# 0-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra.

1 Introduction

Let F be any field. The symmetric group [G.sub.n] naturally acts on the polynomial ring F[X] := F[[x.sub.1], ..., [x.sub.n]] by permuting the variables [x.sub.1], ..., [x.sub.n]. The invariant algebra [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which consists of all the polynomials fixed by this 6n-action, is a polynomial algebra generated by the elementary symmetric functions [e.sub.1], ..., [e.sub.n]. The coinvariant algebra [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is a vector space of dimension n! over F, and when F has characteristic larger than n the coinvariant algebra carries the regular representation of [G.sub.n]. A well known basis for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] consists of the descent monomials. Garsia  obtained this basis by transferring a natural basis from the Stanley-Reisner ring F[[B.sub.n]] of the Boolean algebra [B.sub.n] to the polynomial ring F[X]. Here the Boolean algebra [B.sub.n] is the set of all subsets of [n] := {1,2, ..., n} partially ordered by inclusion, and the Stanley-Reisner ring F[[B.sub.n]] is the quotient of the polynomial algebra F [[y.sub.A]: A [subset or equal to] [n]] by the ideal ([y.sub.A][y.sub.B]: A and B are incomparable in [B.sub.n]).

The 0-Hecke algebra [H.sub.n](0) (of type A) is a deformation of the group algebra of [G.sub.n]. It acts on F[X] by the Demazure operators, also known as the isobaric divided difference operators, having the same invariant algebra as the [G.sub.n]-action on F[X]. In our earlier work , we showed that the coinvariant algebra [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also isomorphic to the regular representation of [H.sub.n](0), for any field F, by constructing another basis for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which consists of certain polynomials whose leading terms are the descent monomials. This and the previously mentioned connection between the Stanley-Reisner ring F[[B.sub.n]] and the polynomial ring F[X] motivate us to define an [H.sub.n](0)-action on F[[B.sub.n]].

It turns out that our [H.sub.n](0)-action on F[[B.sub.n]] has similar properties to the [H.sub.n](0)-action on F[X]. It preserves an [N.sup.n+1]-multigrading of F[[B.sub.n]] and has invariant algebra equal to a polynomial algebra F[[THETA]], where [THETA] is the set of rank polynomials [[theta].sub.i] (the usual analogue of [e.sub.i] in F[[B.sub.n]]). We show that the [H.sub.n](0)-action is [THETA]-linear and thus descends to the coinvariant algebra F[[B.sub.n]]/([THETA]). Using the descent monomials in F[[B.sub.n]] it is not hard to see that F[[B.sub.n]]/([THETA]) carries the regular representation of [H.sub.n](0).

It is well known that every finite dimensional (complex) [G.sub.n]-representation is a direct sum of simple (i.e. irreducible) [G.sub.n]-modules, and the simple [G.sub.n]-modules are indexed by partitions [lambda] of n, which correspond to the Schur functions [s.sub.[lambda]] via the Frobenius characteristic map. Hotta-Springer  and Garsia-Procesi  discovered that the cohomology ring of the Springer fiber indexed by a partition p of n is isomorphic to certain quotient ring of F[X], which admits a graded [G.sub.n]-module structure corresponding to the modified Hall-Littlewood symmetric function [[??].sub.[mu]] (x; t) via the Frobenius characteristic map. The coinvariant algebra of [G.sub.n] is nothing but [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In our previous work  we established a partial analogue of the above result by showing that the [H.sub.n](0)-action on F[X] descends to [R.sub.[mu]] if and only if [mu] = ([1.sup.k], n - k) is a hook, and if so then [R.sub.[mu]] has graded quasisymmetric characteristic equal to HM (x; t) and graded noncommutative characteristic [[??].sub.[mu]] (x; t). Here [[??].sub.[alpha]] (x; t) is the noncommutative modified Hall-Littlewood symmetric function introduced by Bergeron and Zabrocki  for any composition [alpha] of n. Using an analogue of the nabla operator Bergeron and Zabrocki  also introduced a (q, t)-analogue [[??].sub.[alpha]](x; q,t) for any composition a. Now we provide in Theorem 1.1 below a complete representation theoretic interpretation for [[??].sub.[alpha]](x; t) and [[??].sub.[alpha]](x; q, t) by the [H.sub.n](0)-action on F[[B.sub.n]].

To state our result, we first recall the two characteristic maps for representations of [H.sub.n](0) introduced by Krob and Thibon , which we call the quasisymmetric characteristic and the noncommutative characteristic. The simple [H.sub.n](0)-modules are indexed by compositions [alpha] of n and correspond to the fundamental quasisymmetric functions [F.sub.[alpha]] via the quasisymmetric characteristic; the projective indecomposable [H.sub.n](0)-modules are also indexed by compositions [alpha] of n and correspond to the noncommutative ribbon Schur functions sa via the noncommutative characteristic. See [section] 2 for details.

