Outer derivations on blocks of group algebras

TL;DR

This paper investigates the structure of derivations on blocks of group algebras, establishing criteria for their non-vanishing and demonstrating their non-triviality in various significant cases, including principal blocks and groups of Lie type.

Contribution

It provides new group-theoretic criteria for the non-vanishing of Hochschild cohomology and shows that this cohomology is non-zero in broad classes of blocks, especially in prime characteristic greater than 5.

Findings

HH^1(B,B) is non-zero for principal blocks with abelian defect groups

HH^1(B,B) is non-zero for blocks of symmetric and alternating groups

HH^1(B,B) is non-zero for blocks of finite groups of Lie type in defining characteristic

Abstract

Let $G$ be a finite group whose order is divisible by the characteristic of a field $k$. If $B$ is a block of $kG$ with defect group $P$, we prove that the space of derivations on $kP$ which are restrictions of derivations on $kG$, modulo inner derivations, is isomorphic to a subspace of $\operatorname{HH}^1(B,B)$. Using this, we provide various group theoretic criteria for the non-vanishing of $\operatorname{HH}^1(B,B)$. In particular, we show $\operatorname{HH}^1(B,B)\neq 0$ for principal blocks having abelian defect group, for all blocks of the symmetric and alternating groups, for blocks of finite groups of Lie type in defining characteristic, and for blocks of general linear groups in any characteristic. Building on this, we show that if $k$ has prime characteristic $p>5$, and if $B$ is any block of $kG$ with Sylow defect group, then $\operatorname{HH}^1(B,B)\neq 0$. By the same method we also prove that if $k$ has prime characteristic $p>5$, then the first Hochschild cohomology group of any twisted group algebra is non-zero.