We study the coinvariant ring of the complex reflection group $G(r,p,n)$ as a module for the corresponding rational Cherednik algebra $\HH$ and its generalized graded affine Hecke subalgebra $\mathcal{H}$. We construct a basis consisting of non-symmetric Jack polynomials, and using this basis decompose the coinvariant ring into irreducible modules for $\mathcal{H}$. The basis consists of certain non-symmetric Jack polynomials, whose leading terms are the ``descent monomials'' for $G(r,p,n)$ recently studied by Adin, Brenti, and Roichman and Bagno and Biagoli. The irreducible $\mathcal{H}$-submodules of the coinvariant ring are their ``colored descent representations''.
We explain how to use representation theory to give a lower bound on the dimension of the quotient ring by type $B_n$ diagonal invariants that improves upon the current known lower bound $(2n+1)^n$ by a quadratic polynomial in $n$.
We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group $W=G(m,p,n)$ and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category $\cO$ for the rational Cherednik algebra of $W$.
Let $Θ_n = (θ_1, \dots, θ_n)$ and $Ξ_n = (ξ_1, \dots, ξ_n)$ be two lists of $n$ variables and consider the diagonal action of $\mathfrak{S}_n$ on the exterior algebra $\wedge \{ Θ_n, Ξ_n \}$ generated by these variables. Jongwon Kim and Rhoades defined and studied the fermionic diagonal coinvariant ring $FDR_n$ obtained from $\wedge \{ Θ_n, Ξ_n \}$ by modding out by the $\mathfrak{S}_n$-invariants with vanishing constant term. In joint work with Rhoades we gave a basis for the maximal degree components of this ring where the action of $\mathfrak{S}_n$ could be interpreted combinatorially via n…
In this paper we prove that a deformed tensor product of two Lefschetz algebras is a Lefschetz algebra. We then use this result in conjunction with some basic Schubert calculus to prove that the coinvariant ring of a finite reflection has the strong Lefschetz property.
Two actions of the Hecke algebra of type A on the corresponding polynomial ring are studied. Both are deformations of the natural action of the symmetric group on polynomials, and keep symmetric functions invariant. We give an explicit description of these actions, and deduce a combinatorial formula for the resulting graded characters on the coinvariant algebra.
Let $k \leq n$ be nonnegative integers and let $λ$ be a partition of $k$. S. Griffin recently introduced a quotient $R_{n,λ}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which simultaneously generalizes the Delta Conjecture coinvariant rings of Haglund-Rhoades-Shimozono and the cohomology rings of Springer fibers studied by Tanisaki and Garsia-Procesi. We describe the space $V_{n,λ}$ of harmonics attached to $R_{n,λ}$ and produce a harmonic basis of $R_{n,λ}$ indexed by certain ordered set partitions $\mathcal{OP}_{n,λ}$. The combinatorics of this basis is governed by…
The coinvariant algebra is a quotient of the polynomial ring $\mathbb{Q}[x_1,\ldots,x_n]$ whose algebraic properties are governed by the combinatorics of permutations of length $n$. A word $w = w_1 \dots w_n$ over the positive integers is packed if whenever $i > 2$ appears as a letter of $w$, so does $i-1$. We introduce a quotient $S_n$ of $\mathbb{Q}[x_1,\ldots,x_n]$ which is governed by the combinatorics of packed words. We relate our quotient $S_n$ to the generalized coinvariant rings of Haglund, Rhoades, and Shimozono as well as the superspace coinvariant ring.
The 0-Hecke algebra $H_n(0)$ is a deformation of the group algebra of the symmetric group $\SS_n$. We show that its coinvariant algebra naturally carries the regular representation of $H_n(0)$, giving an analogue of the well-known result for $\SS_n$ by Chevalley-Shephard-Todd. By investigating the action of $H_n(0)$ on coinvariants and flag varieties, we interpret the generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. We also study the action of $H_n(0)$ on the cohomology rings of the Springer fibers, and similarly interpret…
We describe a new basis of the ring of quasi-symmetric coinvariants, which is stable by the natural reversal of the set of variables. The indexing set is the set of triangulations of a regular polygon, instead of the set of Dyck paths used for the known basis. On décrit une nouvelle base de l'algèbre des coinvariants quasi-symétriques, qui est stable par l'involution naturelle et indexée par les triangulations d'un polygone régulier.
