Outer derivations on blocks of group algebras
Benjamin Briggs, Lleonard Rubio y Degrassi
TL;DR
This work establishes a blockwise bridge between outer derivations and first Hochschild cohomology, revealing when $HH^1(B,B)$ is nonzero by examining derivations that originate from defect-group algebras. A central criterion uses centralizers: if there exists a $p$-element $x$ with a nonzero map $H_1(C_P(x),\mathbb{F}_p)\to H_1(C_G(x),\mathbb{F}_p)$, then $HH^1(B,B)\neq 0$, enabling broad nonvanishing results for blocks of symmetric groups, GL$_n$, and finite groups of Lie type, and ultimately proving $S(p)$ for all primes $p>5$. The authors confirm Linckelmann's conjecture for blocks of $S_n$ and $A_n$, extend nonvanishing results to twisted group algebras in characteristic $p>5$, and show that $HH^1$ persists under separable and stable Morita equivalences, strengthening connections to Broué-type conjectures. Collectively, the results illuminate the prevalence of outer derivations in modular representation theory and provide practical criteria for predicting nonvanishing of $HH^1$ across many families of groups and their twisted variants.
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.