2}. Define neg(T) := |Neg(T)| and

fmaj (T) := 2 · maj (T) + neg(T).

Example 2. Let T be

2 5 6 3 1 7 4 9 8

We shall also consider T as a pair of tableaux

(T^{1}, T^{2}) = ( \begin{matrix}2 & 5 & 6 \\ 3 \end{matrix} , \begin{matrix}1 & 7 \\ 4 & 9 \\ 8 \end{matrix} ) .

T is a standard Young tableau of shape ((3, 1), (2, 2, 1)), Des(T) = {2, 3, 6, 7}, des(T) = 4, maj (T) = 18, Neg(T) = {1, 4, 7, 8, 9}, neg(T) = 5 and fmaj (T) = 41.

Denote by SY T (λ) the set of all standard Young tableaux of shape λ. Similarly, SSY T (λ) and RSSY T (λ) denote the sets of all semistandard and reverse semistandard Young tableaux of shape λ, respectively. SY T (λ^{1}, λ^{2}) denotes the set of all standard Young tableaux of shape (λ^{1}, λ^{2}).

Let T̂ be a (reverse) semistandard Young tableau. The content vector cont(T̂) = (m_{0}, m_{1}, …) is defined by

m_{i} := |{cells in T̂ with entry i}| (∀i ≥ 0).

2.4 Symmetric Functions

Let Λ (Λ_{n}) be the ring of symmetric functions in infinitely many (resp. n) variables. In this paper we consider three bases for the vector space Λ: The elementary symmetric functions are defined as follows. For a nonempty partition λ = (λ_{1}, λ_{2}, …, λ_{t})

e_{λ}(x̄) := ∏_{i=1}^{t} e_{λ_{i}}(x̄),

where x̄ = (x_{1}, x_{2}, …) and for any k ∈ P

e_{k}(x̄) := ∑_{i_{1}<i_{2}<···<i_{k}} x_{i_{1}} ⋯ x_{i_{k}}.

The power sum symmetric functions are defined as follows. For a nonempty partition λ = (λ_{1}, λ_{2}, …, λ_{t})

p_{λ}(x̄) := ∏_{i=1}^{t} p_{λ_{i}}(x̄),

where for any k ∈ P

p_{k}(x̄) := ∑_{i=1}^{∞} x_{i}^{k}.

The Schur functions are defined in a different manner. For a nonempty partition λ define

s_{λ}(x̄) := ∑_{T ∈SSY T (λ)} x̄^{cont(T)}.

For the empty partition λ = ∅, define e_{∅}(x̄) = p_{∅}(x̄) = s_{∅}(x̄) := 1.

Claim 2.1

s_{λ}(x̄) = ∑_{T ∈RSSY T (λ)} x̄^{cont(T)}.

See [36, Prop. 7.10.4].

Note that corresponding spanning sets for Λ_{n} are obtained by substituting x_{n+1} = x_{n+2} = ⋯ = 0.

The conjugacy classes of the symmetric group S_{n} are described by their cycle type; thus, by partitions of n . The irreducible representations of S_{n} are also indexed by these partitions. See e.g. [30].

Let λ and μ be partitions of n. Denote by χ_{μ}^{λ} the value, at a conjugacy class of cycle type μ, of the character of the irreducible S_{n}-representation corresponding toλ. These character values are entries in the transition matrix between two of the above bases. This fact was discovered and applied by Frobenius as an efficient tool for calculating characters (see e.g. [36, p. 401]).

Frobenius Formula. [36, Corollary 7.17.4] For any partition μ of n

p_{μ}(x̄) = ∑_{λ⊢n} χ_{μ}^{λ}s_{λ}(x̄),

where the sum runs through all partitions λ of n.

Recall that the Weyl group B_{n} may be described as the group of all signed permutations of order n (Subsection 2.2). The conjugacy classes of B_{n} are described by the signed cycle type (i.e., each cycle is described by its length and the product of the signs of its entries). Thus, the conjugacy classes are described by ordered pairs of partitions whose total size is n: the first partition consist of the lengths of the positive cycles and the second consists of the lengths of the negative cycles. The irreducible representations of B_{n} are also indexed by these pairs of partitions. See e.g. [23, §I, Appendix B].

For two partitions μ^{1} and μ^{2}, and two infinite sets of independent variables x̄ = x_{1}, x_{2}, … and ȳ = y_{1}, y_{2}, …, define

p_{μ^{1},μ^{2}}(x̄, ȳ) := ∏_{i}(p_{μ_{i}^{1}}(x̄) + p_{μ_{i}^{1}}(ȳ)) · ∏_{j}(p_{μ_{j}^{2}}(x̄) − p_{μ_{j}^{2}}(ȳ)).

Let χ_{μ^{1},μ^{2}}^{λ^{1},λ^{2}} be the irreducible B_{n}-character indexed by (λ^{1}, λ^{2}) evaluated at a conjugacy class of cycle type (μ^{1}, μ^{2}). Then

Frobenius Formula for B_{n}. [23, p. 178] With the above notations, for any bipartition (μ^{1}, μ^{2}) of n

p_{μ^{1},μ^{2}}(x̄, ȳ) = ∑_{λ^{1},λ^{2}} χ_{μ^{1},μ^{2}}^{λ^{1},λ^{2}}s_{λ^{1}}(x̄)s_{λ^{2}}(ȳ),

where the sum runs through all ordered pairs of partitions (λ^{1}, λ^{2}) of total size |λ^{1}| + |λ^{2}| = |μ^{1}| + |μ^{2}| = n.

2.5 The Coinvariant Algebra

The groups S_{n} and B_{n} have natural actions on the ring of polynomials P_{n} (cf. [21, §3.1]). S_{n} acts by permuting the variables, and B_{n} acts by permuting the variables and multiplying by ±1. The ring of S_{n}-invariant polynomials is Λ_{n}, the ring of symmetric functions in x_{1}, …, x_{n}. Similarly, the ring of B_{n}-invariant polynomials is Λ_{n}^{B}, the ring of symmetric functions in x_{1}^{2}, …, x_{n}^{2}. Let I_{n}, I_{n}^{B} be the ideals of P_{n} generated by the elements of Λ_{n}, Λ_{n}^{B} (respectively) without constant term. The quotient P_{n}/I_{n} (P_{n}/I_{n}^{B}) is called the coinvariant algebra of S_{n} (B_{n}). Each group acts naturally on its coinvariant algebra. The resulting representation is isomorphic to the regular representation. See, e.g., [21, §3.6] and [20, §II.3].

Let R_{k} (0 ≤ k ≤ \binom{n}{2}) be the k-th homogeneous component of the coinvariant algebra of S_{n}: P_{n}/I_{n} = ⊕_{k}R_{k}. Each R_{k} is an S_{n}-module. The following theorem is attributed by Reutenauer to Kraskiewicz and Weyman [29, p. 215].

Kraskiewicz-Weyman Theorem. [29, Theorem 8.8] For any 0 ≤ k ≤ \binom{n}{2} and μ ⊢ n, the multiplicity in R_{k} of the irreducible S_{n}-representation corresponding to μ is

m_{k,μ} = | { T ∈ SY T (μ) | maj (T) = k } |.

The following B-analogue (in different terminology) was proved in [39]. Here R_{k}^{B} is the k-th homogeneous component of the coinvariant algebra of B_{n}.

Stembridge’s Theorem. For any 0 ≤ k ≤ n^{2} and bipartition (μ^{1}, μ^{2}) of n, the multiplicity in R_{k}^{B} of the irreducible B_{n}-representation corresponding to (μ^{1}, μ^{2}) is

m_{k,μ^{1},μ^{2}} = | { T ∈ SY T (μ^{1}, μ^{2}) | fmaj (T) = k } |.

2.6 The Garsia-Stanton Descent Basis (Type A)

For any π ∈ S_{n} define the monomial

a_{π} := ∏_{j∈Des(π)}(x_{π(1)} ⋯ x_{π(j)}).

Using Stanley-Reisner rings Garsia and Stanton showed that the set {a_{π} + I_{n} | π ∈ S_{n}} forms a basis for the coinvariant algebra of type A [17]. This basis will be called the descent basis. Unfortunately, this approach does not give a basis for the coinvariant algebras of other Weyl groups. In this paper we shall find an analogue of the descent basis for B_{n}. Our approach will allow us to obtain refinements of the Kraskiewicz-Weyman and Stembridge theorems.

3 Construction of Descent Representations

In this section we introduce several combinatorial concepts (Subsections 3.1 and 3.2). Then we present a straightening algorithm for expanding an arbitrary monomial in P_{n} in terms of descent basis elements, with coefficients from Λ_{n} (Subsections 3.3 and 3.4). This algorithm is essentially equivalent to the one presented by Allen [4], but our formulation leads naturally to descent representations and may be extended to type B. See Subsections 3.5 and 5.2 below.

3.1 Descent Statistics

In this section we introduce a family of descent statistics for sequences of letters from a linearly ordered alphabet.

Let Σ be a linearly ordered alphabet. For any sequence σ of n letters from Σ define

d_{i}(σ) := |{j ∈ Des(σ) : j ≥ i}| (1 ≤ i ≤ n),

the number of descents in σ from position i on.

Clearly,

(3.1)  d_{1}(σ) = des(σ)

and

(3.2)  ∑_{i=1}^{n} d_{i}(σ) = maj (σ).

Note that d_{n}(σ) = 0.

