Note on the reduction of Alperin's Conjecture

TL;DR

This paper discusses a reduction approach for Alperin's Conjecture, linking previous results and conditions on simple groups to verify the conjecture block by block, emphasizing the role of quasi-simple groups.

Contribution

It introduces a numerical reduction statement for Alperin's Conjecture that simplifies verification to checking conditions on quasi-simple groups, building on prior reduction methods.

Findings

Reduction of Alperin's Conjecture to conditions on quasi-simple groups

Connection between Navarro and Tiep's conditions and the reduction approach

Verification of the conjecture block by block through numerical statements

Abstract

In a recent paper, Gabriel Navarro and Pham Huu Tiep show that the so-called Alperin Weight Conjecture can be verified via the Classification of the Finite Simple Groups, provided any simple group fulfills a very precise list of conditions. Our purpose here is to show to the interested reader that the results in our book "Frobenius categories versus Brauer blocks", Progress in Math. 274(2009), and the reduction arguments in "On the reduction of Alperin's Conjecture to the quasi-simple groups", J. of Algebra 328(2011), suggest a numerical statement - implying Alperin's Conjecture block by block - which can be reduced again to check that the same holds on the quasi-simple groups and, this time, this statement on the quasi-simple groups follows from the list of conditions demanded by Navarro and Tiep.