On indecomposable projective representations of finite groups over fields of characteristic p>0

Abstract

Abstract. Let G be a finite group, F a field of characteristic p with p | |G|, and F λ G the twisted group algebra of the group G and the field F with a 2-cocycle λ ∈ Z 2 (G, F * ). We give necessary and sufficient conditions for F λ G to be of finite representation type. We also introduce the concept of projective F -representation type for the group G (finite, infinite, mixed) and we exhibit finite groups of each type.Introduction. Let F be a field of characteristic p > 0, F * the multiplicative group of the field F , F p = {a p : a ∈ F }, G a finite group of order |G|, where p | |G|, andand Z 2 (G, F * ) the group of all F * -valued normalized 2-cocycles of the group G, where we assume that G acts trivially on F * (see [26, Chapter 1]). Denote by F λ G the twisted group algebra of the group G and the field F with a cocycle λ ∈ Z 2 (G, F * ) and by rad F λ G the radical ofHigman [25] showed that if F is an infinite field and G is an abelian p-group which is neither cyclic nor of order 4, then there exist infinitely many non-isomorphic indecomposable F G-modules of F -dimension n for every natural number n > 1. If G is the non-cyclic group of order 4, then the preceding result is valid for even natural numbers n.

Absolutely indecomposable representations of a twisted group algebra of a finite p-group over a field of characteristic p

“…Given a 2-cocycle λ ∈ Z 2 (G, K * ), we denote by K λ G a twisted group algebra of a finite group G over a field K of characteristic p corresponding to λ (see [18, p. 66]). We recall from [5] that K λ G is of finite representation type if and only if K λ G p is a uniserial algebra.…”

Finite-dimensional twisted group algebras of semi-wild representation type

“…Let G be a finite group, G p a Sylow p-subgroup of G, K a field of characteristic p and λ ∈ Z 2 (G, K * ). We recall from [4] that the twisted group algebra K λ G is of finite representation type if and only if the algebra K λ G p is uniserial.…”

“…Then C = c 1 × c 2 is of type (4,4). Denote by θ j a root of the polynomial X 2 − δ j in the algebraic closure of the field F .…”