Also, for any sequence σ from Σ

(3.3)  d_{i}(σ) ≥ d_{i+1}(σ),  1 ≤ i < n;

(3.4)  d_{i}(σ) = d_{i+1}(σ) ⟹ σ(i) < σ(i + 1);

(3.5)  d_{i}(σ) > d_{i+1}(σ) ⟹ σ(i) > σ(i + 1) and d_{i}(σ) = d_{i+1}(σ) + 1.

It follows from (3.2) and (3.3) that the sequence (d_{1}(σ), d_{2}(σ), …, d_{n}(σ)) is a partition of maj (σ). It follows from (3.4) and (3.5) that this sequence uniquely determines the descent set of σ. More explicitly, the partition conjugate to (d_{1}(σ), …, d_{n}(σ)) is obtained by writing the elements of Des(σ) in decreasing order.

Finally, for a permutation π ∈ S_{n}, let a_{π} be the corresponding descent basis element. Then, clearly,

a_{π} = ∏_{i=1}^{n} x_{π(i)}^{d_{i}(π)}.

3.2 The Index Permutation and the Exponent Partition

The index permutation of a monomial m = ∏_{i=1}^{n} x_{i}^{p_{i}} ∈ P_{n} is the unique permutation π = π(m) ∈ S_{n} such that

(1)  p_{π(i)} ≥ p_{π(i+1)}  (1 ≤ i < n)

and

(2)  p_{π(i)} = p_{π(i+1)} ⟹ π(i) < π(i + 1).

In other words, π reorders the variables x_{i} by (weakly) decreasing exponents, where the variables with a given exponent are ordered by increasing indices.

Let m = ∏_{i=1}^{n} x_{i}^{p_{i}} be a monomial in P_{n}, π = π(m) its index permutation, and a_{π} the corresponding descent basis element. Then m = ∏_{i=1}^{n} x_{π(i)}^{p_{π(i)}} and

a_{π} = ∏_{i=1}^{n} x_{π(i)}^{d_{i}(π)}.

Claim 3.1 The sequence (p_{π(i)} − d_{i}(π))_{i=1}^{n} of exponents in m/a_{π} consists of nonnegative integers, and is weakly decreasing:

p_{π(i)} − d_{i}(π) ≥ p_{π(i+1)} − d_{i+1}(π) (1 ≤ i < n).

Proof. If π(i) < π(i + 1) then d_{i}(π) = d_{i+1}(π) and the claim follows from p_{π(i)} ≥ p_{π(i+1)}. If π(i) > π(i+1) then d_{i}(π) = d_{i+1}(π)+1 and p_{π(i)} > p_{π(i+1)}, so that p_{π(i)} − p_{π(i+1)} ≥ 1 = d_{i}(π) − d_{i+1}(π). This proves monotonicity. Nonnegativity now follows from d_{n}(π) = 0, implying p_{π(n)} − d_{n}(π) ≥ 0.

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For a monomial m = ∏_{i=1}^{n} x_{i}^{p_{i}} with index permutation π ∈ S_{n}, let

λ(m) := (p_{π(1)}, p_{π(2)}, …, p_{π(n)})

be its exponent partition. Note that λ(m) is a partition of the total degree of m.

Define the complementary partition μ(m) of a monomial m to be the partition conjugate to the partition (p_{π(i)} − d_{i}(π))_{i=1}^{n}. Namely,

μ_{j} := |{i | p_{π(i)} − d_{i}(π) ≥ j}| (∀j ≥ 1).

Example. Let m = x_{1}^{2}x_{2}^{4}x_{3}^{2}x_{5}x_{6}^{3} and n = 7. Then m = x_{2}^{4}x_{6}^{3}x_{1}^{2}x_{3}^{2}x_{5}^{1}x_{4}^{0}x_{7}^{0}, λ(m) = (4, 3, 2, 2, 1, 0, 0), π = 2613547 ∈ S_{7}, (d_{1}(π), …, d_{7}(π)) = (2, 2, 1, 1, 1, 0, 0), and μ(m) = (4, 1, 0, 0, 0, 0, 0).

3.3 A Partial Order for Monomials

We shall now define a partial order on the monomials in P_{n}. Comparable monomials will always have the same total degree. Fix such a total degree p; we may assume p ≥ 1.

Definition. For monomials m_{1}, m_{2} of the same total degree p , m_{1} ≺ m_{2} if one of the following holds :

(1) λ(m_{1}) ◁ λ(m_{2}) (strictly smaller in dominance order); or

(2) λ(m_{1}) = λ(m_{2}) and inv(π(m_{1})) > inv(π(m_{2})).

The partial order “≺” may be replaced, in this paper, by any linear extension of it; for example, by the linear extension obtained by replacing the dominance order by lexicographic order in (1). (This is common in the definition of Gröbner bases, which partially motivated our definition.)

3.4 Straightening

Lemma 3.2 Let m ∈ P_{n} be a monomial and let 1 ≤ k ≤ n. Let S^{(k)} be the set of all monomials which appear (with coefficient 1) in the expansion of the polynomial m · e_{k}, the product of m and the k-th elementary symmetric function. Let

m^{(k)} := m · x_{π(1)} · x_{π(2)} ⋯ x_{π(k)},

where π = π(m) is the index permutation of m. Then :

(1) m^{(k)} ∈ S^{(k)}.

(2) m′ ∈ S^{(k)} and m′ ≠ m^{(k)} ⟹ m′ ≺ m^{(k)} .

Proof. (1) is obvious. For (2), let m′ ∈ S^{(k)}. Then m′ = m · x_{i_{1}}x_{i_{2}} ⋯ x_{i_{k}}, where 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ n. Now, let λ_{i}(m) be the i-th part of the exponent partition of the monomial m. Then

λ_{i}(m^{(k)}) − λ_{i}(m) = \{ \begin{matrix}1, & if 1 ≤ i ≤ k; \\ 0, & if i > k, \end{matrix}

and also

λ_{i}(m′) − λ_{i}(m) ∈ {0, 1} (∀i);

∑_{i=1}^{n}[λ_{i}(m′) − λ_{i}(m)] = k.

Thus :

∑_{i=1}^{t} λ_{i}(m′) ≤ ∑_{i=1}^{t}[λ_{i}(m) + 1] = ∑_{i=1}^{t} λ_{i}(m^{(k)}) (1 ≤ t ≤ k),

∑_{i=1}^{t} λ_{i}(m′) ≤ k + ∑_{i=1}^{t} λ_{i}(m) = ∑_{i=1}^{t} λ_{i}(m^{(k)}) (k < t ≤ n).

Therefore

λ(m′) \underline{◁} λ(m^{(k)}).

If λ(m′) ◁ λ(m^{(k)}) then m′ ≺ m^{(k)}, as claimed. Otherwise, λ(m′) = λ(m^{(k)}). Now let

I_{>} := {1 ≤ i ≤ n | λ_{i}(m) > λ_{k}(m)};
I_{=} := {1 ≤ i ≤ n | λ_{i}(m) = λ_{k}(m)};
I_{<} := {1 ≤ i ≤ n | λ_{i}(m) < λ_{k}(m)}.

Namely, i ∈ I_{>} if the exponent (in m) of x_{π(i)} is strictly larger than that of x_{π(k)}; and similarly for I_{=} and I_{<}. By definition,

π(m^{(k)}) = π(m).

Since λ(m′) = λ(m^{(k)}), it follows that π(m′) and π(m) agree on I_{<} ∪ I_{>}, but may differ on I_{=}. Also, π(m^{(k)}) = π(m) is monotone increasing on I_{=}. It follows that if m′ ≠ m^{(k)} then inv(π(m′)) > inv(π(m^{(k)})) and thus m′ ≺ m^{(k)} in the monomial partial order.

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Lemma 3.3 Let m_{1} and m_{2} be monomials in P_{n} of the same total degree, and let e_{k} be the k-th elementary symmetric function, 1 ≤ k ≤ n. If m_{1} ≺ m_{2} then

m_{1}^{(k)} ≺ m_{2}^{(k)},

where m_{i}^{(k)} (i = 1, 2) is the ≺-maximal summand in m_{i}e_{k} whose existence is guaranteed by Lemma 4.2.

Proof. By the definition of m_{i}^{(k)}: π(m_{i}^{(k)}) = π(m_{i}) and λ(m_{i}^{(k)}) = λ(m_{i}) + δ^{(k)}, where δ^{(k)} = (1, …, 1, 0, …, 0) (k ones, n − k zeroes). Using this with the definition of ≺ shows that m_{1} ≺ m_{2} ⟹ m_{1}^{(k)} ≺ m_{2}^{(k)}.

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Corollary 3.4 Let m ∈ P_{n} be a monomial, π = π(m) its index permutation, and μ = μ(m) the complementary partition defined in Subsection 3.2. Let S be the set of monomials which appear (with nonzero coefficient) in the expansion of a_{π} · e_{μ}. Then :

(1) m ∈ S.

(2) m′ ∈ S and m′ ≠ m ⟹ m′ ≺ m .

Proof. Iterative applications of Lemma 3.2, using π(m) = π(a_{π(m)}) and Lemma 3.3.

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A straightening algorithm follows.

A Straightening Algorithm : For a monomial m ∈ P_{n}, let π = π(m) be its index permutation, a_{π} the corresponding descent basis element, and μ = μ(m) the corresponding complementary partition. Write (by Corollary 3.4)

m = a_{π} · e_{μ} − Σ,

where Σ is a sum of monomials m′ ≺ m. Repeat the process for each m′.