Theorem 1.1 Let [alpha] be a composition of n. Then there exists a homogeneous [H.sub.n](0)-invariant ideal [I.sub.[alpha]] of the multigraded algebra F[[B.sub.n]] such that the quotient algebra F[[B.sub.n]]/[I.sub.[alpha]] becomes a projective [H.sub.n](0)-module with multigraded noncommutative characteristic equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One has [[??].sub.[alpha]](x; t, [t.sup.2], ..., [t.sup.n-1]) = [[??].sub.[alpha]](x; t), and obtains [[??].sub.[alpha]](x; q, t) from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by taking [t.sub.i] = [t.sup.i] for all i [member of] D([alpha]), and [t.sub.i] = [q.sup.n-i] for all i [member of] [n - 1]\D([alpha]).

Here D([alpha]) is the set of partial sums of [alpha], the notation [beta][??][alpha] means [alpha] and [beta] are compositions of n with D([beta]) [subset or equal to] D([alpha]), and [[t.bar].sup.S] denotes the product [[PI].sub.i[member of]S][t.sub.i] over all elements i in a multiset S, including the repeated ones. Taking [alpha] = ([1.sup.n]) shows that F[[B.sub.n]]/([THETA]) carries the regular representation of [H.sub.n](0).

Specializations of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] include not only [[??].sub.[alpha]](x; q, t), but also a more general family of noncommutative symmetric functions depending on parameters associated with paths in binary trees introduced recently by Lascoux, Novelli, and Thibon .

Next we study the quasisymmetric characteristic of F[[B.sub.n]]. We combine the usual [N.sub.n+1]-multigrading of F[[B.sub.n]] (recorded by [t.bar] := [t.sub.0], ..., [t.sub.n]) with the length filtration of [H.sub.n](0) (recorded by q) and obtain an N x [N.sup.n+1]-multigraded quasisymmetric characteristic for F[[B.sub.n]].

Theorem 1.2 The N x Nn+1-multigraded quasisymmetric characteristic of F[[B.sub.n]] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here we identify [F.sub.I] with [F.sub.[alpha]] if D([alpha]) = I [subset or equal to] [n - 1]. The set Com(n, k) consists of all weak compositions of n with length k, i.e. all the sequences [alpha] = ([[alpha].sub.1], ..., [[alpha].sub.k]) of k nonnegative integers with [absolute value of [alpha]] := [[summation].sup.k.sub.i=1] [[alpha].sub.i] = n. The descent multiset of the weak composition [alpha] is the multiset

D([alpha]) := {[[alpha].sub.1], [[alpha].sub.1] + [[alpha].sub.2], ..., [[alpha].sub.1] + ... + [[alpha].sub.k-1]}.

We also define [G.sup.[alpha]] := {w [member of] [G.sub.n]: D(w) [subset or equal to] D([alpha])}. The set [[k + 1].sup.n] consists of all words of length n on the alphabet [k + 1]. Given p = ([p.sub.1], ..., [p.sub.n]) [member of] [[k + 1].sup.n], we write [p'.sub.i] := #{j: [p.sub.j] [less than or equal to] i}, inv(p) := #{(i,j): 1 [less than or equal to] i [less than or equal to] j [less than or equal to] n: [p.sub.i] > [p.sub.j]}, and D(p) := {i: [p.sub.i] > [p.sub.i+1]}.

Let [ps.sub.q;l]([F.sub.[alpha]]) := [F.sub.[alpha]](1, q, [q.sup.2], ..., [q.sup.l-1], 0, 0, ...). Applying the linear transformation [[summation].sub.l[greater than or equal to]0][u.sup.l.sub.1][ps.sub.q1;l+1] and the specialization [t.sub.i] = [q.sup.i.sub.2] [u.sub.2] for all i = 0,1, ..., n to Theorem 1.2, we recover a result of Garsia and Gessel [8, Theorem 2.2] on the generating function of the joint distribution of five permutation statistics:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Here [(u; q).sub.n]: [[PI].sub.0[less than or equal to]i[less than or equal to]n](1 - [q.sup.i]u) the set B(l, k) consists of pairs of weak compositions [lambda] = ([[lambda].sub.1], ..., [[lambda].sub.n]) and [mu] = ([[mu].sub.1], ..., [[mu].sub.n]) satisfying the conditions l [greater than or equal to] [[lambda].sub.1] [greater than or equal to] ... [greater than or equal to] [[lambda].sub.n], max{[[mu].sub.i]: 1 [less than or equal to] i [less than or equal to] n} [less than or equal to] k, and [[lambda].sub.i] = [[lambda].sub.i+1] [right arrow] [u.sub.i] [greater than or equal to] [[mu].sub.i+1] (such pairs ([lambda], [mu]) are sometimes called bipartite partitions), and inv([mu]) is the number of inversion pairs in [mu]. Some further specializations of Theorem 1.2 imply identities of Carlitz-MacMahon [6, 17] and Adin-Brenti-Roichman .

