Dillon Hanson:

TITLE:

Invariant Theory of Modular Reflection Groups

ABSTRACT:

We consider finite reflection groups acting over fields of arbitrary characteristic. Many arguments of classical invariant theory break down in the modular setting, when the characteristic of the field divides the order of the group. Here, we examine the action on differential derivations, which arise in Catalan combinatorics. We develop an analogue of Saito's freeness criterion that distinguishes the characteristic 2 case and obtain an explicit description of the invariant differential derivations for special linear groups and general linear groups over finite fields.

Blake Norman:

Title: An introduction to sharp permutation groups

Abstract: Let (G,X) be a finite permutation group with associated permutation character θ. If L = {θ(g) | g ∈ G-{1}}, then we say θ is of type L. In 1904, Blichfeldt proved that |G| divides the product, over all l ∈ L, θ(1)-θ(l). In particular, |G| is less than or equal to the product, over all l ∈ L, θ(1)-θ(l), and when equality holds, we say θ is a sharp character of type L. When the sharp permutation character θ is the permutation character associated to the permutation group (G,X), we say (G,X) is a sharp permutation group of type L. In this talk, we will discuss results for sharp permutation groups with a focus on the case |L| = 2.