Example. Let m = x_{1}^{2}x_{2}x_{3} ∈ P_{3}. Then π(m) = 123, a_{π(m)} = 1 and μ(m) = (3, 1). Hence

x_{1}^{2}x_{2}x_{3} = 1 · e_{(3,1)} − Σ = (x_{1}x_{2}x_{3})(x_{1} + x_{2} + x_{3}) − Σ,

so Σ = x_{1}x_{2}^{2}x_{3} + x_{1}x_{2}x_{3}^{2}. For the first summand π(x_{1}x_{2}^{2}x_{3}) = 213 and μ(x_{1}x_{2}^{2}x_{3}) = (3). Indeed x_{1}x_{2}^{2}x_{3} = a_{213} · e_{3} = x_{2} · (x_{1}x_{2}x_{3}). Similarly, for the second summand π(x_{1}x_{2}x_{3}^{2}) = 312 and μ(x_{1}x_{2}x_{3}^{2}) = (3), and x_{1}x_{2}x_{3}^{2} = a_{312} · e_{3} = x_{3} · (x_{1}x_{2}x_{3}). We have obtained

x_{1}^{2}x_{2}x_{3} = a_{123} · e_{(3,1)} − a_{213} · e_{3} − a_{312} · e_{3}.

This algorithm implies

Lemma 3.5 (Straightening Lemma) Each monomial m ∈ P_{n} has an expression

m = e_{μ(m)} · a_{π(m)} + ∑_{m′≺m} n_{m′,m}e_{μ(m′)} · a_{π(m′)},

where n_{m′,m} are integers.

The following corollary appears in [4].

Corollary 3.6 The set {a_{π} + I_{n} | π ∈ S_{n}} forms a basis for the coinvariant algebra P_{n}/I_{n}.

Proof. By Lemma 3.5, {a_{π} + I_{n} | π ∈ S_{n}} spans P_{n}/I_{n} as a vector space over Q. Since dim (P_{n}/I_{n}) = |S_{n}| [21, §3.6], this is a basis.

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3.5 Descent Representations

Let λ = (λ_{1}, …, λ_{n}) be a partition with at most n parts. Let P \underline{◁} ; λ be the subspace of the polynomial ring P_{n} = Q[x_{1}, …, x_{n}] spanned by all monomials whose exponent partition is dominated by λ:

P \underline{◁} ; λ := span_{Q}\{m | λ(m) \underline{◁} λ\}.

Similarly, define P_{λ}^{◁} by strict dominance:

P_{λ}^{◁} := span_{Q}{m | λ(m) ◁ λ}.

These subspaces are S_{n}-modules consisting of homogeneous polynomials of degree k = |λ|.

Consider now the canonical projection of P_{n} onto the coinvariant algebra

ψ : P_{n} ⟶ P_{n}/I_{n}.

Define R_{λ} to be a quotient of images under this map:

R_{λ} := ψ(P \underline{◁} ; λ)/ψ(P_{λ}^{◁}).

Then R_{λ} is also an S_{n}-module. We will show that every homogeneous component of the coinvariant algebra may be decomposed into a direct sum of certain R_{λ}-s.

For any subset S ⊆ {1, …, n} define a partition

λ_{S} := (λ_{1}, …, λ_{n})

by

λ_{i} := | S ∩ {i, …, n} | (1 ≤ i ≤ n).

For example, for any π ∈ S_{n}

λ_{Des(π)} = (d_{1}(π), …, d_{n}(π)).

Hence Claim 3.7 For any permutation π ∈ S_{n}

λ(a_{π}) = λ_{Des(π)}.

The Straightening Lemma (Lemma 3.5) implies

Lemma 3.8 For any two permutations τ and π in S_{n}, the action of τ on the monomial a_{π} ∈ P_{n} has the expression

τ(a_{π}) = ∑_{\{w∈S_{n} | λ(a_{w}) \underline{◁} λ(a_{π})\}} n_{w}a_{w} + p,

where n_{w} ∈ Z and p ∈ I_{n}.

Proof. Apply the Straightening Lemma (Lemma 3.5) to m = τ(a_{π}). Note that e_{μ(m′)} ∉ I_{n} iff μ(m′) = ∅, and then m′ = a_{π(m′)}. Denoting w := π(m′) we get : m′ ⪯ m ⟹ λ(a_{w}) = λ(m′) \underline{◁} λ(m) = λ(τ(a_{π})) = λ(a_{π}).

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Another description of the S_{n}-module ψ(P \underline{◁} ; λ) follows.

Lemma 3.9 For every partition λ

ψ(P \underline{◁} ; λ) = span_{Q}\{a_{π} + I_{n} | π ∈ S_{n}, λ(a_{π}) \underline{◁} λ \}.

Proof. Clearly, for any π ∈ S_{n} with λ(a_{π}) \underline{◁} λ, a_{π} + I_{n} = ψ(a_{π}) ∈ ψ(P \underline{◁} ; λ). The opposite inclusion follows from Lemma 3.5.

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Let

J\underline{◁} ; λ := span_{Q}\{a_{π} + I_{n} | π ∈ S_{n}, λ(a_{π}) \underline{◁} λ \}

and

J_{λ}^{◁} := span_{Q}{a_{π} + I_{n} | π ∈ S_{n}, λ(a_{π}) ◁ λ }.

By Lemma 3.9

R_{λ} = J\underline{◁} ; λ/J_{λ}^{◁}.

Hence

Corollary 3.10 The following conditions on a partition λ = (λ_{1}, …, λ_{n}) are equivalent :

(1) R_{λ} ≠ 0.

(2) λ = λ(a_{π}) for some π ∈ S_{n}.

(3) λ = λ_{S} for some S ⊆ [n − 1].

(4) The difference between consecutive parts of λ is either 0 or 1, i.e., (denoting λ_{n+1} := 0):

λ_{i} − λ_{i+1} ∈ {0, 1} (1 ≤ i ≤ n).

Proof. R_{λ} ≠ 0 if and only if J\underline{◁} ; λ ≠ J_{λ}^{◁}. This happens if and only if there exists a permutation π ∈ S_{n} for which λ(a_{π}) = λ (since, by Corollary 3.6, the various a_{π} + I_{n} are linearly independent). By Claim 3.7, λ(a_{π}) = λ_{S} for S = Des(π). Conversely, any subset of [n − 1] is a descent set for an appropriate permutation. This shows the equivalence of the first three conditions. The equivalence of (3) and (4) follows easily from the definition of λ_{S}.

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From now on denote R_{S} := R_{λ_{S}}.

For π ∈ S_{n}, denote by ā_{π} the image of the descent basis element a_{π}+I_{n} ∈ J\underline{◁} ; λ_{S} in the quotient R_{S}.

Corollary 3.11 For any subset S ⊆ [n − 1], the set

{ ā_{π} | π ∈ S_{n}, Des(π) = S }

forms a basis of R_{S}.

Proof. Follows by elementary linear algebra from Corollary 3.6, Claim 3.7, and the definitions of J\underline{◁} ; S, J_{S}^{◁} and ā_{π}.

□

Recall the notation R_{k} for the k-th homogeneous component of the coinvariant algebra P_{n}/I_{n}.

Theorem 3.12 For every 0 ≤ k ≤ \binom{n}{2},

R_{k} ≅ ⨁_{S} R_{S}

as S_{n}-modules, where the sum is over all subsets S ⊆ [n − 1] such that

∑_{i∈S} i = k.

Proof. Note that a_{π} + I_{n} ∈ R_{k} ⟺ maj (π) = k. It follows, by Corollary 3.6, that {a_{π} + I_{n}|maj (π) = k} is a basis for R_{k}; so that, by Corollary 3.11, R_{k} ≅ ⨁_{S} R_{S} (sum over all S ⊆ [n−1] with ∑_{i∈S} i = k) as vector spaces over Q. By Maschke’s Theorem, if V is a finite dimensional G-module for a finite group G (over a field of characteristic zero) and W ⊆ V is a G-submodule, then V ≅ W ⊕ (V/W) as G-modules. Apply this to the poset

\{J\underline{◁} ; S | S ⊆ [n − 1] , ∑_{i∈S} i = k\}

ordered by the dominance order on the partitions λ_{S}. Using the definition of R_{S}, we get by induction on the poset the required isomorphism.

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4 Decomposition of Descent Representations

In this section we refine the Kraskiewicz-Weyman Theorem.

Theorem 4.1 For any subset S ⊆ [n − 1] and partition μ ⊢ n, the multiplicity in R_{S} of the irreducible S_{n}-representation corresponding to μ is

m_{S,μ} := | { T ∈ SY T (μ) | Des(T) = S } |.

The proof applies an argument of Stanley (cf. [29, Theorem 8.8] and references therein), using symmetric functions. Its use is possible here due to the Straightening Lemma. A key lemma in the proof is an extension of Stanley’s formula.

4.1 A Multivariate Extension of Stanley’s Formula

In this subsection we state and prove a multivariate extension of a well known formula of Stanley for the principal specialization of a Schur function.

Recall that s_{λ}(x_{1}, …, x_{n}) denotes the Schur function corresponding to a partition λ of n. Then

Proposition 4.2 (Stanley’s Formula) [36, Prop. 7.19.11] If λ is a partition of n then

s_{λ}(1, q, q^{2}, …, q^{n−1}) = \frac{∑_{T ∈SY T (λ)} q^{maj (T)}}{(1 − q)(1 − q^{2}) ⋯ (1 − q^{n})},

where T runs through all standard Young tableaux of shape λ.