The structure of this paper is as follows. Section 2 reviews the representation theory of the 0-Hecke algebra. Section 3 studies the Stanley-Reisner ring of the Boolean algebra. Section 4 defines a 0-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra. The noncommutative and quasisymmetric characteristics are discussed in Section 5 and Section 6. Finally we give some remarks and questions for future research in Section 7, including a generalization to an action of the Hecke algebra of any finite Coxeter group on the Stanley-Reisner ring of the Coxeter complex.

2 Representation theory of the 0-Hecke algebra

We review the representation theory of the 0-Hecke algebra in this section. The (type A) Hecke algebra [H.sub.n](q) is the associative F(q)-algebra generated by [T.sub.1], ..., [T.sub.n-1] with relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

It has an F(q)-basis {[T.sub.w]: w [member of] [G.sub.n]} where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a reduced expression.

Specializing q = 1 gives the group algebra of [G.sub.n], with [s.sub.i] = [T.sub.i[absolute value of q=1] and w = [T.sub.w[absolute value of q=1]]. Let w [member of] [S.sub.n]. The length of w equals inv(w) := #{(i, j): 1 [less than or equal to] i < j [less than or equal to] n, w(i) > w(j)}, and the descent set of w is D(w) = {i: 1 [less than or equal to] i [less than or equal to] n - 1, w(i) > w(i + 1)}. We write des(w) := [absolute value of D(w)] and maj(w) := [[sumamtion].sub.i[member of]D(w)]i.

Let [alpha] be a (weak) composition of n, and let [[alpha].sup.c] be the composition of n with D([[alpha].sup.c]) = [n - 1]\D([alpha]). The parabolic subgroup [G.sub.[alpha]] is the subgroup of [G.sub.n] generated by {[s.sub.i]: i [member of] D([[alpha].sup.c])}. The set of all minimal [G.sub.[alpha]]-coset representatives is [G.sup.[alpha]] := {w [member of] [G.sub.n]: D(w) [subset or equal to] D([alpha])}. The descent class of a consists of the permutations in [G.sub.n] with descent set equal to D([alpha]), and turns out to be an interval under the (left) weak order of [G.sub.n], denoted by [[w.sub.0]([alpha]), [w.sub.1]([alpha])]. One sees that [w.sub.0]([alpha]) is the longest element of the parabolic subgroup [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the longest element in [G.sup.[alpha]] (c.f. Bjorner and Wachs [5, Theorem 6.2]).

Another interesting specialization of [H.sub.n](q) is the 0-Hecke algebra [H.sub.n](0), with generators [[bar.[pi]].sub.i] = [T.sub.i|q=0] for i = 1, ..., n - 1, and an F-basis {[[bar.[pi]].sub.w] = [T.sub.w|q=0]: w [member of] [G.sub.n]}. Let [[pi].sub.i] := [[bar.[pi]].sub.i] + 1. Then [[pi].sub.1], ..., [[pi].sub.n-1] form another generating set for [H.sub.n](0), with the same relations as (2) except [[pi].sup.2.sub.i] = [[pi].sub.i], 1 [less than or equal to] i [less than or equal to] n - 1. The element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is well defined for any w [member of] [G.sub.n] with a reduced expression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is another F-bases for [H.sub.n](0). One can check that [[pi].sub.w] equals the sum of [[bar.[pi]].sub.u] over all u less than or equal to w in the Bruhat order of [G.sub.n]. In particular, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the sum of [[bar.[pi]].sub.u] for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Norton  decomposed the 0-Hecke algebra [H.sub.n](0) into a direct sum of projective indecomposable submodules [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [alpha] |= n (i.e. compositions of n). Each [P.sub.[alpha]] has an F-basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Its radical rad Pa is the unique maximal [H.sub.n](0)-submodule spanned by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Although [P.sub.[alpha]] itself is not necessarily simple, its top [C.sub.[alpha]] := [P.sub.[alpha]]/rad [P.sub.[alpha]] is a one-dimensional simple [H.sub.n](0)-module with the action of [H.sub.n](0) given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It follows from general representation theory of algebras (see e.g. [2, [section] I.5]) that {[P.sub.[alpha]]: [alpha] |= n} and {[C.sub.[alpha]]: [alpha] |= n} are the complete lists of pairwise non-isomorphic projective indecomposable and simple [H.sub.n](0)-modules, respectively.

