Alternating super-polynomials and super-coinvariants of finite reflection groups
Joshua P Swanson
备注：
18 pages, 1 table. Superseded by joint work with Nolan Wallach, arXiv:2001.06076, which is more technical but type-independent and which contains additional references to the existing literature
Motivated by a recent conjecture of Zabrocki, Wallach described the alternants in the super-coinvariant algebra of the symmetric group in one set of commuting and one set of anti-commuting variables under the diagonal action. We give a type-independent generalization of Wallach's result to all real reflection groups $G$. As an intermediate step, we explicitly describe the alternating super-polynomials in $k[V] \otimes Λ(V)$ for all complex reflection groups, providing an analogue of a classic result of Solomon which describes the invariant super-polynomials in $k[V] \otimes Λ(V^*)$. Using our construction, we explicitly describe the alternating harmonics and coinvariants for all real reflection groups.
For finite complex reflexion groups, we consider the graded $W$-modules of diagonally harmonic polynomials in $r$ sets of variables, and show that associated Hilbert series may be described in a global manner, independent of the value of $r$.
The purpose of this paper is to give an explicit description of the trivial and alternating components of the irreducible representation decomposition of the bigraded module obtained as the tensor square of the coinvariant space for hyperoctahedral groups.
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.
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 $W$.
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 $\mathcal{H}$-subm…
Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W := \wedge (V \oplus V^*)/ \langle \wedge (V \oplus V^*)^{W}_+ \rangle$ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = \mathfrak{S}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped $\mathfrak{S}_n$-modules. We relate the Hilbert series of…
This paper studies three results that describe the structure of the super-coinvariant algebra of pseudo-reflection groups over a field of characteristic $0$. Our most general result determines the top component in total degree, which we prove for all Shephard--Todd groups $G(m, p, n)$ with $m \neq p$ or $m=1$. Our strongest result gives tight bi-degree bounds and is proven for all $G(m, 1, n)$, which includes the Weyl groups of types $A$ and $B$/$C$. For symmetric groups (i.e. type $A$), this provides new evidence for a recent conjecture of Zabrocki related to the Delta Conjecture of Haglund--…
We introduce two new partial orders on the standard Young tableaux of a given partition shape, in analogy with the strong and weak Bruhat orders on permutations. Both posets are ranked by the major index statistic offset by a fixed shift. The existence of such ranked poset structures allows us to classify the realizable major index statistics on standard tableaux of arbitrary straight shape and certain skew shapes. By a theorem of Lusztig--Stanley, this classification can be interpreted as determining which irreducible representations of the symmetric group exist in which homogeneous component…
In this paper, we continue our study of the Hilbert polynomials of coinvariants begun in our previous work math.QA/0205324 (paper I). We describe the sl_n-fusion products for symmetric tensor representations following the method of Feigin and Feigin, and show that their Hilbert polynomials are A_{n-1}-supernomials. We identify the fusion product of arbitrary irreducible sl_n-modules with the fusion product of their resctriction to sl_{n-1}. Then using the equivalence theorem from paper I and the results above for sl_3, we give a fermionic formula for the Hilbert polynomials of a class of affin…
We analyze the structure of the algebra N of symmetric polynomials in non-commuting variables in so far as it relates to its commutative counterpart. Using the "place-action" of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of N analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.