Define Q[[p_{1}, p_{1}p_{2}, …]] to be the ring of formal power series in countably many variables p_{1}, p_{1}p_{2}, …, p_{1}p_{2} ⋯ p_{k}, …; a linear basis for it consists of the monomials p^{λ} := p_{1}^{λ_{1}} ⋯ p_{n}^{λ_{k}} for all partitions λ = (λ_{1}, …, λ_{k}) (λ_{1} ≥ … ≥ λ_{k} ≥ 0). Let Q[[q_{1}, q_{1}q_{2}, …]] be similarly defined.

Let ι : Q[[p_{1}, p_{1}p_{2}, …]] ⟶ Q[[q_{1}, q_{1}q_{2}, …]] be defined by

(4.1)  ι(p^{λ}) := q^{λ′}  (∀λ),

where λ′ is the partition conjugate to λ. Extend ι by linearity. Note that ι is not a ring homomorphism.

For any standard Young tableau T define

d_{i}(T) := | { j ≥ i | j ∈ Des(T) } | (1 ≤ i ≤ n),

where Des(T) is the set of all descents in T (as in Subsection 2.3).

Lemma 4.3 (Multivariate Formula) If λ is a partition of n then

ι[s_{λ}(1, p_{1}, p_{1}p_{2}, …, p_{1}p_{2} ⋯ p_{i}, …)] = \frac{∑_{T ∈SY T (λ)} ∏_{i=1}^{n} q_{i}^{d_{i}(T)}}{∏_{i=1}^{n}(1 − q_{1}q_{2} ⋯ q_{i})},

where T runs through all standard Young tableaux of shape λ.

Note that Stanley’s formula is the special case obtained by substituting q_{1} = q_{2} = … = q_{n} = q.

Proof. By Claim 2.1

s_{λ}(x_{0}, x_{1}, x_{2}, …) = ∑_{T̂∈RSSY T (λ)} ∏_{i=0}^{∞} x_{i}^{m_{i}(T̂)},

where T̂ runs through all reverse semi-standard Young tableaux, with nonnegative entries, of shape λ; and

m_{i}(T̂) := |{cells in T̂ with entry i}| (∀i ≥ 0).

Letting x_{0} = 1 and

x_{i} = p_{1}p_{2} ⋯ p_{i} (i ≥ 1)

we get

s_{λ}(1, p_{1}, p_{1}p_{2}, …) = ∑_{T̂∈RSSY T (λ)} ∏_{i=1}^{∞} p_{i}^{m_{≥i}(T̂)},

where

m_{≥i}(T̂) := |{cells in T̂ with entry ≥ i}| (∀i).

Of course,

m_{≥0}(T̂) = n.

Let μ(T̂) be the vector (m_{≥1}(T̂), m_{≥2}(T̂), …). Then μ(T̂) is a partition with largest part at most n. Note that the conjugate partition

μ(T̂)′ = (T̂_{1}, T̂_{2}, …, T̂_{n})

is the vector of entries of T̂ in weakly decreasing order.

Thus

(4.2)  ι[s_{λ}(1, p_{1}, p_{1}p_{2}, …)] = ∑_{T̂∈RSSY T (λ)} ∏_{i=1}^{n} qT̂_{i} ; i.

Recall that SY T (λ) and RSSY T (λ) are the sets of standard Young tableaux and reverse semi-standard Young tableaux of shape λ, respectively. For any λ ⊢ n define a map

ϕ_{λ} : RSSY T (λ) ⟶ SY T (λ) × N^{n}

by

ϕ_{λ}(T̂) := (T, Δ),

where, for T̂ ∈ RSSY T (λ), T is a standard Young tableau of shape λ and Δ = (Δ_{1}, …, Δ_{n}) is a sequence of nonnegative integers, defined as follows:

(1) Let (T̂_{1}, …, T̂_{n}) be the vector of entries of T̂, in weakly decreasing order. Then T is the standard Young tableau, of the same shape as T̂, having entry i (1 ≤ i ≤ n) in the same cell in which T̂ has entry T̂_{i}. If some of the entries of T̂ are equal then they necessarily belong to distinct columns, and the corresponding entries of T are then chosen increasing from left to right (i.e., with increasing column indices).

(2) Define

Δ_{i} := T̂_{i} − d_{i}(T) − T̂_{i+1} + d_{i+1}(T) (1 ≤ i ≤ n),

where, by convention, T̂_{n+1} = d_{n+1}(T) := 0.

Example. Let λ = (3, 2, 2) and

T̂ = 6 3 3 3 1 2 0 ∈ RSSY T (λ).

Computing ϕ_{λ}(T̂) = (T, Δ), the first step yields

T = 1 3 4 2 6 5 7 ∈ SY T (λ),

so that Des(T) = {1, 4, 6}. Therefore, (T̂_{1} − d_{1}(T), …, T̂_{7} − d_{7}(T)) = (3, 1, 1, 1, 1, 0, 0) and Δ = (2, 0, 0, 0, 1, 0, 0).

The following claim is easy to verify.

Claim 4.4 (1) ϕ_{λ} is a bijection.

(2) If (T̂_{1}, …, T̂_{n}) is the vector of entries of T̂, in weakly decreasing order,

then

T̂_{i} = d_{i}(T) + ∑_{j≥i} Δ_{j} (1 ≤ i ≤ n),

where (T, Δ) = ϕ_{λ}(T̂).

By Claim 4.4, for any T̂ ∈ RSSY T (λ)

(4.3)  ∏_{i=1}^{n} qT̂_{i} ; i = ∏_{i=1}^{n} q_{i}^{d_{i}(T)} · ∏_{j=1}^{n}(q_{1} ⋯ q_{j})^{Δ_{j}},

where (T, Δ) = ϕ_{λ}(T̂).

Substituting (4.3) into (4.2) we get the formula in the statement of Lemma 4.3.

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4.2 Proof of Theorem 4.1

For a permutation τ ∈ S_{n} let the (graded) trace of its action on the polynomial ring P_{n} be

Tr_{P_{n}}(τ) := ∑_{m}⟨τ(m), m⟩ · q̄^{λ(m)},

where the sum is over all monomials m in P_{n}, λ(m) is the exponent partition of the monomial m, and the inner product is such that the set of all monomials is an orthonormal basis for P_{n}. In particular, ⟨τ(m), m⟩ ∈ {0, 1} (∀τ ∈ S_{n}).

Claim 4.5 If τ ∈ S_{n} is of cycle type μ then the trace of its action on P_{n} is

Tr_{P_{n}}(τ) = ι[p_{μ}(1, p_{1}, p_{1}p_{2}, …)],

where p_{μ} is the power sum symmetric function corresponding to μ, and p_{1}, p_{2}, … are independent variables.

Proof. For a monomial m in P_{n}, ⟨τ(m), m⟩ = 1 (i.e., τ(m) = m) if and only if, for each cycle of τ, the variables x_{i} with indices in that cycle all have the same exponent in m. If μ = (μ_{1}, …, μ_{t}) is the cycle type of τ we thus obtain, for each sequence (e_{1}, …, e_{t}) ∈ N^{t}, a unique monomial m with τ(m) = m and e_{j} as the common exponent for cycle number j (1 ≤ j ≤ t). Thus λ(m) consists of μ_{j} copies of e_{j} (1 ≤ j ≤ t), reordered; and

ι^{−1}[q̄^{λ(m)}] = p̄^{λ(m)′} = ∏_{j=1}^{t}(p_{1} ⋯ p_{e_{j}})^{μ_{j}}.

Summing over all choices of e_{1}, …, e_{t} gives p_{μ}(1, p_{1}, p_{1}p_{2}, …).

□

Recall that for the coinvariant algebra P_{n}/I_{n} we have the descent basis {a_{π} + I_{n} | π ∈ S_{n}}. Define, for τ ∈ S_{n}:

Tr_{P_{n}/I_{n}}(τ) := ∑_{π∈S_{n}}⟨τ(a_{π} + I_{n}), a_{π} + I_{n}⟩ · q̄^{λ(a_{π})},

where the inner product is such that the descent basis is orthonormal. From the Straightening Lemma (Lemma 3.5) it follows that

Claim 4.6 For every n ≥ 1 and every τ ∈ S_{n}

Tr_{P_{n}}(τ) = Tr_{P_{n}/I_{n}}(τ) · ∑_{λ} q̄^{λ},

where the sum is over all partitions λ with at most n parts.

Proof. Replace the monomial basis of P_{n} by the homogeneous polynomial basis {a_{π}e_{μ}| π ∈ S_{n}, μ is a partition with largest part at most n}. The trace Tr_{P_{n}}(τ) is not changed, provided that we now use the inner product for which the new basis is orthonormal. Note that τ(e_{μ}) = e_{μ} (∀τ ∈ S_{n}). Also note that, by the Straightening Lemma, the ≺-maximal monomial m in a_{π}e_{μ} has an exponent partition λ(m) = λ(a_{π}) + μ′, where addition of partitions is componentwise.

□

Observation 4.7

∑_{λ} q̄^{λ} = ∏_{i=1}^{n} \frac{1}{1 − q_{1} ⋯ q_{i}} .

Using the Frobenius formula, the multivariate extension of Stanley’s formula (Lemma 4.3) and Claim 4.5, we obtain for every permutation τ ∈ S_{n} of cycle type μ:

Tr_{P_{n}}(τ) = ι[p_{μ}(1, p_{1}, p_{1}p_{2}, …)] =
= ∑_{λ⊢n} χ_{μ}^{λ}ι[s_{λ}(1, p_{1}, p_{1}p_{2}, …)] =

= ∑_{λ⊢n} χ_{μ}^{λ} · \frac{∑_{T ∈SY T (λ)} ∏_{i=1}^{n} q_{i}^{d_{i}(T)}}{∏_{i=1}^{n}(1 − q_{1}q_{2} ⋯ q_{i})}.