Krob and Thibon  introduced a correspondence between [H.sub.n](0)-representations and the dual Hopf algebras QSym and NSym, which we review next. The Hopf algebra QSym has a free Z-basis of fundamental quasisymmetric functions [F.sub.[alpha]], and the dual Hopf algebra NSym has a dual basis of non- commutative ribbon Schur functions [s.sub.[alpha]], for all compositions [alpha].

Let M = [M.sub.0] [contains or equal to] [M.sub.1] [contains or equal to] ... [contains or equal to] [M.sub.k] [contains or equal to] [M.sub.k+1] = 0 be a composition series of [H.sub.n](0)-modules with simple factors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then the quasisymmetric characteristic of M is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The noncommutative characteristic of a projective [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is not hard to extend these characteristic maps to [H.sub.n](0)-modules with gradings and filtrations.

3 Stanley-Reisner ring of the Boolean algebra

In this section we study the Stanley-Reisner ring of the Boolean algebra. The Boolean algebra [B.sub.n] is the ranked poset of all subsets of [n] := {1,2, ..., n} ordered by inclusion, with minimum element [phi] and maximum element [n]. The rank of a subset of [n] is defined as its cardinality. The Stanley-Reisner ring F[[B.sub.n]] of the Boolean algebra [B.sub.n] is the quotient of the polynomial algebra F [[y.sub.A]: A [subset or equal to] [n]] by the ideal ([y.sub.A][y.sub.B]: A and B are incomparable in [B.sub.n]). It has an F-basis {[y.sub.M]} indexed by the multichains M in [B.sub.n], and is multigraded by the rank multisets r(M) of the multichains M.

The symmetric group [G.sub.n] acts on the Boolean algebra [B.sub.n] by permuting the integers 1, ..., n. This induces an [G.sub.n]-action on the Stanley-Reisner ring F[[B.sub.n]], preserving its multigrading. The invariant algebra [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] consists of all elements in F[[B.sub.n]] invariant under this 6n- action. One can show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [THETA] := {[[theta].sub.0], ..., [[theta].sub.n]}. Garsia  showed that F[[B.sub.n]] is a free F[[THETA]]-module on the basis of descent monomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

There is an analogy between the Stanley-Reisner ring F[[B.sub.n]] and the polynomial ring F[X] via the transfer map [tau]: F[[B.sub.n]] [right arrow] F[X] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all multichains M = ([A.sub.1] [subset or equal to] ... [subset or equal to] [A.sub.k]) in [B.sub.n]. It is not a ring homomorphism (e.g. [y.sub.{1}][y.sub.{2}] = 0 but [x.sub.1][x.sub.2] [not equal to] 0). Nevertheless, it induces an isomorphism [tau]: F[[B.sub.n]]/([[theta].sub.0]) [??] F[X] of [G.sub.n]-modules. Moreover, it sends the rank polynomials [[theta].sub.1], ..., [[theta].sub.n] to the elementary symmetric polynomials [e.sub.1], ..., [e.sub.n], and sends the descent monomials [Y.sub.w] in F[[B.sup.*.sub.n]] defined by (3) to the corresponding descent monomials in F[X] for all w [member of] [S.sub.n].

Example 3.1 The Boolean algebra [B.sub.3] consists of all subsets of {1, 2, 3}. Its Stanley-Reisner ring F[[B.sub.3]] is a free F[[THETA]]-module with a basis of descent monomials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [THETA] consists of the rank polynomials [[theta].sub.0] = [y.sub.[phi]], [[theta].sub.1] := [y.sub.1] + [y.sub.2] + [y.sub.13], [[theta].sub.2] := [y.sub.12] + [y.sub.13] + [y.sub.23], [[theta].sub.3] := [y.sub.123]. The transfer map [tau] sends [[theta].sub.1], [[theta].sub.2], [[theta].sub.3] to [e.sub.1], [e.sub.2], [e.sub.3], and sends the six descent monomials in F[[B.sub.3]] to the six descent monomials 1, [x.sub.2], [x.sub.3], [x.sub.1][x.sub.3], [x.sub.2] [x.sub.3], [x.sub.2][x.sub.2] in F[[x.sub.1], [x.sub.2], [x.sub.3]].

