arXiv subjects

David A. Craven

Publications and source records attributed to David A. Craven.

26 records · Page 2Linked to original sources

arXiv2012

Perverse Equivalences and Broué's Conjecture II: The Cyclic Case

We study Broué's abelian defect group conjecture for groups of Lie type using the recent theory of perverse equivalences and Deligne--Lusztig varieties. Our approach is to analyze the perverse equivalence induced by certain Deligne--Lusztig varieties (the geometric form of Broué's conjecture) directly; this uses the cohomology of these varieties, together with information from the cyclotomic Hecke algebra. We start with a conjecture on the cohomology of these Deligne--Lusztig varieties, prove various desirable properties about it, and then use this to prove the existence of the perverse equivalences predicted by the geometric form of Broué's conjecture whenever the defect group is cyclic (except possibly for two blocks whose Brauer tree is unknown). This is a necessary first step to proving Broué's conjecture in general, as perverse equivalences are built up inductively from various Levi subgroups. This article is the latest in a series by Raphael Rouquier and the author with the eventual aim of proving Broué's conjecture for unipotent blocks of groups of Lie type.

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arXiv2010

Perverse Equivalences and Broué's Conjecture

We give a new approach to the construction of derived equivalences between blocks of finite groups, based on perverse equivalences, in the setting of Broué's conjecture. We provide in particular local and global perversity data describing the principal blocks and the derived equivalences for a number of finite simple groups with Sylow subgroups C3xC3. We also examine extensions to automorphism groups in a general setting.

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arXiv2010

On the Unit Conjecture for Supersoluble Group Rings, I

We introduce structure theorems for the study of the unit conjecture for supersoluble group rings and apply our results to the (Passman) fours group G. We show that over any field K, the group algebra KG has no non-trivial units of length at most 3, and find that the Promislow set can never be the support of a unit in KG. We conclude our work with an introduction to the theory of "consistent chains" toward a preliminary analysis of units of higher length in KG.

math.RA↗

arXiv2010

Fusion systems on small p-groups

In this article we study several classes of `small' 2-groups: we complete the classification, started in [Stancu, 2006], of all saturated fusion systems on metacyclic p-groups for all primes p. We consider Suzuki 2-groups, and classify all center-free saturated fusion systems on 2-groups of 2-rank 2. We end by classifying all possible F-centric, F-radical subgroups in saturated fusion systems on 2-groups of 2-rank 2.

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arXiv2008

Simple Modules for Groups with Abelian Sylow 2-Subgroups are Algebraic

Let G be a finite group and let p be a prime. A module for G over a field of characteristic p is called algebraic if it satisfies a polynomial, with addition and multiplication given by direct sum and tensor product. In some sense, having this property is equivalent to the tensor structure being 'nice' for that module. In this paper we prove that if G is a group with abelian Sylow 2-subgroups, and p=2, then all simple modules for G are algebraic. We include the conjecture that this result holds for all abelian 2-blocks.

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arXiv2008

Algebraic Modules and the Auslander--Reiten Quiver

Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander--Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity.

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arXiv2008

On the Tensor Products of Modules for Dihedral 2-Groups

Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that if L is a component of the (stable) Auslander-Reiten quiver for a dihedral 2-group consisting of non-periodic modules, then there is at most one algebraic module on L.

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arXiv2007

Symmetric Group Character Degrees and Hook Numbers

In this article we prove the following result: that for any two natural numbers k and j, and for all sufficiently large symmetric groups Sym(n), there are k disjoint sets of j irreducible characters of Sym(n), such that each set consists of characters with the same degree, and distinct sets have different degrees. In particular, this resolves a conjecture most recently made by Moretó. The methods employed here are based upon the duality between irreducible characters of the symmetric groups and the partitions to which they correspond. Consequently, the paper is combinatorial in nature.

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