By Claim 4.6 and Observation 4.7 we now get

Tr_{P_{n}/I_{n}}(τ) = ∑_{λ⊢n} χ_{μ}^{λ} ∑_{T ∈SY T (λ)} ∏_{i=1}^{n} q_{i}^{d_{i}(T)}.

We conclude that the graded multiplicity in P_{n}/I_{n} of the irreducible S_{n}- representation corresponding to λ is

∑_{T ∈SY T (λ)} ∏_{i=1}^{n} q_{i}^{d_{i}(T)} = ∑_{T ∈SY T (λ)} q̄^{λ_{Des(T)}}.

Consider now the claim of Theorem 3.12. Its proof shows that

P_{n}/I_{n} ≅ ⨁_{S⊆[n−1]} R_{S}

actually holds as an isomorphism of graded S_{n}-modules. By Corollary 3.11 and Claim 3.7, R_{S} is the homogeneous component of multi-degree λ_{S} in P_{n}/I_{n}. It thus follows that the multiplicity in R_{S} of the irreducible S_{n}- representation corresponding to λ is

| {T ∈ SY T (λ) | Des(T) = S} |,

and the proof of Theorem 4.1 is complete.

□

5 Descent Representations for Type B

In this section we give B-analogues of the concepts and results in Sections 3 and 4.

5.1 The Signed Descent Basis

For any signed permutation σ ∈ B_{n} let

Des(σ) := {i ∈ [n − 1] : σ(i) > σ(i + 1)}

be the set of descents in σ with respect to the standard linear order on the integers, and let

d_{i}(σ) := |{j ∈ Des(σ) : j ≥ i}| (1 ≤ i ≤ n)

be the number of descents in σ from position i on.

Also let

ε_{i}(σ) := \{ \begin{matrix}1, & if σ(i) < 0, \\ 0, & otherwise, \end{matrix}

and

f_{i}(σ) := 2d_{i}(σ) + ε_{i}(σ).

Note that

∑_{i=1}^{n} f_{i}(σ) = fmaj (σ).

Finally, associate to σ the monomial

b_{σ} := ∏_{i=1}^{n} x_{|σ(i)|}^{f_{i}(σ)}.

We will show that the set {b_{σ} + I_{n}^{B} | σ ∈ B_{n}} forms a linear basis for the coinvariant algebra of type B. We call it the signed descent basis.

5.2 Straightening

The signed index permutation of a monomial m = ∏_{i=1}^{n} x_{i}^{p_{i}} ∈ P_{n} is the unique signed permutation σ = σ(m) ∈ B_{n} such that

(1)  p_{|σ(i)|} ≥ p_{|σ(i+1)|}  (1 ≤ i < n),
(2)  p_{|σ(i)|} = p_{|σ(i+1)|} ⟹ σ(i) < σ(i + 1)

and

(3)  p_{|σ(i)|} ≡ 0(mod 2) ⟺ σ(i) > 0.

In other words, σ reorders the variables x_{i} as does the corresponding index permutation of type A (see Subsection 3.2), after attaching a minus sign to (indices of) variables with odd exponents. Note that this reverses the order of “negative” indices.

Example. m = x_{1}^{2}x_{2}^{3}x_{3}^{2}x_{5}x_{6}^{3} and n = 7. Then m = x_{2}^{3}x_{6}^{3}x_{1}^{2}x_{3}^{2}x_{5}^{1}x_{4}^{0}x_{7}^{0} and σ(m) = [−6, −2, 1, 3, −5, 4, 7] ∈ B_{7}.

Let m = ∏_{i=1}^{n} x_{i}^{p_{i}} be a monomial in P_{n}, σ = σ(m) its signed index permutation, and b_{σ} the corresponding signed descent basis element. In analogy with Claim 3.1 (for type A) we have

Claim 5.1 The sequence (p_{|σ(i)|} − f_{i}(σ))_{i=1}^{n} of exponents in m/b_{σ} consists of nonnegative even integers, and is weakly decreasing:

p_{|σ(i)|} − f_{i}(σ) ≥ p_{|σ(i+1)|} − f_{i+1}(σ) (1 ≤ i < n).

Proof. By condition (3) above ε_{i}(σ) = 0 ⟺ p_{|σ(i)|} ≡ 0(mod 2). The numbers p_{|σ(i)|} −f_{i} = p_{|σ(i)|} −ε_{i}(σ)−2d_{i}(σ) are therefore even integers. Also d_{n}(σ) = 0 so that p_{|σ(n)|} − f_{n} = p_{|σ(n)|} − ε_{n}(σ) ≥ 0. It remains to show that the sequence is weakly decreasing.

If σ(i) < σ(i + 1) then d_{i}(σ) = d_{i+1}(σ), and p_{|σ(i+1)|} − ε_{i+1}(σ) ≥ p_{|σ(i)|} − ε_{i}(σ) since their difference is an even integer ≥ −1.

If σ(i) > σ(i + 1) and ε_{i}(σ) = ε_{i+1}(σ) then d_{i}(σ) = d_{i+1}(σ) + 1; so (p_{|σ(i+1)|} − f_{i+1}(σ)) − (p_{|σ(i)|} − f_{i}(σ)) = p_{|σ(i+1)|} − p_{|σ(i)|} − 2 ≥ 0 since p_{|σ(i+1)|} ≡ p_{|σ(i+1)|}(mod 2) and because of conditions (1) and (2) above.

Finally, if σ(i) > σ(i + 1) and ε_{i}(σ) ≠ ε_{i+1}(σ) then σ(i) > 0 > σ(i + 1) so that ε_{i}(σ) = 0, ε(i + 1) = 1 and d_{i}(σ) = d_{i+1}(σ) + 1. Thus (p_{|σ(i+1)|} − f_{i+1}(σ)) − (p_{|σ(i)|} − f_{i}(σ)) = p_{|σ(i+1)|} − p_{|σ(i)|} − 1 ≥ 0.

□

Denote by μ_{B}(m) the partition conjugate to (\frac{p_{|σ(i)|}−f_{i}(σ)}{2})_{i=1}^{n} .

Definition: For monomials m_{1}, m_{2} ∈ P_{n} of the same total degree, m_{1} ≺_{B} m_{2} if, for every 1 ≤ i ≤ n, the exponents of x_{i} in m_{1} and m_{2} have the same parity; and also either

(1) λ(m_{1}) ◁ λ(m_{2}) (strictly smaller in dominance order); or

(2) λ(m_{1}) = λ(m_{2}) and inv(π(m_{1})) > inv(π(m_{2})).

Here π(m_{i}) is the (unsigned) index permutation of m_{i} as in Subsection 3.2.

Imitating the proof of Lemma 3.5 we obtain

Corollary 5.2 (Straightening Lemma) Each monomial m ∈ P_{n} has an expression

m = e_{μ_{B}(m)}(x_{1}^{2}, …, x_{n}^{2}) · b_{σ(m)} + ∑_{m′≺_{B}m} n_{m′,m}e_{μ_{B}(m′)}(x_{1}^{2}, …, x_{n}^{2}) · b_{σ(m′)},

where n_{m′,m} are integers.

5.3 Construction of Descent Representations

Recall (from Subsection 2.5) the definition of I_{n}^{B}.

Corollary 5.3 The set

{b_{σ} + I_{n}^{B} | σ ∈ B_{n}}

forms a basis for the coinvariant algebra P_{n}/I_{n}^{B}.

Proof. Similar to the proof of Corollary 3.6.

□

Remark. A different proof of Corollary 5.3 may be obtained by combining and extending the proofs of [5, Theorems 1(3) and 2].

Corollary 5.4 For any pair of elements τ and σ in B_{n}, the action of τ on b_{σ} ∈ P_{n} has the expression

τ(b_{σ}) = ∑_{\{w∈B_{n} | λ(b_{w}) \underline{◁} λ(b_{σ})\}} n_{w}b_{w} + p,

where n_{w} ∈ Z and p ∈ I_{n}^{B}.

Proof. The same as the proof of Lemma 3.8. □

Note that, actually, n_{w} ≠ 0 in Corollary 5.4 only if b_{w} and b_{σ} have the same number of odd exponents.

Note: Unlike the case of type A, for σ ∈ B_{n}, λ(b_{σ}) and λ_{Des(σ)} are not necessarily equal. Indeed, for subsets S_{1} ⊆ [n − 1] and S_{2} ⊆ [n] define a vector λ_{S_{1},S_{2}} by

λ_{S_{1},S_{2}} := 2λ_{S_{1}} + 1_{S_{2}},

where λ_{S_{1}} is as in Subsection 3.5 above, 1_{S_{2}} ∈ {0, 1}^{n} is the characteristic vector of S_{2}, and addition is componentwise. Thus

λ_{S_{1},S_{2}}(i) = 2 · |{j ≥ i| j ∈ S_{1}}| + 1_{S_{2}}(i).

It should be noted that λ_{S_{1},S_{2}} is not necessarily a partition.

Claim 5.5 For any σ ∈ B_{n}

λ(b_{σ}) = λ_{S_{1},S_{2}}

where Des(σ) = S_{1} and Neg(σ) = S_{2}.

