The $p=2$ case of the multiplicity-free solvability criterion for $\mathrm{HH}^1$ of blocks with abelian defect groups (Linckelmann–Wang, 2025)

Linckelmann and Wang (2025) proved that, for a block $B$ of a finite group algebra over a field of characteristic $p$ whose defect group $P$ is abelian and whose inertial quotient $E$ satisfies $[P,E]=P$, solvability of the Lie algebra $\mathrm{HH}^1(B)$ is governed by the multiplicity-freeness of the $\mathbb{F}_pE$-module $P/\Phi(P)$: if $P/\Phi(P)$ is multiplicity free then $\mathrm{HH}^1(B)$ is solvable, and the converse is proved only under the hypothesis that $p$ is odd. The case $p=2$ is left unresolved by the source paper: it is not known whether a non-multiplicity-free $\mathbb{F}_2E$-module structure of $P/\Phi(P)$ forces the Lie algebras $\mathrm{HH}^1(kP)^E$, $\mathrm{HH}^1(k_\alpha(P\rtimes E))$, and — via stable equivalences of Morita type — $\mathrm{HH}^1(B)$ to be non-solvable. This problem asks to settle the $p=2$ case.

Background

Let $p$ be a prime and $k$ a field of characteristic $p$. A block $B$ of a finite group algebra $kG$ is an indecomposable direct summand of $kG$ as an algebra; a defect group $P$ of $B$ is a maximal $p$-subgroup of $G$ such that $kP$ occurs as a direct summand of $B$ as a $kP$-$kP$-bimodule. Blocks are the basic objects of the modular representation theory of finite groups, and the inertial quotient $E = N_G(P,e)/PC_G(P)$ (where $e$ is a primitive idempotent of $kG$ belonging to $B$ whose defect group is $P$) is a $p'$-group, that is, a finite group of order prime to $p$, which acts by conjugation on $P$; the subgroup $[P,E]$ generated by all commutators $(^{e}u)u^{-1}$ with $u \in P$, $e \in E$ is called the hyperfocal subgroup of $P$ in $P \rtimes E$.

For any finite-dimensional $k$-algebra $A$, the first Hochschild cohomology $\mathrm{HH}^1(A) = \mathrm{Der}(A)/\mathrm{IDer}(A)$ — the space of derivations of $A$ modulo the inner derivations $[a,-]$ — is a Lie algebra under the commutator bracket $[f,g] = fg - gf$. A Lie algebra is solvable if its derived series eventually reaches zero. Because $\mathrm{HH}^1$ is preserved by Morita equivalences (equivalences of module categories induced by tensoring with a suitable bimodule) and, as shown by König, Liu, and Zhou, by stable equivalences of Morita type (equivalences of stable module categories, taken modulo morphisms factoring through projective modules, induced by suitable bimodules), the Lie algebra $\mathrm{HH}^1(B)$ is an invariant through which the local action of $E$ on $P$ can influence — and in favorable cases determine — the global structure of the block. Prior work had clarified various aspects of this Lie algebra: Eisele and Raedschelders gave a general quiver-theoretic approach to solvability of $\mathrm{HH}^1(A)$; Linckelmann and Rubio y Degrassi related solvability and simplicity to the Ext-quiver of $A$, and showed that when $\mathrm{HH}^1(A)$ is simple with at most two parallel arrows, the characteristic is not 2; Murphy computed $\mathrm{HH}^1$ of the blocks of the Mathieu groups.

