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Reduction theorems for a conjecture on basis in source algebras of blocks of finite groups

Tiberiu Coconet, Constantin-Cosmin Todea

TL;DR

The paper investigates Barker and Gelvin's conjecture stating that source algebras of blocks with non-trivial defect groups admit a unit group containing a $D\times D$-stable basis, i.e., a uniform source $D$-algebra. It develops reduction theorems showing that establishing uniformity for blocks of quasi-simple groups suffices to deduce it for all $\mathcal{F}$-reduction simple blocks, with lifts along normal subgroups and central extensions under $p'$-conditions. The work also provides concrete verification for many blocks of the first sporadic groups, leveraging known block structures and fusion-system methods. The results narrow the verification of the conjecture to a finite, manageable set of quasi-simple cases, offering a pathway toward a complete resolution in general.

Abstract

The aim of this short research note is to present some results about a conjecture of Barker and Gelvin claiming that any source algebra of a block of a finite group has the unit group containing a basis stabilised by the left and right actions of the defect group. We obtain some reduction theorems for the existence of stable unital basis in source algebras of block algebras. Along the way we investigate this problem for the blocks of some finite simple groups.

Reduction theorems for a conjecture on basis in source algebras of blocks of finite groups

TL;DR

The paper investigates Barker and Gelvin's conjecture stating that source algebras of blocks with non-trivial defect groups admit a unit group containing a -stable basis, i.e., a uniform source -algebra. It develops reduction theorems showing that establishing uniformity for blocks of quasi-simple groups suffices to deduce it for all -reduction simple blocks, with lifts along normal subgroups and central extensions under -conditions. The work also provides concrete verification for many blocks of the first sporadic groups, leveraging known block structures and fusion-system methods. The results narrow the verification of the conjecture to a finite, manageable set of quasi-simple cases, offering a pathway toward a complete resolution in general.

Abstract

The aim of this short research note is to present some results about a conjecture of Barker and Gelvin claiming that any source algebra of a block of a finite group has the unit group containing a basis stabilised by the left and right actions of the defect group. We obtain some reduction theorems for the existence of stable unital basis in source algebras of block algebras. Along the way we investigate this problem for the blocks of some finite simple groups.
Paper Structure (4 sections, 5 theorems, 18 equations)

This paper contains 4 sections, 5 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Uniform source algebras of blocks of normal subgroups
  3. Proofs of Theorem \ref{['thmreduck']} and \ref{['thm:12']}
  4. Uniform source algebras of blocks of sporadic groups

Key Result

Theorem 1.2

With the above notations if $\bar{A}$ is a uniform source $D$-algebra of $\bar{b}$ then $A$ is a uniform source $D$-algebra of $b.$

Theorems & Definitions (11)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • proof
  • Remark 3.1
  • proof
  • ...and 1 more