Define

R_{λ}^{B} := ψ^{B}(P \underline{◁} ; λ) / ψ^{B}(P_{λ}^{◁}),

where ψ^{B} : P_{n} ⟶ P_{n}/I_{n}^{B} is the canonical map from P_{n} onto the coinvariant algebra of type B, and P \underline{◁}, P^{◁} are as in Subsection 3.5.

By arguments similar to those given in the proof of Corollary 3.9, R_{λ}^{B} may be described via the signed descent basis :

R_{λ}^{B} = J\underline{◁} ; λ,B/J_{λ,B}^{◁},

where

J\underline{◁} ; λ,B := span_{Q}\{b_{σ} + I_{n}^{B} | σ ∈ B_{n}, λ(b_{σ}) \underline{◁} λ\},

and

J_{λ,B}^{◁} := span_{Q}{b_{σ} + I_{n}^{B} | σ ∈ B_{n}, λ(b_{σ}) ◁ λ}.

Corollary 5.6 The following conditions on a partition λ = (λ_{1}, …, λ_{n}) are equivalent :

(1) R_{λ}^{B} ≠ 0.

(2) λ = λ(b_{σ}) for some σ ∈ B_{n}.

(3) λ = λ_{S_{1},S_{2}} for some S_{1} ⊆ [n−1] and S_{2} ⊆ [n], and λ_{S_{1},S_{2}} is a partition.

(4) The difference between consecutive parts of λ is either 0, 1 or 2, i.e.,

(denoting λ_{n+1} := 0):

λ_{i} − λ_{i+1} ∈ {0, 1, 2} (1 ≤ i ≤ n).

Proof. Similar to the proof of Corollary 3.10. □

From now on, denote R_{S_{1},S_{2}}^{B} := R_{λ_{S_{1},S_{2}}} . For σ ∈ B_{n}, denote by b̄_{σ} the image of the signed descent basis element b_{σ} +I_{w}^{B} ∈ J\underline{◁} ; λ_{S_{1},S_{2}},B in the quotient R_{S_{1},S_{2}}. Corollary 5.7 For any S_{1} ⊆ [n − 1] and S_{2} ⊆ [n], the set

{ b̄_{σ} | σ ∈ B_{n}, Des(σ) = S_{1}, Neg(σ) = S_{2}}

forms a basis of R_{S_{1},S_{2}}.

Recall the notation R_{k}^{B} for the k-th homogeneous component of P_{n}/I_{n}^{B}. Theorem 5.8 For every 0 ≤ k ≤ n^{2},

R_{k}^{B} ≅ ⨁_{S_{1},S_{2}} R_{S_{1},S_{2}},

as B_{n}-modules, where the sum is over all subsets S_{1} ⊆ [n − 1] and S_{2} ⊆ [n] such that λ_{S_{1},S_{2}} is a partition and

2 · ∑_{i∈S_{1}} i + |S_{2}| = k.

Proof. Similar to the proof of Theorem 3.12. □

5.4 Decomposition of Descent Representations

In this section we give a type B analogue of Theorem 4.1. Theorem 5.9 For any pair of subsets S_{1} ⊆ [n − 1], S_{2} ⊆ [n], and a bipartition (μ^{1}, μ^{2}) of n, the multiplicity of the irreducible B_{n}-representation corresponding to (μ^{1}, μ^{2}) in R_{S_{1},S_{2}} is

m_{S_{1},S_{2},μ^{1},μ^{2}} := | { T ∈ SY T (μ^{1}, μ^{2}) | Des(T) = S_{1} and Neg(T) = S_{2} } |.

Again, we need a (type B) multivariate extension of Proposition 4.2 (Stanley’s formula). Recall the definition (4.1) of the mapping ι.

Let (λ^{1}, λ^{2}) be a bipartition of n. Recall the definitions of a standard Young tableau T = (T^{1}, T^{2}) of shape (λ^{1}, λ^{2}) and the sets Des(T) and Neg(T) from Subsection 2.3. Denote the set of all standard tableaux of shape (λ^{1}, λ^{2}) by SY T (λ^{1}, λ^{2}). A reverse semi-standard Young tableau of shape (λ^{1}, λ^{2}) is a pair (T̂^{1}, T̂^{2}) of reverse semi-standard Young tableaux, where: T̂^{i} has shape λ^{i} (i = 1, 2); the entries of T̂^{i} are congruent to i − 1 (mod 2); the entries in each row are weakly decreasing; and the entries in each column are strictly decreasing. Denote by RSSY T (λ^{1}, λ^{2}) the set of all such tableaux.

For any standard Young tableau T = (T^{1}, T^{2}) of shape (λ^{1}, λ^{2}) define

d_{i}(T) := |{j ≥ i | i ∈ Des(T)},
ε_{i}(T) := \{ \begin{matrix}1, & if i ∈ Neg(T), \\ 0, & otherwise, \end{matrix}

and

f_{i}(T) := 2d_{i}(T) + ε_{i}(T).

Then

fmaj (T) = ∑_{i=1}^{n} f_{i}(T).

Lemma 5.10 If (λ^{1}, λ^{2}) is a bipartition of n then

ι[s_{λ^{1}}(1, p_{1}p_{2}, p_{1}p_{2}p_{3}p_{4}, …) · s_{λ^{2}}(p_{1}, p_{1}p_{2}p_{3}, …)] =

= \frac{∑_{T ∈SY T (λ^{1},λ^{2})} ∏_{i=1}^{n} q_{i}^{f_{i}(T)}}{∏_{i=1}^{n}(1 − q_{1}^{2}q_{2}^{2} ⋯ q_{i}^{2})} ,

where T runs through all standard Young tableaux of shape (λ^{1}, λ^{2}).

Proof. By Claim 2.1

s_{λ}(x_{0}, x_{1}, x_{2}, …) = ∑_{T̂∈RSSY T (λ)} ∏_{i=0}^{∞} x_{i}^{m_{i}(T̂)},

where T̂ runs through all reverse semi-standard Young tableaux of shape λ, and

m_{i}(T̂) := |{cells in T̂ with entry i}| (∀i ≥ 0).

Recall that, for (T̂^{1}, T̂^{2}) ∈ RSSY T (λ^{1}, λ^{2}) we have T̂^{1}/2 ∈ RSSY T (λ^{1}), where T̂^{1}/2 is obtained from T̂^{1} by dividing each entry by 2, and similarly (T̂^{2} − 1)/2 ∈ RSSY T (λ^{2}). Thus

s_{λ^{1}}(x_{0}, x_{1}, …) · s_{λ^{2}}(y_{0}, y_{1}, …) = ∑_{(T̂^{1},T̂^{2})∈RSSY T (λ^{1},λ^{2})} ∏_{i=1}^{∞} x_{i}^{m_{2i}(T̂^{1})}y_{i}^{m_{2i+1}(T̂^{2})}.

Letting

x_{0} = 1

and

x_{i} = p_{1}p_{2} ⋯ p_{2i−1}p_{2i} (i ≥ 1)

for s_{λ^{1}}, while

y_{i} = p_{1}p_{2} ⋯ p_{2i+1} (i ≥ 0)

for s_{λ^{2}}, we get

s_{λ^{1}}(1, p_{1}p_{2}, …) · s_{λ^{2}}(p_{1}, p_{1}p_{2}p_{3}, …) = ∑_{(T̂^{1},T̂^{2})∈RSSY T (λ^{1},λ^{2})} ∏_{i=1}^{∞} p_{i}^{m_{≥i}(T̂^{1},T̂^{2})},

where

m_{≥i}(T̂^{1}, T̂^{2}) := |{cells in (T̂^{1}, T̂^{2}) with entry ≥ i}| (∀i ≥ 0).

Let μ(T̂^{1}, T̂^{2}) be the vector (m_{≥1}(T̂^{1}, T̂^{2}), m_{≥2}(T̂^{1}, T̂^{2}), …). Then μ(T̂^{1}, T̂^{2}) is a partition with largest part at most n. The conjugate partition

μ(T̂^{1}, T̂^{2})′ = ((T̂^{1}, T̂^{2})_{1}, (T̂^{1}, T̂^{2})_{2}, …, (T̂^{1}, T̂^{2})_{n})

is the vector of entries of (T̂^{1}, T̂^{2}) in weakly decreasing order.

Thus

(5.1) ι[s_{λ^{1}}(1, p_{1}p_{2}, …) · s_{λ^{2}}(p_{1}, …)] = ∑_{(T̂^{1},T̂^{2})∈RSSY T (λ^{1},λ^{2})} ∏_{i=1}^{n} q_{i}^{(T̂^{1},T̂^{2})_{i}}.

For any bipartition (λ^{1}, λ^{2}) of n define a map

ϕ_{(λ^{1},λ^{2})} : RSSY T (λ^{1}, λ^{2}) ⟶ SY T (λ^{1}, λ^{2}) × N^{n}

by

ϕ_{(λ^{1},λ^{2})}(T̂^{1}, T̂^{2}) := ((T^{1}, T^{2}), Δ),

where, for (T̂^{1}, T̂^{2}) ∈ RSSY T (λ^{1}, λ^{2}), (T^{1}, T^{2}) is a standard Young tableau of shape (λ^{1}, λ^{2}) and Δ = (Δ_{1}, …, Δ_{n}) is a sequence of nonnegative integers, defined as follows:

(1) Let ((T̂^{1}, T̂^{2})_{1}, …, (T̂^{1}, T̂^{2})_{n}) be the vector of entries of (T̂^{1}, T̂^{2}), in weakly decreasing order. Then (T^{1}, T^{2}) is the standard Young tableau, of the same shape as (T̂^{1}, T̂^{2}), with entry i (1 ≤ i ≤ n) in the same cell in which (T̂^{1}, T̂^{2}) has entry (T̂^{1}, T̂^{2})_{i}. If some of the entries of (T̂^{1}, T̂^{2}) are equal then they necessarily belong to distinct columns (in the same tableau), and the corresponding entries of (T^{1}, T^{2}) are then chosen as consecutive integers, increasing from left to right (i.e., with increasing column indices).

