Modular representations of p groups

MINI-COURSE: p-MODULAR REPRESENTATIONS OF p-ADIC GROUPS

1.1. The case ` 6= p. In this case, we are in the setting of the classical Local Langlands Correspondence, which can be stated (roughly) as follows: Let n ≥ 1. We then have an injective map (1)  continuous representations of Gal(Qp/Qp) on n-dimensional Q`-vector spaces, up to isomorphism  ↪−→  irreducible, admissible representations of GLn(Qp) on Q`-vector spaces, up to isomor...

Resolutions of p-Modular TQFT’s and Representations of Symmetric Groups

Modular Representations of Symmetric Groups

Over the last few weeks, we have seen lots of things in representation theory—we began by looking at an onslaught of examples to see explicitly what representations look like, we discussed character theory and looked at character tables, we talked about representations from a module-theoretic perspective, and then we saw how we can model the representation theory of the symmetric group via the ...

Modular Representations of Profinite Groups

Our aim is to transfer several foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We are thus interested in profinite modules over the completed group algebra k[[G]] of a profinite group G, where k is a finite field of characteristic p. We define the concept of relative projectivity for a profinite k[[G]]-module. We prove a cha...

Minimal polynomials of elements of order p in p-modular projective representations of alternating groups

Let F be an algebraically closed field of characteristic p > 0 and G be a quasi-simple group with G/Z(G) ∼= An. We describe the minimal polynomials of elements of order p in irreducible representations of G over F . If p = 2 we determine the minimal polynomials of elements of order 4 in 2modular irreducible representations of An, Sn, 3 ·A6, 3 ·S6, 3 ·A7, and 3 ·S7.