We give a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Our "top-down" approach uses the methods of Cartan's exterior calculus and is in some sense dual to related work of Solomon, Orlik--Solomon, and Shepler describing (semi-)invariant differential forms. We apply our results to a recent conjecture of Zabrocki which provides a representation theoretic-model for the Delta conjecture of Haglund--Remmel--Wilson in terms of a certain non-commutative coinvariant algebra for the s…
Let $r$ be a positive integer and let $G_n$ be the reflection group of $n \times n$ monomial matrices whose entries are $r^{th}$ complex roots of unity and let $k \leq n$. We define and study two new graded quotients $R_{n,k}$ and $S_{n,k}$ of the polynomial ring $\mathbb{C}[x_1, \dots, x_n]$ in $n$ variables. When $k = n$, both of these quotients coincide with the classical coinvariant algebra attached to $G_n$. The algebraic properties of our quotients are governed by the combinatorial properties of $k$-dimensional faces in the Coxeter complex attached to $G_n$ (in the case of $R_{n,k}$) and…
We define a bijection that transforms an alternating sign matrix A with one -1 into a pair (N,E) where N is a (so called) ``neutral'' alternating sign matrix (with one -1) and E is an integer. The bijection preserves the classical parameters of Mills, Robbins and Rumsey as well as three new parameters (including E). It translates vertical reflection of A into vertical reflection of N. A hidden symmetry allows the interchange of E with one of the remaining two new parameters. A second bijection transforms (N,E) into a configuration of lattice paths called ``mixed configuration''.
The cluster complex on one hand, parking functions on the other hand, are two combinatorial (po)sets that can be associated to a finite real reflection group. Cluster parking functions are obtained by taking an appropriate fiber product (over noncrossing partitions). There is a natural structure of simplicial complex on these objects, and our main goal is to show that it has the homotopy type of a (pure) wedge of spheres. The unique nonzero homology group (as a representation of the underlying reflection group) is a sign-twisted parking representation, which is the same as Gordon's quotient of…
Shephard groups are unitary reflection groups arising as the symmetries of regular complex polytopes. For a Shephard group, we identify the representation carried by the principal ideal in the coinvariant algebra generated by the image of the product of all linear forms defining reflecting hyperplanes. This representation turns out to have many equivalent guises making it analogous to the sign representation of a finite Coxeter group. One of these guises is (up to a twist) the cohomology of the Milnor fiber for the isolated singularity at 0 in the hypersurface defined by any homogeneous invari…
In this paper, we investigate statistics on alternating words under correspondence between ``possible reflection paths within several layers of glass'' and ``alternating words''. For $v=(v_1,v_2,\cdots,v_n)\in\mathbb{Z}^{n}$, we say $P$ is a path within $n$ glass plates corresponding to $v$, if $P$ has exactly $v_i$ reflections occurring at the $i^{\rm{th}}$ plate for all $i\in\{1,2,\cdots,n\}$. We give a recursion for the number of paths corresponding to $v$ satisfying $v \in \mathbb{Z}^n$ and $\sum_{i\geq 1} v_i=m$. Also, we establish recursions for statistics around the number of paths corr…
The partition algebra $\mathsf{P}_k(n)$ and the symmetric group $\mathsf{S}_n$ are in Schur-Weyl duality on the $k$-fold tensor power $\mathsf{M}_n^{\otimes k}$ of the permutation module $\mathsf{M}_n$ of $\mathsf{S}_n$, so there is a surjection $\mathsf{P}_k(n) \to \mathsf{Z}_k(n) := \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k}),$ which is an isomorphism when $n \ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\mathsf{Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities …
In 2005 J.L. Waldspurger proved the following theorem: given a finite real reflection group $W$, the closed positive root cone is tiled by the images of the open weight cone under the action of the linear transformations $id-w$. Shortly thereafter E. Meinrencken extended the result to affine Weyl groups. P.V. Bibikov and V.S. Zhgoon then gave a uniform proof for a discrete reflection group acting on a simply-connected space of constant curvature. In this paper we show that the Waldspurger and Meinrenken theorems of type A give a new perspective on the combinatorics of the symmetric group. In p…
We study the intersection lattice of the arrangement $\mathcal{A}^G$ of subspaces fixed by subgroups of a finite linear group $G$. When $G$ is a reflection group, this arrangement is precisely the hyperplane reflection arrangement of $G$. We generalize the notion of finite reflection groups. We say that a group $G$ is generated (resp. strictly generated) in codimension $k$ if it is generated by its elements that fix point-wise a subspace of codimension at most $k$ (resp. precisely $k$). If $G$ is generated in codimension two, we show that the intersection lattice of $\mathcal{A}^G$ is atomic. …