Linckelmann and Wang study the case where $P$ is abelian and $[P,E] = P$. The Frattini subgroup $\Phi(P)$ is the intersection of the maximal subgroups of $P$; the quotient $P/\Phi(P)$ is elementary abelian and is naturally a module for $\mathbb{F}_pE$ (the group algebra of $E$ over the field with $p$ elements), semisimple because $p$ does not divide $|E|$. It is called multiplicity free if no simple summand occurs more than once in its direct-sum decomposition. Their Theorem 1.2 states that if $[P,E] = P$ and $P/\Phi(P)$ is multiplicity free, then the Lie algebra $\mathrm{HH}^1(kP)^E$ of $E$-stable classes (that is, classes of derivations of $kP$ fixed by the action of $E$ induced from its conjugation action on $P$) is solvable, with the converse proved when $p$ is odd; Theorem 1.3 proves the analogous criterion for twisted group algebras $k_\alpha(P\rtimes E)$ (algebras like the group algebra of the semidirect product but with multiplication twisted by a 2-cocycle $\alpha$ of $E$, that is, a map $\alpha: E \times E \to k^\times$ satisfying the cocycle identity), again with the converse for odd $p$; and Theorem 1.1 transfers the criterion to blocks $B$ whose inertial quotient acts freely on $P \setminus \{1\}$, using a stable equivalence of Morita type due to Puig (for arbitrary blocks with abelian defect, Külshammer's theorem first gives a Morita equivalence with $k_\alpha(P\rtimes E)$ when $P$ is normal). The key mechanism is a surjective Lie algebra homomorphism from $\mathrm{HH}^1(kP)^E$ onto a product of general linear Lie algebras $\prod_j \mathfrak{gl}_{n_j}(\mathbb{F}_j)$, where $n_j$ are the multiplicities of the simple $kE$-summands of $J(kP)/J(kP)^2$ ($J$ denoting the Jacobson radical) and $\mathbb{F}_j$ are extension fields of $k$; the kernel is nilpotent. When $p$ is odd, any factor $\mathfrak{gl}_n(\mathbb{F})$ with $n \ge 2$ is non-solvable, which yields the converse. At $p = 2$ this argument fails precisely for $n = 2$, because $\mathfrak{gl}_2(\mathbb{F})$ is solvable in characteristic 2 (its derived algebra $\mathfrak{sl}_2$ becomes a nilpotent Heisenberg-type algebra), and the paper leaves the entire $p = 2$ converse unresolved. The authors also note that uniform application of the criterion to all blocks with abelian defect groups would require the stable equivalences predicted by Broué's abelian defect group conjecture — the famous conjecture that every block with abelian defect group is derived equivalent to its Brauer correspondent (the corresponding block of the normalizer of a defect group) — which remains open in general, though it has been verified for additional classes of blocks in recent work of Q. Zhou and K. Zhang and of Hu, Zhang, and Zhou.

Problem Statement

Settle the case $p=2$ of the multiplicity-free solvability criteria of Linckelmann–Wang for Hochschild cohomology Lie algebras of blocks with abelian defect groups. Let $k$ be a field of characteristic 2 (for block-level statements, assumed large enough for the subgroups involved), let $P$ be a non-trivial finite abelian 2-group, and let $E$ be a finite group of odd order acting on $P$ with hyperfocal condition $[P,E] = P$.

(i) Decide whether the converse of the source criteria holds at $p=2$: if the $\mathbb{F}_2E$-module $P/\Phi(P)$ is not multiplicity free, is the Lie algebra $\mathrm{HH}^1(kP)^E$ necessarily non-solvable? By the reductions in the source paper, which prove the equivalence between $\mathrm{HH}^1(k_\alpha(P \rtimes E))$ and $\mathrm{HH}^1(kP)^E$ only when $E$ is abelian, a related but formally weaker question is whether $\mathrm{HH}^1(k_\alpha(P \rtimes E))$ is non-solvable for every 2-cocycle $\alpha$ inflated from $E$ when $E$ is abelian; settling that restricted case does not by itself settle (i) for non-abelian $E$.

(ii) More generally, determine whether the solvability of $\mathrm{HH}^1(kP)^E$ at $p=2$ is governed by the multiplicity structure of the $\mathbb{F}_2E$-module $P/\Phi(P)$ alone — that is, whether a necessary and sufficient criterion in terms of the multiplicities of its simple summands exists — and, under the hypotheses for which the source paper proves the reduction to $\mathrm{HH}^1(kP)^E$ (for instance $E$ abelian, or $E$ acting freely on $P \setminus \{1\}$), determine the corresponding criterion for the solvability of $\mathrm{HH}^1(k(P \rtimes E))$ and $\mathrm{HH}^1(k_\alpha(P \rtimes E))$ when $p=2$.

