arXiv subjects

Stephen Griffeth

Publications and source records attributed to Stephen Griffeth.

23 records · Page 2Linked to original sources

arXiv2009

Unitary representations of rational Cherednik algebras

We study unitarity of lowest weight irreducible representations of rational Cherednik algebras. We prove several general results, and use them to determine which lowest weight representations are unitary in a number of cases. In particular, in type A, we give a complete description of the unitarity locus (justified in Section 5 and appendix by Stephen Griffeth), and resolve a question by Cherednik on the unitarity of the irreducible subrepresentation of the polynomial representation. Also, as a by-product, we establish Kasatani's conjecture in full generality (the previous proof by Enomoto assumes that the parameter c is not a half-integer).

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arXiv2008

Orthogonal functions generalizing Jack polynomials

The rational Cherednik algebra $\HH$ is a certain algebra of differential-reflection operators attached to a complex reflection group $W$. Each irreducible representation $S^λ$ of $W$ corresponds to a standard module $M(λ)$ for $\HH$. This paper deals with the infinite family $G(r,1,n)$ of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra $\ttt$ of $\HH$ discovered by Dunkl and Opdam. In this case, the irreducible $W$-modules are indexed by certain sequences $λ$ of partitions. We first show that $\ttt$ acts in an upper triangular fashion on each standard module $M(λ)$, with eigenvalues determined by the combinatorics of the set of standard tableaux on $λ$. As a consequence, we construct a basis for $M(λ)$ consisting of orthogonal functions on $\CC^n$ with values in the representation $S^λ$. For $G(1,1,n)$ with $λ=(n)$ these functions are the non-symmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of $M(λ)$ in the case in which the orthogonal functions are all well-defined.

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arXiv2008

Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r,p,n)

The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a self-contained and elementary proof of the analog for the groups G(r,p,n), with r>1, of Gordon's theorem (previously Haiman's conjecture) on the diagonal coinvariant ring. We impose no restriction on p; the result for p<r has been proved by Vale using a technique analogous to Gordon's. Because of the combinatorial application to Haiman's conjecture, the paper is logically self-contained except for standard facts about complex reflection groups. The main results should be accessible to mathematicians working in algebraic combinatorics who are unfamiliar with the impressive range of ideas used in Gordon's proof of his theorem.

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arXiv2008

Jack polynomials and the coinvariant ring of $G(r,p,n)$

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''.

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arXiv2004

Affine Hecke algebras and the Schubert calculus

Using a combinatorial approach which avoids geometry, this paper studies the ring structure of K_T(G/B), the T-equivariant K-theory of the (generalized) flag variety G/B. Here the data is a complex reductive algebraic group (or symmetrizable Kac-Moody group) G, a Borel subgroup B, and a maximal torus T, and K_T(G/B) is the Grothendieck group of T-equivariant coherent sheaves on G/B. We prove "Pieri-Chevalley" formulas for the products of a Schubert class by a homogeneous line bundle (dominant or anti-dominant) and for products of a Schubert class by a codimension 1 Schubert class. All of these Pieri-Chevalley formulas are given in terms of the combinatorics of the Littelmann path model. We give explicit computations of products of Schubert classes for the rank two cases and this data allows us to make a "positivity conjecture" generalizing the theorems of Brion and Graham, which treat the cases K(G/B) and H_T^*(G/B), respectively.

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