Abstract:

Lewis, Reiner, and Stanton conjectured a combinatorial description for the Hilbert series of invariants in the polynomial ring modulo Frobenius powers under the action of the general linear group over arbitrary finite fields. We will discuss the case of modular reflection groups in general. 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. We will describe the invariant ring for these Landweber-Stong groups reflecting about a fixed hyperplane.