Related papers: On the inductive blockwise Alperin weight conditio…

We study combinatorial blocks of multipartitions, exploring further the notions of weight, hub and core block introduced by the author in earlier papers. We answer the question of which pairs (w,theta) occur as the weight and hub of a…

This paper proposes a conjectural framework for Alperin's Main Problem of Block Theory from 1976. The character sets considered here are defined by nonvanishing at given elements, not only by degree conditions. From this point of view,…

In this paper we prove that Brou\'{e}'s abelian defect group conjecture is true for the finite odd-dimensional orthogonal groups $\SO_{2n+1}(q)$ at linear primes with $q$ odd. We first make use of the reduction theorem of…

The McKay--Navarro conjecture is a refinement of the McKay conjecture that additionally takes the action of some Galois automorphisms into account. We verify the inductive McKay--Navarro condition in the defining characteristic for the…

This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the…

We determine the blocks of the walled Brauer algebra in characteristic zero. These can be described in terms of orbits of the action of a Weyl group of type $A$ on a certain set of weights. In positive characteristic we give a linkage…

In this paper, we classify all irreducible weight modules with finite-dimensional weight spaces over the affine-Virasoro Lie algebra of type $A_1$.

Many open conjectures in the representation theory of finite groups can be studied by reducing them to related questions about quasi-simple groups. In such studies, $p$-radical subgroups typically play a critical role. To classify the…

For a prime $\ell$, the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with $\ell'$-degree and the corresponding set for the normalizer of a Sylow $\ell$- subgroup. Navarro's refinement…

We exhibit abelian topological groups admitting no nontrivial strongly continuous irreducible representations in Banach spaces. Among them are some abelian Banach-Lie groups and some monothetic subgroups of the unitary group of a separable…

We prove that for forms of U(3) which are compact at infinity and split at places dividing a prime p, in generic situations the Serre weights of a mod p modular Galois representation which is irreducible when restricted to each…

Let $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of positive characteristic $p$. Let $\cd$ be an involution of the algebra $FG$ which is a linear extension of an anti-automorphism of the group $G$ to $FG$. If…

We characterise the Morita equivalence classes of blocks with extraspecial defect groups $p_+^{1+2}$ for $p \geq 5$, and so show that Donovan's conjecture and the Alperin-McKay conjecture hold for such $p$-groups. For $p=3$ we reduce…

The Alperin--McKay conjecture relates irreducible characters of a block of an arbitrary finite group to those of its $p$-local subgroups. A refinement of this conjecture was stated by the author in a previous paper. We prove that this…

We prove that every element of the special linear group can be represented as the product of at most six block unitriangular matrices, and that there exist matrices for which six products are necessary, independent of indexing. We present…

Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\bar\kappa$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $<\bar\kappa$ type definable…

This paper establishes a classification of the critical numbers for restricted sumsets in finite abelian groups, determining them exactly for even-order groups and bounding them for odd-order groups, while revealing a fundamental structural…

The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem. Let G be a countable discrete group, f one…

For any algebra $A$ over an algebraically closed field $\mathbb{F}$, we say that an $A$-module $M$ is Schurian if $\mathrm{End}_A(M) \cong \mathbb{F}$. We say that $A$ is Schurian-finite if there are only finitely many isomorphism classes…

In this paper we determine the torsion free rank of the group of endotrivial modules for any finite group of Lie type, in both defining and non-defining characteristic. On our way to proving this, we classify the maximal rank $2$ elementary…