Blocks with the hyperfocal subgroup $Z_{2^n}\times Z_{2^n}$

TL;DR

This paper computes character counts for blocks of finite groups with a specific hyperfocal subgroup structure, confirming Alperin's weight conjecture in these cases.

Contribution

It provides explicit calculations of characters for blocks with hyperfocal subgroup $Z_{2^n} imes Z_{2^n}$ and verifies Alperin's weight conjecture under these conditions.

Findings

Confirmed Alperin's weight conjecture for the studied blocks.

Calculated the number of irreducible characters in these blocks.

Analyzed cases based on control by the normalizer and subgroup containment.

Abstract

In this paper, we calculate the numbers of irreducible ordinary characters and irreducible Brauer characters in a block of a finite group $G$, whose associated fusion system over a 2-subgroup $P$ of $G$ (which is a defect group of the block) has the hyperfocal subgroup $\mathbb Z_{2^n}\times \mathbb Z_{2^n}$ for some $n\geq 2$, when the block is controlled by the normalizer $N_G(P)$ and the hyperfocal subgroup is contained in the center of $P$, or when the block is not controlled by $N_G(P)$ and the hyperfocal subgroup is contained in the center of the unique essential subgroup in the fusion system. In particular, Alperin's weight conjecture holds in the considered cases.