Outer derivations on blocks of group algebras

Abstract

Let $B$ be a block of a finite group algebra $kG$. Linckelmann has conjectured that if the defect group $P$ of $B$ is nontrivial then $B$ admits a $k$-linear outer derivation, or in other words $\operatorname{HH}^1(B)\neq 0$. We provide various criteria for the non-vanishing of $\operatorname{HH}^1(B)$ in terms of the subgroup $P\subseteq G$. In particular, we show that Linckelmann's conjecture holds 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. Our main tool relates the non-vanishing of the Batalin-Vilkovisky operator $\Delta\colon \operatorname{HH}^1(B)\to \operatorname{HH}^0(B)$ to the existence of a certain elements of $P$, called extra-strong non-Schur elements. We show that extra-strong non-Schur $p$-elements exist for all groups having order divisible by a prime $p>5$, extending work of Fleischmann, Janiszczak, and Lempken. Using this, we prove that if $k$ has characteristic $p>5$, then Linckelmann's conjecture holds for all blocks of $kG$ with Sylow defect group. Likewise, when $k$ has characteristic $p>5$ we deduce that the first Hochschild cohomology is non-zero for any nontrivial twisted group algebra over $k$.