The homogeneous components of F[[B.sub.n]] are indexed by multisets with elements in {0, ..., n}, or equivalently by weak compositions [alpha] of n. The [alpha]-homogeneous component F[[[B.sub.n]].sub.[alpha]] has an F- basis {[y.sub.M]: r(M) = D([alpha])}. Denote by Com(n, k) the set of all weak compositions of n with length k. If M = ([A.sub.1] [subset or equal to] ... [subset or equal to] [A.sub.k]) is a multichain of length k in [B.sub.n] then we set [A.sub.0] := [phi] and [A.sub.k+1] := [n] by convention. Define [alpha](M) := ([[alpha].sub.1], ..., [[alpha].sub.k+1]), where [[alpha].sub.i] = [absolute value of [A.sub.i]] - [absolute value of [A.sub.i-1]] for all i [member of] [k + 1]. Then [alpha](M) [member of] Com(n, k + 1) and D([alpha](M)) = r(M), i.e. [alpha](M) indexes the homogeneous component containing [y.sub.M]. Define [sigma](M) to be the minimal element in [G.sub.n] which sends the standard multichain [[[alpha].sub.1]] [subset or equal to] [[[alpha].sub.1] + [[alpha].sub.2]] [subset or equal to] ... [subset or equal to] [[[alpha].sub.1] + ... + [[alpha].sub.k]] with rank multiset D([alpha](M)) to M. Then [sigma](M) [member of] [G.sup.[alpha](M)].

The map M [right arrow] ([alpha](M), [sigma](M)) is a bijection between multichains of length k in [B.sub.n] and the pairs ([alpha], [sigma]) of [alpha] [member of] Com(n, k + 1) and [sigma] [member of] [G.sup.[alpha]]. A short way to write down this encoding of M is to insert bars at the descent positions of [sigma](M). For example, the length-4 multichain {2} [subset or equal to] {2} [subset or equal to] {1,2,4} [subset or equal to]  in B4 is encoded by 2[parallel] 14[absolute value of 3].

There is another way to encode the multichain M. Let [p.sub.i](M) := min{j[member of]G [k + 1]: i [member of] [A.sub.j]}, 1 [less than or equal to] i [less than or equal to] n. So [p.sub.i](M) is the first position where i appears in M. One checks that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

The map M [right arrow] p(M) := ([p.sub.1](M), ..., [p.sub.n](M)) is an bijection between the set of multichains with length k in [B.sub.n] and the set [[k + 1].sup.n] of all words of length n on the alphabet [k +1], for any fixed integer k [greater than or equal to] 0.

Let p(M) = p = ([p.sub.1], ..., [p.sub.n]) [member of] [[k + 1].sup.n]. Then inv(p(M)) = inv([sigma](M)). Let p' := ([p'.sub.1], ..., [p'.sub.k]) where [p'.sub.i] := [absolute value of {j: [p.sub.j](M) [less than or equal to] i}] = [absolute value of [A.sub.i]]. Then the rank multiset of M consists of [p'.sub.1], ..., [p'.sub.k]. Define

D(p) := {i [member of] [n - 1]: [p.sub.i] > [p.sub.i+1]}. For example, the multichain 3[absolute value of 14][absolute value of 2]5 corresponds to p = (2,4,1,2,5) [member of] [.sup.5], and one has p' = (1, 3, 3, 4), D(2, 5, 1, 2, 4) = {2}.

These two encodings (with slightly different notation) were already used by Garsia and Gessel  in their work on generating functions of multivariate distributions of permutation statistics.

4 0-Hecke algebra action

We saw an analogy between F[[B.sub.n]] and F[X] in the last section. The usual [H.sub.n](0)-action on the polynomial ring F[X] is via the Demazure operators

[[bar.[pi]].sub.i](f) := [x.sub.i+1]f - [x.sub.i+1][s.sub.i]f/[x.sub.i] - [x.sub.i+1], [for all f [member of] F[X], 1 [less than or equal to] i [less than or equal to] n - 1. (5)

The above definition is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Here m is any monomial in F[X] containing neither [x.sub.i] nor [x.sub.i+1]. Denote by [[bar.[pi]]'.sub.i] the operator obtained from (6) by taking only the leading term (underlined) in the lexicographic order of the result. Then [[bar.[pi]]'.sub.1], ..., [[bar.[pi]]'.sub.n-1] realize another [H.sub.n](0)-action on F[X]. We call it the transferred [H.sub.n](0)-action because it can be obtained by applying the transfer map [tau] to our [H.sub.n](0)-action on F[[B.sub.n]], which we now define.

Let M = ([A.sub.1] [subset or equal to] ... [subset or equal to] [A.sub.k]) be a multichain in [B.sub.n]. Recall that [p.sub.i](M) := min{j [member of] [k + 1]: i [member of] [A.sub.j]}, 1 [less than or equal to] i [less than or equal to] n. We define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

for i = 1, ..., n - 1. Applying the transfer map [tau] one recovers [[bar.[pi]]'.sub.i]. For instance, when n = 4 one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One can check that [[bar.[pi]].sub.1], ..., [[bar.[pi]].sub.n-1] realize an [H.sub.n](0)-action on F[[B.sub.n]] preserving the multigrading of F[[B.sub.n]]. If one set [t.sub.i] = [t.sup.i] for i = 1, ..., n, then there is an isomorphism F[[B.sub.n]]/([empty set]) [??] F[X] of graded [H.sub.n](0)-modules (which can be given explicitly, but not via the transfer map t).

