For a finite abelian p-group A and a subgroup $\Gamma \le \operatorname {\mathrm {Aut}}(A)$, we say that the pair $(\Gamma ,A)$ is fusion realizable if there is a saturated fusion system ${\mathcal {F}}$ over a finite p-group $S\ge A$ such that $C_S(A)=A$, $\operatorname {\mathrm {Aut}}_{{\mathcal {F}}}(A)=\Gamma $ as subgroups of $\operatorname {\mathrm {Aut}}(A)$, and . In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for $p=2$ or $3$ and $\Gamma $ one of the Mathieu groups, that the only ${\mathbb {F}}_p\Gamma $-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian p-groups with an abelian subgroup of index p. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete p-toral groups with an abelian subgroup of index p. In all of these cases (finite or infinite), we reduce the problem to one of listing all 𝔽pG-modules (for G finite) satisfying certain conditions: a problem which was solved in the earlier paper [15] using the classification of finite simple groups.

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$ -group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$ -adic vector space associated with  $M$ . This leads to our second variation, which is a study of the finite linear groups that can arise.

From the mid-1990s onwards, the main focus of L. G. Kovács’ research was on Lie powers. This brief survey presents some of the key results on Lie powers obtained by Kovács and his collaborators, and discusses some subsequent developments and applications of this work.

We study basic properties of the category of smooth representations of a p-adic group G with coefficients in any commutative ring R in which p is invertible. Our main purpose is to prove that Hecke algebras are Noetherian whenever R is; this question arose naturally with Bernstein's fundamental work for R = ℂ, in which case he proved this Noetherian property. In a first step, we prove that Noetherianity would follow from a generalization of the so-called second adjointness property between parabolic functors, also due to Bernstein for complex representations. Then, to attack this second adjointness, we introduce and study ‘parahoric functors’ between representations of groups of integral points of smooth integral models of G and of their ‘Levi’ subgroups. Applying our general study to Bruhat-Tits parahoric models, we get second adjointness for minimal parabolic groups. For non-minimal parabolic subgroups, we have to restrict to classical and linear groups, and use smooth models associated with Bushnell-Kutzko and Stevens semi-simple characters. The same strategy should apply to ‘tame’ groups, using Yu's smooth models and generic characters.

Let $L(V)$ be the free Lie algebra on a finite-dimensional vector space $V$ over a field $K$, with homogeneous components $L^n(V)$ for $n \geq 1$. If $G$ is a group and $V$ is a $KG$-module, the action of $G$ extends naturally to $L(V)$, and the $L^n(V)$ become finite-dimensional $KG$-modules, called the Lie powers of $V$. In the decomposition problem, the aim is to identify the isomorphism types of indecomposable $KG$-modules, with their multiplicities, in unrefinable direct decompositions of the Lie powers. This paper is concerned with the case where $G$ has prime order $p$, and $K$ has characteristic $p$. As is well known, there are $p$ indecomposables, denoted here by $J_1,\dots,J_p$, where $J_r$ has dimension $r$. A theory is developed which provides information about the overall module structure of $L(V)$ and gives a recursive method for finding the multiplicities of $J_1,\dots,J_p$ in the Lie powers $L^n(V)$. For example, the theory yields decompositions of $L(V)$ as a direct sum of modules isomorphic either to $J_1$ or to an infinite sum of the form $J_r \oplus J_{p-1} \oplus J_{p-1} \oplus \ldots $ with $r \geq 2$. Closed formulae are obtained for the multiplicities of $J_1,\dots,J_p$ in $L^n(J_p)$ and $L^n(J_{p-1})$. For $r < p-1$, the indecomposables which occur with non-zero multiplicity in $L^n(J_r)$ are identified for all sufficiently large $n$.

2000 Mathematical Subject Classification: 17B01, 20C20.

We prove the quantum version - for Hecke algebras $H(A_n)$ of type $A$ at roots of unity - of Kleshchev's modular branching rule for symmetric groups. This result describes the socle of the restriction of an irreducible $H(A_n)$-module to the subalgebra $H(A_{n-1})$. As a consequence, we describe the quantum version of the Mullineux involution describing the irreducible module obtained on twisting an irreducible module with the sign representation.

1991 Mathematics Subject Classification: 20C05, 20G05.

In this paper we find the multiplicities $\dim L(\lambda)_{\lambda-\alpha}$ where $\alpha$ is an {\em arbitrary} root and $L(\lambda)$ is an irreducible $SL_n$-module with highest weight $\lambda$. We provide different bases of the corresponding weight spaces and outline some applications to the symmetric groups. In particular we describe certain composition multiplicities in the modular branching rule.

1991 Mathematics Subject Classification: 20C05, 20G05.

Every irreducible ordinary character in a p-block of a finite metabelian group is of height 0 if and only if the defect group of the p-block is abelian.