Stephen Griffeth's preprints and publications |
Jack polynomials as fractional quantum Hall states and the Betti numbers of the m-equals ideal (with C. Berkesch and S. Sam) We prove conjectures of Bernevig and Haldane on the vanishing properties of Jack polynomials by relating them to representations of Cherednik algebras, and make our own conjecture calculating the Betti numbers of the m-equals ideal in terms of the combinatorics of cores and quotients. We prove our conjecture if m is large enough. |
Systems of parameters and holonomicity of A-hypergeometric systems (with C. Berkesch and E. Miller) We give a short, direct and elementary proof of the holonomicity of the A-hypergeometric system. |
Macdonald polynomials as characters of Cherednik algebra modules I prove that Haiman's n! theorem, giving an isomorphism between the Hilbert scheme of points in the plane and a certain equivariant Hilbert scheme, is equivalent to a bigraded character formula for irreducible Cherednik algebra modules. |
Generalized Jack polynomials and the representation theory of rational Cherednik algebras (with Charles Dunkl, in Selecta Mathematica) By using the family of orthogonal functions studied in "Orthogonal functions generalizing Jack polynomials", we determine when the Cherednik algebra of type G(r,1,n) is equivalent to its spherical subalgebra. As another application of the same technique, we show that the ordering on category O studied by I. Gordon in "Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras" is an ordering of O as a highest weight category. |
Appendix to ``Supports of irreducible spherical representations of rational Cherednik algebras of finite Coxeter groups'' (by P. Etingof) The paper by Etingof determines the support of the irreducible lowest weight module with trivial lowest weight for the rational Cherednik algebra of a Coxeter group. In the tiny note of an appendix, I prove that in the case of constant parameters, Etingof's determination of when this module is finite dimensional agrees with that of Varagnolo-Vasserot. |
Unitary representations of rational Cherednik algebras, II In this paper I describe an algorithm that takes as input an irreducible representation of one of the complex reflections groups in the infinite series G(r,p,n), and outputs, from small codimension to large codimension, a semilinear set of parameters for which the contravariant form on the corresponding irreducible lowest weight module for the Cherednik algebra is positive definite. |
Cell modules and canonical basic sets for Hecke algebras from Cherednik algebras (with Maria Chlouveraki and Iain Gordon, Contemporary Mathematics vol. 562, 2012) We label irreducible represetations of the finite Hecke algebra of a complex reflection group W by subsets of the irreducible complex linear representations of W; the Cherednik algebra associated to a choice of logarithms of the parameters of the finite Hecke algebra tells us how to do this. |
Catalan numbers for complex reflection groups (with Iain Gordon, to appear in American Journal of Math) Modulo a technical assumption about the Hecke algebra, we construct a (q,t)-Catalan number for each complex reflection group and confirm a conjecture of Bessis and Reiner on q-Catalan numbers. The method relates the q-Catalan numbers to quotients of the diagonal coinvariant ring. Our construction uses the rational Cherednik algebra, the Knizhnik-Zamolodchikov differential equations, and the Hecke algebra. The paper is built upon previous work of Opdam and Rouquier. |
Appendix to ``Unitary representations of rational Cherednik algebras'' (by P. Etingof and E. Stoica, Represent. Theory 13 (2009), 349--370) The paper by Etingof and Stoica initiates the study of unitary representations of rational Cherednik algebras, and as a consequence proves the Dunkl-Kasatani conjecture on the submodule structures of the polynomial representation of the rational Cherednik algebra of type A in full generality (a previous proof by Enomoto assumed the parameter was not a half integer). In the appendix, I use results of I. Cherednik and T. Suzuki to complete the classification of unitary modules for the rational Cherednik algebra of type A that Etingof and Stoica had conjectured in a preliminary version of the paper. |
Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces (with D. Anderson and E. Miller, to appear in Journal of the European Math Society, arXiv:0808.2785) We prove the Griffeth-Ram and Graham-Kumar conjectures on the ``positivity'' of the structure constants of equivariant K-theory of homogeneous spaces. This generalizes theorems of Graham in equivariant cohomology and Brion in ordinary K-theory. The main tools are Edidin-Graham's approximate mixing space approach to equivariant K-theory, Sierra's homological transversality theorem, and Brion's implementation of Kawamata-Viehweg vanishing for calculating structure constants in ordinary K-theory. |
Orthogonal functions generalizing Jack polynomials (to appear in Transactions of the AMS, arxiv.org/pdf/math.RT/0707.0251v1) This paper diagonalizes the standard (sometimes called ``Verma'') modules for the rational Cherednik algebra of type G(r,1,n) with respect to the Dunkl-Opdam subalgebra, and as an application determines the submodule structure in the case when the eigenspaces are one-dimensional. The eigenbasis obtained is a generalization of the non-symmetric Jack polynomials, and the combinatorics that controls the submodule structure is built upon (and analogous to) the Jucys-Murphy tableaux combinatorics that controls representations of the symmetric group. |
Jack polynomials and the coinvariant ring of G(r,p,n) (Proc. Amer. Math. Soc. 137 (2009), no. 5, 1621--1629, arXiv:0806.3292 ) This paper contains a strengthening of the results in my thesis. It shows that the descent representations, first consdiered by Garsia-Stanton for the symmetric group and later by Adin-Brenti-Roichman and Bagno-Biagioli for the groups G(r,p,n), are realizable as irreducible Hecke-algebra submodules of the coinvariant ring. The basic idea is to write down a particular basis for the coinvariant ring consisting of non-symmetric Jack polynomials and study the action of the rational Cherednik algebra on that basis. |
Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r,p,n) (to appear in Proceedings of the Edinburgh Mathematical Society, arxiv.org/pdf/math.RT/0612733) This paper builds a machinery of intertwining operators for the rational Cherednik algebra of type G(r,p,n) analogous to Cherednik's machinery for double affine Hecke algebras of Weyl groups. As an application, I prove the analog of Gordon's theorem (previously Haiman's conjecture) on the diagonal coinvariant ring for the groups G(r,p,n). I have tried to make this paper self-contained enough that mathematicians in algebraic combinatorics who are unfamiliar with rational Cherednik algebras can read it. |
Affine Hecke algebras and the Schubert calculus (with A. Ram, in the European Journal of Combinatorics 25 (2004), no. 8 1263--1283, arXiv:math/0405333) We studied the torus equivariant K-theory of homogeneous spaces by using Hecke algebras to obtain multiplication formulas for Schubert classes by codimension one Schubert classes, and to write down multiplication tables for all rank two groups. We make a positivity conjecture supported by these data. |
My thesis. (At the University of Wisconsin-Madison, August 2006.) I studied connections between coinvariant rings and various types of modules for rational Cherednik algebras. Some of the results of this thesis appeared in the papers ``Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r,p,n)'' and ''Jack polynomials and the coinvariant ring of G(r,p,n)'', but there are no inclusions among the three papers. |