It is not hard to show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the invariant algebra of the [H.sub.n](0)-action on F[[B.sub.n]], defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 4.1 The [H.sub.n](0)-action on F[[B.sub.n]] is [THETA]-linear.

Therefore the coinvariant algebra F[[B.sub.n]]/([THETA]) is a multigraded [H.sub.n](0)-module, which is isomorphic to the regular representation of [H.sub.n](0) by Theorem 1.1. This cannot be obtained simply by applying the transfer map [tau], since [tau] is not a map of [H.sub.n](0)-modules.

5 Noncommutative characteristic

In this section we use the [H.sub.n](0)-action on the Stanley-Reisner ring F[[B.sub.n]] of the Boolean algebra [B.sub.n] to provide a noncommutative analogue of the following remarkable result.

Theorem 5.1 (Hotta-Springer , Garsia-Procesi ) For any partition [mu] = (0 < [[mu].sub.1] [less than or equal to] ... [less than or equal to] [[mu].sub.k]) of n, there exists an [G.sub.n]-invariant ideal JM of C[X] such that C[X]/[J.sub.[mu]] is isomorphic to the cohomology ring of the Springer fiber indexed by p and has graded Frobenius characteristic equal to the modified Hall-Littlewood symmetric function

[[??].sub.[mu]](X; t) = [summation over [lambda]][t.sup.n(mu])][K.sub.[lambda][mu]]([t.sup.-1])[s.sub.[lambda]] inside Sym[t]

where n([mu]) := [[mu].sub.k-1] + 2[[mu].sub.k-2] + ... + (k - 1)[[mu].sub.1] and [K.sub.[lambda][mu]](t) is the Kostka-Foulkespolynomial.

Example 5.2 Tanisaki  gives a construction for the ideal [J.sub.[mu]]. If p = ([1.sup.k], n - k) is a hook then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is generated by [e.sub.1], ..., [e.sub.k] and all monomials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with 1 [less than or equal to] [i.sub.1] < ... < [i.sub.k+1] [less than or equal to] n.

Now consider a composition [alpha] = ([[alpha].sub.1], ..., [[alpha].sub.l]). The major index of [alpha] is maj([alpha]) := [[summation].sub.i[member of]D([alpha])] i. Viewing a partition [??] = (0 < [[mu].sub.1] [less than or equal to] [[mu].sub.2] [less than or equal to] ...) as a composition one has maj([mu]) = n([mu]). Recall that [??] := ([[alpha].sub.l], ..., [[alpha].sub.l]) and ac is the composition of n with D([[alpha].sup.c]) = [n - 1]\D([alpha]). We define [alpha]' := [[??].sub.c] = [([??]).sup.c]. One can identify a with a ribbon diagram, i.e. a connected skew Young diagram without 2 by 2 boxes, which has row lengths [[alpha].sub.1], ..., [[alpha].sub.l], ordered from bottom to top. Note that a ribbon diagram is a Young diagram if and only if it is a hook. One can check that [alpha]' is the transpose of a; see the example below.

[alpha] = (2, 3,1, 1) [[alpha].sup.c] = (1, 2,1, 3) [alpha]' = (3,1, 2, 1)

Bergeron and Zabrocki  introduced a noncommutative modified Hall-Littlewood symmetric function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

and a (q, t)-analogue

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

for every composition [alpha], where [s.sub.[beta]] is the noncommutative ribbon Schur function indexed by [beta], and c([alpha], [beta]) := [[summation].sub.i[member of]D([alpha])[intersection]D([beta])] i. In our earlier work  we provided a partial representation theoretic interpretation for [[??].sub.a](x; t) when [alpha] = ([1.sup.k], n - k) is a hook, using the [H.sub.n](0)-action on the polynomial ring F[X] by the Demazure operators.

Theorem 5.3 () The ideal [J.sub.[mu]]of F[X] is [H.sub.n](0)-invariant if and only if [mu] = ([1.sup.n-k], k) is a hook, and if that holds then F[X]/[J.sub.[mu]] becomes a graded projective [H.sub.n](0)-module with

[ch.sub.t](F[X]/[J.sub.[mu]]) = [[??].sub.[mu]](x; t),

[Ch.sub.t](F[X]/[J.sub.[mu]]) = [[??].sub.[mu]](x; t).

Now we switch to the Stanley-Reisner ring F[[B.sub.n]] and define [I.sub.[alpha]] to be its ideal generated by

[[THETA].sub.[alpha]] := {[[theta].sub.i]: i [member of] D([alpha]) [union] {n}} and {[y.sub.A]: A [subset or equal to] [n], [absolute value of A] [not member of] D([alpha]) [union] {n}}

for any composition [alpha] of n. The following result is a restatement of Theorem 1.1.

