Algebra and Combinatorics Seminar

Date: January 31, 2020

Time: 3:00PM - 3:50PM

Location: BLOC 628

Speaker: Chelsea Drescher , University of North Texas

Title: Invariants of polynomials modulo Frobenius powers and Hilbert series

Abstract: In 2017, Lewis, Reiner, and Stanton conjectured a connection between the modular general linear group and (q,t)-binomial coefficients. This conjecture was an analog to a theorem connecting the representation theory of rational Cherednik algebras for Coxeter groups and Catalan numbers. We will describe a local case - the invariant ring for groups reflecting about a fixed hyperplane acting on a polynomial ring modulo Frobenius powers. When the characteristic of the underlying field divides the order of the group, the subgroup fixing a reflecting hyperplane is a semi-direct product of diagonalizable reflections and transvections. The resulting Hilbert series counts the number of orbits of the group acting on a vector space, solving a special case of the Lewis, Reiner, Stanton conjecture.