(2) Define

Δ_{i} := ((T̂^{1}, T̂^{2})_{i}−f_{i}(T^{1}, T^{2})−(T̂^{1}, T̂^{2})_{i+1}+f_{i+1}(T^{1}, T^{2}))/2 (1 ≤ i ≤ n),

where, by convention, (T̂^{1}, T̂^{2})_{n+1} := 0.

Example. Let (λ^{1}, λ^{2}) = ((3, 1), (2, 1)) be a bipartition of 7, and let

(T̂^{1}, T̂^{2}) = ( \begin{matrix}10 & 6 & 2 \\ 2 \end{matrix} , \begin{matrix}9 & 7 \\ 1 \end{matrix} ) ∈ RSSY T (λ^{1}, λ^{2}).

Computing ϕ_{(λ^{1},λ^{2})}(T̂^{1}, T̂^{2}) = ((T^{1}, T^{2}), Δ), the first step yields

(T^{1}, T^{2}) = ( \begin{matrix}1 & 4 & 6 \\ 5 \end{matrix} , \begin{matrix}2 & 3 \\ 7 \end{matrix} ) ∈ SY T (λ^{1}, λ^{2}),

so that Des(T^{1}, T^{2}) = {1, 4, 6} and Neg(T^{1}, T^{2}) = {2, 3, 7}. Therefore, (f_{1}(T^{1}, T^{2}), …, f_{7}(T^{1}, T^{2})) = (6, 5, 5, 4, 2, 2, 1), ((T̂^{1}, T̂^{2})_{1} − f_{1}(T^{1}, T^{2}), …, (T̂^{1}, T̂^{2})_{7} − f_{7}(T^{1}, T^{2})) = (4, 4, 2, 2, 0, 0, 0), and Δ = (0, 1, 0, 1, 0, 0, 0).

The following claim is easy to verify

Claim 5.11 (1) ϕ_{(λ^{1},λ^{2})} is a bijection.

(2) If ((T̂^{1}, T̂^{2})_{1}, …, (T̂^{1}, T̂^{2})_{n}) is the vector of entries of (T̂^{1}, T̂^{2}), in weakly decreasing order, then

(T̂^{1}, T̂^{2})_{i} = f_{i}(T^{1}, T^{2}) + ∑_{j≥i} 2Δ_{j} (1 ≤ i ≤ n)

where ((T^{1}, T^{2}), Δ) = ϕ_{(λ^{1},λ^{2})}(T̂^{1}, T̂^{2}).

By Claim 5.11, for any (T̂^{1}, T̂^{2}) ∈ RSSY T (λ^{1}, λ^{2})

(5.2)  ∏_{i=1}^{n} q_{i}^{(T̂^{1},T̂^{2})_{i}} = ∏_{i=1}^{n} q_{i}^{f_{i}(T^{1},T^{2})} · ∏_{j=1}^{n}(q_{1} ⋯ q_{j})^{2Δ_{j}}.

Substituting (5.2) into (5.1) we get the formula in the statement of Lemma 5.10.

□

5.5 Proof of Theorem 5.9

For a permutation τ ∈ B_{n} let the trace of its action on the polynomial ring P_{n} be

Tr_{P_{n}}(τ) := ∑_{m}⟨τ(m), m⟩ q̄^{λ(m)},

where the sum is over all monomials m in P_{n}, λ(m) is the exponent partition of the monomial m, and the inner product is such that the set of all monomials is an orthonormal basis for P_{n}.

Claim 5.12 If τ ∈ B_{n} is of cycle type (μ^{1}, μ^{2}) then the trace of its action on P_{n} is

Tr_{P_{n}}(τ) = ι[p_{μ^{1},μ^{2}}(x̄, ȳ)],

where

x̄ := (1, p_{1}p_{2}, p_{1}p_{2}p_{3}p_{4}, …)

and

ȳ:= (p_{1}, p_{1}p_{2}p_{3}, …).

Proof. Let μ^{1} = (1^{α_{1}}, 2^{α_{2}}, …) and μ^{2} = (1^{β_{1}}, 2^{β_{2}}, …). Then ⟨τ(m), m⟩ ≠ 0 (and is ±1) iff all variables x_{i} with i in the same cycle of τ have equal exponents in m. The sign depends on the number of negative cycles with odd exponent. Thus

ι^{−1}[Tr_{P_{n}}(τ)] = ∏_{i≥1} (∑_{t≥0}(p_{1} ⋯ p_{t})^{i})^{α_{i}} · ∏_{j≥1} (∑_{s≥0}(−1)^{s}(p_{1} ⋯ p_{s})^{j})^{β_{j}} =

= p_{μ^{1},μ^{2}}(x̄, ȳ).

□

Proof of Theorem 5.9 From the Straightening Lemma for type B (Corollary 5.2) it follows that for any τ ∈ B_{n}

Tr_{R_{n}^{B}}(τ) · ∏_{i=1}^{n} \frac{1}{1 − (q_{1} ⋯ q_{i})^{2}} = Tr_{P_{n}}(τ).

Inserting Claim 5.12 and the Frobenius formula for type B (see Subsection 2.4) we obtain

Tr_{R_{n}^{B}}(τ) = ∏_{i=1}^{n}[1 − (q_{1} ⋯ q_{i})^{2}] · ι[p_{μ^{1},μ^{2}}(x̄, ȳ)] =

= ∏_{i=1}^{n}[1 − (q_{1} ⋯ q_{i})^{2}] · ι[ ∑_{λ^{1},λ^{2}} χ_{μ^{1},μ^{2}}^{λ^{1},λ^{2}}s_{λ^{1}}(x̄)s_{λ^{2}}(ȳ)].

Applying the linearity of ι and Lemma 5.10 gives

Tr_{R_{n}^{B}}(τ) = ∑_{λ^{1},λ^{2}} χ_{μ^{1},μ^{2}}^{λ^{1},λ^{2}} ∑_{(T^{1},T^{2})∈SY T (λ^{1},λ^{2})} ∏_{i=1}^{n} q_{i}^{f_{i}(T^{1},T^{2})}.

□

6 Combinatorial Identities

In this section we apply the above algebraic setting to obtain new combinatorial identities. The basic algebraic-combinatorial tool here is the Hilbert series of polynomial rings with respect to multi-degree rearranged into a weakly decreasing sequence (i.e., a partition). The computation of the Hilbert series follow in general the one presented in [29, §8.3] for total degree; the major difference is the fact that total degree gives a grading of the polynomial ring, whereas rearranged multi-degree only leads to a filtration: P_{n} = ∪_{λ}P \underline{◁} ; λ and P \underline{◁} ; λ · P \underline{◁} ; μ ⊆ P \underline{◁} ; λ+μ. For this reason, not every homogeneous basis for the coinvariant algebra will do: the numerators of the RHS in Theorems 6.2 and 6.5 below are generating functions for the descent basis (signed descent basis, respectively), but other bases (e.g., Schubert polynomials) have different generating functions, and are therefore not appropriate for use here. The success of the argument relies on properties of the descent basis.

6.1 Main Combinatorial Results

For any signed permutation σ ∈ B_{n} let d_{i}(σ), n_{i}(σ), and ε_{i}(σ) have the same meaning as in Subsection 1.2.1.

The main result of this section is

Theorem 6.1 Let n ∈ P. Then

∑_{σ∈B_{n}} ∏_{i=1}^{n} q_{i}^{d_{i}(σ)+n_{i}(σ^{−1})} = ∑_{σ∈B_{n}} ∏_{i=1}^{n} q_{i}^{2d_{i}(σ)+ε_{i}(σ)}.

Proof. Theorem 6.1 is an immediate consequence of Theorems 6.5 and 6.7 below.

□

For any partition λ = (λ_{1}, …, λ_{n}) with at most n positive parts let

m_{j}(λ) := | {1 ≤ i ≤ n | λ_{i} = j} | (∀j ≥ 0),

and let \binom{n}{m̄(λ)} denote the multinomial coefficient \binom{n}{m_{0}(λ),m_{1}(λ),...}.

By considering the Hilbert series of the polynomial ring P_{n} with respect to weakly decreasing multi-degree we obtain

Theorem 6.2 Let n ∈ P. Then

∑_{ℓ(λ)≤n} \binom{n}{m̄(λ)} ∏_{i=1}^{n} q_{i}^{λ_{i}} = \frac{∑_{π∈S_{n}} ∏_{i=1}^{n} q_{i}^{d_{i}(π)}}{∏_{i=1}^{n}(1 − q_{1} ⋯ q_{i})}

in Z[[q_{1}, …, q_{n}]], where the sum on the left-hand side is taken over all partitions with at most n parts.

The theorem will be proved using the following lemma.

Recall from Subsection 3.2 the definitions of the index permutation π(m) and the complementary partition μ(m) of a monomial m ∈ P_{n}.

Lemma 6.3 The mapping m ⟼ (π(m), μ(m)′) is a bijection between the set of all monomials in P_{n} and the set of all pairs (π, μ̃), where π ∈ S_{n} and μ̃ is a partition with at most n parts.