Theorem 5.4 Let [alpha] be a composition of n. Then F[[B.sub.n]]/[I.sub.[alpha]] is a projective [H.sub.n](0)-module with multi-graded noncommutative characteristic equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One has [[??].sub.[alpha]](x; t, [t.sup.2], ..., [t.sup.n-1]) = [[??].sub.[alpha]](x; t), and one obtains [[??].sub.[alpha]](x; q, t) from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by taking [t.sub.i] = [t.sup.i] for all i [member of] D([delta]), and [t.sub.i] = [q.sup.n-i] for all i [member of] [n - 1]\D([alpha]).

Proof: There is an F-basis for F[[B.sub.[alpha]]]/([[THETA].sub.[alpha]]) given by the descent monomials [Y.sub.w] defined in (3) for all w [member of] [G.sup.[alpha]]. The result follows from the [H.sub.n](0)-action on this basis and (4).

The proof of this theorem is actually simpler than the proof of our partial interpretation for Ha (x; t) in . This is because [[bar.[pi]].sub.i] sends a descent monomial in F[[B.sub.n]] to either 0 or [+ or -] 1 times a descent monomial, but sends a descent monomial in F[X] to a polynomial in general (whose leading term is still a descent monomial). We view the Stanley-Reisner ring F[[B.sub.n]] (or F[[B.sub.n]]/([empty set])) as a q = 0 analogue of the polynomial ring F[X]. For an odd (i.e. q = -1) analogue, see Lauda and Russell .

Remark 5.5 If [alpha] = ([1.sup.k], n - k) is a hook, one can check that the ideal [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has generators [[theta].sub.1], ..., [[theta].sub.k] and all [y.sub.A] with A [subset or equal to] [n] and [absolute value of A] [not member of] [k]. By Example 5.2, the images of these generators under the transfer map t are the Tanisaki generators for the ideal [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], but [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For any composition [alpha] |= n, one can view [[??].sub.[alpha]](x; [t.sub.1], ..., [t.sub.n-1]) as a modified version of

[H.sub.[alpha]] = [H.sub.[alpha]](x; [t.sub.1], ..., [t.sub.n-1]) := [summation over B[??][alpha]] [[t.bar].sup.D([alpha])\D([beta])][s.sub.[beta]].

Below are some properties satisfied by [H.sub.[alpha]], generalizing the properties of [H.sub.[alpha]](x; t) given in .

Proposition 5.6 Let [alpha] and [beta] be two compositions.

(I) [H.sub.[alpha]](0, ..., 0) = [s.sub.[alpha]], [H.sub.[alpha]](1, ..., 1) = [h.sub.[alpha]].

(ii) [[union].sub.n[greater than or equal to]0] {[H.sub.[alpha]]: [alpha] |= n} is a basis for NSym[[t.sub.1], [t.sub.2], ...].

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any pair of compositions a and ft.

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(v) If n = [absolute value of [alpha]] and t|n := ([t.sub.1], ..., [t.sub.n-1], 1, [t.sub.1], ..., [t.sub.n-1], 1, ...) then

[H.sub.[alpha]](x; [t.sub.1], ..., [t.sub.n-1])[H.sub.[beta]](x; t|n) = [H.sub.[alpha][beta]](t|n).

6 Quasisymmetric characteristic

Now we study the quasisymmetric characteristic of F[[B.sub.n]]. The following lemma follows easily from (4).

Lemma 6.1 Let [alpha] be a weak composition of n. Then the [alpha]-homogeneous component F[[[B.sub.n]].sub.a] of the Stanley-Reisner ring F[[B.sub.n]] is an [H.sub.n](O)-submodule of F[[B.sub.n]] with homogeneous multigrading [[t.bar].sup.D([alpha])] and isomorphic to the cyclic module [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since F[[[B.sub.n]].sub.[alpha]] is a cyclic multigraded [H.sub.n](0)-module, we get an N x [N.sup.n+1]-multigraded quasisymmetric characteristic

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where q keeps track of the length filtration and [t.bar] keeps track of the multigrading of F[[[B.sub.n]].sub.[alpha]]. This defines an N x [N.sup.n+1]-multigraded quasisymmetric characteristic for the Stanley-Reisner ring F[[B.sub.n]], which is explicitly given in Theorem 1.2 and restated below.

Theorem 6.2 The N x [N.sup.n+1]-multigraded quasisymmetric characteristic of F[[B.sub.n]] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof: Use the two encodings of the multichains in [B.sub.n] as well as the free F[[THETA]]-basis {[Y.sub.w]: w [member of] [G.sub.n]} of descent monomials for F[[B.sub.n]] discussed in Section 3.