The $p=2$ case can be resolved by a proof that the converse holds at $p=2$, by a criterion as requested in (ii) (including a characterization of the failure cases, if any), or by an explicit counterexample: a configuration $(k, P, E, \alpha)$ at $p=2$ with $[P,E]=P$ in which $P/\Phi(P)$ is not multiplicity free yet $\mathrm{HH}^1(kP)^E$ is solvable (or, when $E$ is abelian, yet $\mathrm{HH}^1(k_\alpha(P \rtimes E))$ is solvable for some 2-cocycle $\alpha$ inflated from $E$). The corresponding block-level question, under the hypotheses of the source criteria — a non-trivial abelian defect group $P$ with inertial quotient $E$ acting freely on $P \setminus \{1\}$ (Theorem 1.1, via Puig's stable equivalence of Morita type), $P$ normal in $G$ (Külshammer's Morita equivalence), or $E$ abelian with a stable equivalence of Morita type between $B$ and its Brauer correspondent (Corollary 7.7) — is whether non-multiplicity-freeness of $P/\Phi(P)$ forces $\mathrm{HH}^1(B)$ to be non-solvable at $p=2$.

References

1. Markus Linckelmann and Jialin Wang, On the solvability of the Lie algebra HH^1(B) for blocks of finite groups, Journal of the London Mathematical Society 112 (2025), no. 6, DOI 10.1112/jlms.70407, https://doi.org/10.1112/jlms.70407; preprint arXiv:2503.16697 (2025), https://arxiv.org/abs/2503.16697
2. Burkhard Külshammer, Crossed products and blocks with normal defect groups, Communications in Algebra 13 (1985). DOI 10.1080/00927878508823154, https://doi.org/10.1080/00927878508823154
3. Lluis Puig, Une correspondance de modules pour les blocs à groupes de défaut abéliens, Geometriae Dedicata 37 (1991). DOI 10.1007/BF00150403, https://doi.org/10.1007/BF00150403
4. Steffen König, Yuming Liu, and Guodong Zhou, Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander-Reiten conjecture, Transactions of the American Mathematical Society 364 (2012). DOI 10.1090/S0002-9947-2011-05358-4, https://doi.org/10.1090/S0002-9947-2011-05358-4
5. Radha Kessar and Markus Linckelmann, On blocks with Frobenius inertial quotient, Journal of Algebra 249 (2002). DOI 10.1006/jabr.2001.9058, https://doi.org/10.1006/jabr.2001.9058
6. Markus Linckelmann and Lleonard Rubio y Degrassi, Block algebras with HH^1 a simple Lie algebra, The Quarterly Journal of Mathematics 69 (2018). DOI 10.1093/qmath/hay017, https://doi.org/10.1093/qmath/hay017
7. Markus Linckelmann and Lleonard Rubio y Degrassi, On the Lie algebra structure of HH^1(A) of a finite-dimensional algebra A, Proceedings of the American Mathematical Society 148 (2020), arXiv:1903.08484. DOI 10.1090/proc/14875, https://arxiv.org/abs/1903.08484
8. William Murphy, The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups, Journal of Group Theory 26 (2023), arXiv:2110.02941. DOI 10.1515/jgth-2021-0176, https://arxiv.org/abs/2110.02941
9. Florian Eisele and Theo Raedschelders, On solvability of the first Hochschild cohomology of a finite-dimensional algebra, Transactions of the American Mathematical Society 373 (2020), DOI 10.1090/tran/8064, https://doi.org/10.1090/tran/8064; preprint arXiv:1903.07380, https://arxiv.org/abs/1903.07380
10. Matthew Antrobus, On solvability of the first Hochschild cohomology of odd p-groups, arXiv:2512.14555 (2025). https://arxiv.org/abs/2512.14555
11. Benjamin Briggs and Lleonard Rubio y Degrassi, Outer derivations on blocks of group algebras, arXiv:2601.09602 (2026). https://arxiv.org/abs/2601.09602
12. Qianhu Zhou and Kun Zhang, 2-blocks with abelian defect groups and inertial quotient of prime order, arXiv:2604.09974 (2026). https://arxiv.org/abs/2604.09974
13. Xueqin Hu, Kun Zhang, and Yuanyang Zhou, Hyperfocal subalgebras of hyperfocal abelian Frobenius blocks, arXiv:2602.20613 (2026). https://arxiv.org/abs/2602.20613