### A class of projective representations of hyperoctahedral groups and Schur Q-functions

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Let K be a field of characteristic p > 0, K* the multiplicative group of K and $G={G}_{p}\times B$ a finite group, where ${G}_{p}$ is a p-group and B is a p’-group. Denote by ${K}^{\lambda}G$ a twisted group algebra of G over K with a 2-cocycle λ ∈ Z²(G,K*). We give necessary and sufficient conditions for G to be of OTP projective K-representation type, in the sense that there exists a cocycle λ ∈ Z²(G,K*) such that every indecomposable ${K}^{\lambda}G$-module is isomorphic to the outer tensor product V W of an indecomposable ${K}^{\lambda}{G}_{p}$-module V and a simple...

Let S be a commutative complete discrete valuation domain of positive characteristic p, S* the unit group of S, Ω a subgroup of S* and $G={G}_{p}\times B$ a finite group, where ${G}_{p}$ is a p-group and B is a p’-group. Denote by ${S}^{\lambda}G$ the twisted group algebra of G over S with a 2-cocycle λ ∈ Z²(G,S*). For Ω satisfying a specific condition, we give necessary and sufficient conditions for G to be of OTP projective (S,Ω)-representation type, in the sense that there exists a cocycle λ ∈ Z²(G,Ω) such that every indecomposable...

Let G be a finite group, K a field of characteristic p > 0, and ${K}^{\lambda}G$ the twisted group algebra of G over K with a 2-cocycle λ ∈ Z²(G,K*). We give necessary and sufficient conditions for ${K}^{\lambda}G$ to be of semi-wild representation type in the sense of Drozd. We also introduce the concept of projective K-representation type for a finite group (tame, semi-wild, purely semi-wild) and we exhibit finite groups of each type.

Let G be a noncyclic abelian p-group and K be an infinite field of finite characteristic p. For every 2-cocycle λ ∈ Z²(G,K*) such that the twisted group algebra ${K}^{\lambda}G$ is of infinite representation type, we find natural numbers d for which G has infinitely many faithful absolutely indecomposable λ-representations over K of dimension d.

Let G be a finite group, F a field of characteristic p with p||G|, and ${F}^{\lambda}G$ the twisted group algebra of the group G and the field F with a 2-cocycle λ ∈ Z²(G,F*). We give necessary and sufficient conditions for ${F}^{\lambda}G$ 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.

Our aim is to determine necessary and sufficient conditions for a finite nilpotent group to have a faithful irreducible projective representation over a field of characteristic p ≥ 0.