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 solve a local case of the conjecture by considering subgroups of the general linear group that fix one hyperplane. 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 provide an explicit description of the invariant ring for these Landweber-Stong groups reflecting about a fixed hyperplane. In addition, we will show that the Hilbert series of the invariant space counts orbits, solving a special case of the Lewis-Reiner-Stanton conjecture.
- Courses and Schedules
- News & Events