Let $n,k,$ and $r$ be nonnegative integers and let $S_n$ be the symmetric group. We introduce a quotient $R_{n,k,r}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which carries the structure of a graded $S_n$-module. When $r \geq n$ or $k = 0$ the quotient $R_{n,k,r}$ reduces to the classical coinvariant algebra $R_n$ attached to the symmetric group. Just as algebraic properties of $R_n$ are controlled by combinatorial properties of permutations in $S_n$, the algebra of $R_{n,k,r}$ is controlled by the combinatorics of objects called {\em tail positive words}. We calcul…
Let R_n be the ring of coinvariants for the diagonal action of the symmetric group S_n. It is known that the character of R_n as a doubly-graded S_n module can be expressed using the Frobenius characteristic map as \nabla e_n, where e_n is the n-th elementary symmetric function, and \nabla is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for \nabla e_n and prove that it has many desirable properties which support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. Th…
Let the symmetric group $\mathfrak{S}_n$ act on the polynomial ring $\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n]$ by variable permutation. The coinvariant algebra is the graded $\mathfrak{S}_n$-module $R_n := {\mathbb{Q}[\mathbf{x}_n]} / {I_n}$, where $I_n$ is the ideal in $\mathbb{Q}[\mathbf{x}_n]$ generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient $R_{n,k}$ of the polynomial ring $\mathbb{Q}[\mathbf{x}_n]$ depending on two positive integers $k \leq n$ which reduces to the classical coinvariant algebra of the s…
FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of S_n-representations. In this paper we prove the Noetherian property for FI-modules over arbitrary Noetherian rings: any sub-FI-module of a finitely generated FI-module is finitely generated. This lets us extend many of the results of [CEF] to representations in positive characteristic, and even to integral coefficients. We focus on three ma…
This article is devoted to the computation of Jack connection coefficients, a generalization of the connection coefficients of two classical commutative subalgebras of the group algebra of the symmetric group: the class algebra and the double coset algebra. The connection coefficients of these two algebraic structures are of significant interest in the study of Schur and zonal polynomials as well as the irreducible characters of the symmetric group and the zonal spherical functions. Furthermore they play an important role in combinatorics as they give the number of factorizations of a permutat…
For each pair of positive integers (k,r) such that k+1,r-1 are coprime, we introduce an ideal $I^{(k,r)}_n$ of the ring of symmetric polynomials. The ideal $I^{(k,r)}_n$ has a basis consisting of Jack polynomials with parameter $β=-(r-1)/(k+1)$, and admits an action of a family of differential operators of Dunkl type including the positive half of the Virasoro algebra. The space $I^{(k,2)}_n$ coincides with the space of all symmetric polynomials in $n$ variables which vanish when $k+1$ variables are set equal. The space $I_n^{(2,r)}$ coincides with the space of correlation functions of an abel…
Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity free. If V_λ is an irreducible representation of K, let S_λ denote the corresponding isotypic component of S. Write S_λ S_μ for the subspace of S spanned by products of S_λ and S_μ. If V_ν occurs as an irreducible constituent of the tensor product of V_λ and V_μ, is it true that S_ν is contained in S_λ S_μ? We investigate this question for representations arising in the context of Hermitian symmetric pairs. We show that the answer…
We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases $t=1$ and $q=0$, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and $q$-Whittaker polynomials.
In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most ca…
For each pair (k,r) of positive integers with r>1, we consider an ideal I^(k,r)_n of the ring of symmetric polynomials in n variables. The ideal I_n^(k,r) has a basis consisting of Macdonald polynomials P(x_1,...,x_n;q,t) at t^{k+1}q^{r-1}=1, and is a deformed version of the one studied earlier in the context of Jack polynomials. In this paper we give a characterization of I^(k,r)_n in terms of explicit zero conditions on the k-codimensional shifted diagonals of the form x_{2}=tq^{s_1}x_1,...,x_{k+1}=tq^{s_k}x_k. The ideal I^(k,r)_n may be viewed as a deformation of the space of correlation fu…