Proof of lemma 6.3. The indicated mapping is clearly into the claimed set, since the largest part of the complementary partition μ(m) is at most n. Conversely, to each pair (π, μ̃) as in the statement of the lemma, associate the ≺-maximal monomial m in the expansion of a_{π} · e_{}_{μ̃′}. Then π(m) = π and μ(m)′ = μ̃. Thus the mapping is a bijection.

□

Proof of Theorem 6.2. Recall the notation λ(m) for the exponent partition of a monomial m ∈ P_{n}. For any partition with n parts denote by q̄^{λ} the product ∏_{i=1}^{n} q_{i}^{λ_{i}}.

For any partition λ with at most n parts, \binom{n}{m̄(λ)} is the number of monomials in P_{n} with exponent partition equal to λ. Therefore the Hilbert series of the polynomial ring P_{n} by exponent partition is equal to the LHS of the theorem.

On the other hand, since λ(m) = λ(a_{π(m)}) + μ(m)′, using Lemma 6.3 we get

∑_{m∈P_{n}} q̄^{λ(m)} = ∑_{m∈P_{n}} q̄^{λ(a_{π(m)})+μ(m)′} = ∑_{π∈S_{n}} q̄^{λ(a_{π})} · ∑_{μ} q̄^{μ},

where μ in the last sum runs through all partitions having at most n parts. By Claim 3.7, this product is equal to the RHS of the theorem.

□

A well known result, attributed to Gessel [18] by Garsia [14], follows.

Corollary 6.4 Let n ∈ P. Then

\frac{∑_{π∈S_{n}} t^{des(π)}q^{maj (π)}}{∏_{i=0}^{n}(1 − tq^{i})} = ∑_{r≥0}[r + 1]_{q}^{n}t^{r}.

in Z[q][[t]].

Proof. Substitute, in Theorem 6.2, q_{1} = qt and q_{2} = q_{3} = … = q_{n} = q, divide both sides by 1 − t, and apply (3.1) and (3.2). We obtain

\frac{∑_{π∈S_{n}} t^{des(π)}q^{maj (π)}}{∏_{i=0}^{n}(1 − tq^{i})} = ∑_{ℓ(λ)≤n} \binom{n}{m̄(λ)} · q^{Σ_{i}λ_{i}} · \frac{t^{λ_{1}}}{1 − t} =

= ∑_{r=0}^{∞} ∑_{ℓ(λ)≤n , λ_{1}≤r} \binom{n}{m̄(λ)}q^{Σ_{i}λ_{i}}t^{r}.

The coefficient of t^{r} is

∑_{ℓ(λ)≤n , λ_{1}≤r} \binom{n}{m̄(λ)}q^{Σ_{i}λ_{i}} = ∑_{(ℓ_{1},...,ℓ_{n})∈[0,r]^{n}} q^{Σ_{i}ℓ_{i}} = (∑_{j=0}^{r} q^{j})^{n} = [r + 1]_{q}^{n}.

□

The Hilbert series of P_{n} rearranged by multi-degree may be computed in a different way, by considering the signed descent basis for the coinvariant algebra of type B and applying the Straightening Lemma for this type.

Theorem 6.5 With notations as in Theorem 6.2

∑_{ℓ(λ)≤n} \binom{n}{m̄(λ)} ∏_{i=1}^{n} q_{i}^{λ_{i}} = \frac{∑_{σ∈B_{n}} ∏_{i=1}^{n} q_{i}^{2d_{i}(σ)+ε_{i}(σ)}}{∏_{i=1}^{n}(1 − q_{1}^{2} ⋯ q_{i}^{2})}

in Z[[q_{1}, …, q_{n}]], where the sum on the left-hand side runs through all partitions with at most n parts.

Recall from Subsection 5.2 the definitions of the signed index permutation σ(m) ∈ B_{n} and the complementary partition μ_{B}(m) of a monomial m ∈ P_{n}.

Lemma 6.6 The mapping m ⟼ (σ(m), μ_{B}(m)′) is a bijection between the set of all monomials in P_{n} and the set of all pairs (σ, μ̃), where σ ∈ B_{n} and μ̃ is a partition with at most n parts.

Proof of lemma 6.6. Similar to the proof of Lemma 6.3. Here to each pair (σ, μ̃) associate the ≺-maximal monomial m in b_{σ} · e_{}_{μ̃′}(x_{1}^{2}, …, x_{n}^{2}), and then σ(m) = σ and μ_{B}(m)′ = μ̃.

□

Proof of Theorem 6.5. Again, the LHS of the theorem is equal to the Hilbert series of the polynomial ring P_{n} by exponent partition. On the other hand, since λ(m) = λ(b_{σ(m)}) + 2μ_{B}(m)′, Lemma 6.6 gives

∑_{m∈P_{n}} q̄^{λ(m)} = ∑_{m∈P_{n}} q̄^{λ(b_{σ(m)})+2μ_{B}(m)′} = ∑_{σ∈B_{n}} q̄^{λ(b_{σ})} · ∑_{μ} q̄^{2μ},

where μ in the last sum runs through all partitions having at most n parts. By Claim 5.5, this product is equal to the RHS of the theorem.

□

Direct combinatorial arguments imply

Theorem 6.7 With notations as in Theorem 6.2

∑_{ℓ(λ)≤n} \binom{n}{m̄(λ)} ∏_{i=1}^{n} q_{i}^{λ_{i}} = \frac{∑_{σ∈B_{n}} ∏_{i=1}^{n} q_{i}^{d_{i}(σ)+n_{i}(σ^{−1})}}{∏_{i=1}^{n}(1 − q_{1}^{2} ⋯ q_{i}^{2})}

in Z[[q_{1}, …, q_{n}]].

Proof. Define the subset T ⊆ B_{n} by

T := {π ∈ B_{n} : des(π) = 0}.

It is clear from our definitions that d_{i}(σu) = d_{i}(u) and n_{i}(u^{−1}σ^{−1}) = n_{i}(σ^{−1}) for all σ ∈ T, u ∈ S_{n} and 1 ≤ i ≤ n. Therefore

∑_{π∈B_{n}} ∏_{i=1}^{n} q_{i}^{d_{i}(π)+n_{i}(π^{−1})} = ∑_{u∈S_{n}} ∑_{σ∈T} ∏_{i=1}^{n} q_{i}^{d_{i}(σu)+n_{i}((σu)^{−1})}

= ∑_{u∈S_{n}} ∑_{σ∈T} ∏_{i=1}^{n} q_{i}^{d_{i}(u)+n_{i}(σ^{−1})}

= ∑_{u∈S_{n}} ∏_{i=1}^{n} q_{i}^{d_{i}(u)} · ∑_{σ∈T} ∏_{i=1}^{n} q_{i}^{n_{i}(σ^{−1})}.

An element σ ∈ T is uniquely determined by the set Neg(σ^{−1}). Hence

∑_{σ∈T} ∏_{i=1}^{n} q_{i}^{n_{i}(σ^{−1})} = ∏_{i=1}^{n}(1 + q_{1} ⋯ q_{i}).

Theorem 6.2 completes the proof. □

6.2 Negative and Flag Statistics

In this subsection we present a central result from [1] and show that it may be obtained as a special case of results from the previous subsection.

For any σ ∈ B_{n} define the negative descent multiset by

NDes(σ) := Des(σ) ⨄ Neg(σ^{−1}),

where Neg(σ) is the set of positions of negative entries in σ, defined in Subsection 1.2.1. Note that NDes(σ) can be defined rather naturally also in purely Coxeter group theoretic terms. In fact, for i ∈ [n] let η_{i} ∈ B_{n} be defined by

η_{i} := [1, 2, …, i − 1, −i, i + 1, …, n],

so η_{1} = s_{0}. Then η_{1}, …, η_{n} are reflections of B_{n} (in the Coxeter group sense; see, e.g., [21]). Clearly

NDes(σ) = {i ∈ [n − 1] : l(σs_{i}) < l(σ)} ⨄{i ∈ [n] : l(σ^{−1}η_{i}) < l(σ^{−1})}.

For σ ∈ B_{n} let ndes(σ) := |NDes(σ)|

and

nmaj (σ) := ∑_{i∈NDes(σ)} i.

Recall the notation

fmaj(σ) = 2 maj(σ) + |Neg(σ)|.

and let

fdes(σ) := 2 des(σ) + ε_{1}(σ).

Then the two pairs of statistics (fdes, fmaj ) and (ndes, nmaj ) are equidistributed over B_{n}.

Corollary 6.8 [1, Corollary 4.5] Let n ∈ P. Then

∑_{σ∈B_{n}} t^{ndes(σ)}q^{nmaj (σ)} = ∑_{σ∈B_{n}} t^{fdes(σ)}q^{fmaj (σ)}.

Proof. Substitute in Theorem 6.1 : q_{1} = qt, q_{2} = … = q_{n} = q, and apply the identities

d_{1}(σ) + n_{1}(σ^{−1}) = ndes(σ);

∑_{ı=1}^{n}(d_{i}(σ) + n_{i}(σ^{−1})) = nmaj (σ);

and

∑_{ı=1}^{n}(2 · d_{i}(σ) + ε_{i}(σ)) = fmaj (σ).

□

Acknowledgments. This paper grew out of stimulating discussions with Dominique Foata and Ira Gessel during the conference “Classical Combinatorics” in honor of Foata’s 65^{th} birthday. The authors are indebted to Ira Gessel for the idea of using the coinvariant algebra in the study of multivariate statistics. Thanks are also due to Eli Bagno and Richard Stanley for useful comments.

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