Next we explain here how this theorem specializes to (1), a result of Garsia and Gessel [8, Theorem 2.2] on the multivariate generating function of the permutation statistics inv(w), maj(w), des(w), maj([w.sup.-1]), and des([w.sup.-1]) for all w [member of] [G.sub.n]. First recall that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [ps.sub.q;l]([F.sub.[alpha]]) := [F.sub.[alpha]](1, q, [q.sup.2], ..., [q.sup.l-1], 0, 0, ...), and [(u; q).sub.n] := (1 - u)(1 - qu)(1 - [q.sup.2]u) ... (1 - [q.sup.n]u). It is not hard to check (see Gessel and Reutenauer [10, Lemma 5.2]) that

[summation over l[greater than or equal to]0] [u.sup.l][ps.sub.q;l+1]([F.sub.[alpha]]) = [q.sup.maj([alpha])][u.sup.des([alpha])]/[(u; q).sub.n]

Then applying the linear transformation 0 [[summation].sub.l[greater than or equal to]0][u.sup.l.sub.1][ps.sub.q1;l+1] and the specialization [t.sub.i] = [q.sup.i.sub.2][u.sup.2] for all i = 0,1, ..., n to Theorem 6.2 we recover (1).

A further specialization of Theorem 6.2 gives a well known result which is often attributed to Carlitz  but actually dates back to MacMahon [17, Volume 2, Chapter 4].

Corollary 6.3 (Carlitz-MacMahon) Let [[k + 1].sub.q] := 1 + q + [q.sup.2] + ... + [q.sup.k]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 6.2 also implies the following result, which was obtained by Adin, Brenti, and Roichman  from the Hilbert series of the coinvariant algebra [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 6.4 (Adin-Brenti-Roichman) Let Par(n) be the set of weak partitions [lambda] = ([[lambda].sub.1], ..., [[lambda].sub.n]) with [[lambda].sub.1] [greater than or equal to] ... [greater than or equal to] [[lambda].sub.n] [greater than or equal to] 0, and let m([lambda]) = ([m.sub.0]([lambda]), [m.sub.1]([lambda]), ...), where [m.sub.j]([lambda]) := #{1 [less than or equal to] i [less than or equal to] n: [[lambda].sub.i] = j}.

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

7 Remarks and questions for future research

7.1 Hecke algebra action

It is well known that the symmetric group Sn is the Coxeter group of type [A.sub.n-1]. The Stanley-Reisner ring of [B.sub.n] is essentially the Stanley-Reisner ring of the Coxeter complex of [G.sub.n]. The Hecke algebra [H.sup.W](q) can be defined for any finite Coxeter group W. We can generalize our action [H.sub.n](0)-action on F[[B.sub.n]] to an [H.sub.W](q)-action on the Stanley-Reisner ring F(q)[[DELTA](W)] of the Coxeter complex [DELTA](W) of any finite Coxeter group W. We show similar results for this [H.sub.W](q)-action.

7.2 Gluing the group algebra and the 0-Hecke algebra

The group algebra FW of a finite Coxeter group W naturally admits both actions of W and [H.sub.W](0). Hivert and Thiery  defined the Hecke group algebra of W by gluing these two actions. In type A, one can also glue the usual actions of [G.sub.n] and [H.sub.n](0) on the polynomial ring F[X], but the resulting algebra is different from the Hecke group algebra of [G.sub.n].

Now one has a W-action and an [H.sub.W](0)-action on the Stanley-Reisner ring F[[DELTA](W)]. What can we say about the algebra generated by the operators [s.sub.i] and [[bar.[pi]].sub.i] on F[[DELTA](W)]? Is it the same as the Hecke group algebra of W? If not, what properties (dimension, bases, presentation, simple and projective indecomposable modules, etc.) does it have?

7.3 Tits Building

Let [DELTA](G) be the Tits building of the general linear group G = GL(n, [F.sub.q]) and its usual BN-pair over a finite field [F.sub.q]; see e.g. Bjorner . The Stanley-Reisner ring F[[DELTA](G)] is a q-analogue of F[[B.sub.n]]. The nonzero monomials in F[[DELTA](G)] are indexed by multiflags of subspaces of [F.sup.n.sub.q], and there are [q.sup.inv(w)] many multiflags corresponding to a given multichain M in [B.sub.n], where w = [sigma](M). Can one obtain the multivariate quasisymmetric function identities in Theorem 1.2 by defining a nice [H.sub.n](0)-action on F[[DELTA](G)]?

Acknowledgements

The author is grateful to Victor Reiner for providing valuable suggestions. He also thanks Ben Braun and Jean-Yves Thibon for helpful conversations and email correspondence.

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Jia Huang *

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

* Email: jiahuangustc@gmail.com. Supported by NSF grant DMS-1001933.
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