$2$-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle

TL;DR

This paper classifies blocks of finite groups with specific 2-group defect structures and inertial quotients containing Singer cycles, providing explicit Morita equivalence classes based on the parameters.

Contribution

It offers a complete classification of Morita equivalence classes for blocks with homocyclic defect groups and Singer cycle-involving inertial quotients, extending known results.

Findings

For m=1, blocks are Morita equivalent to principal blocks of specific groups.

For m>1, blocks are Morita equivalent to a semidirect product D ⋊ E.

Explicit classification depending on parameters n and m.

Abstract

We consider a block $B$ of a finite group with defect group $D \cong (C_{2^m})^n$ and inertial quotient $\mathbb{E}$ containing a Singer cycle (an element of order $2^n-1$). This implies $\mathbb{E} = E \rtimes F$, where $E \cong C_{2^n-1}$, $F \leq C_n$, and $E$ acts transitively on the elements in $D$ of order $2$, and freely on $D \backslash \{1\}$. We classify the basic Morita equivalence classes of $B$ over a complete discrete valuation ring $\mathcal{O}$: when $m=1$, $B$ is basic Morita equivalent to the principal block of one of $SL_2(2^n) \rtimes F$, $D \rtimes \mathbb{E}$, or $J_1$ (where $J_1$ occurs only when $n=3$). When $m>1$, $B$ is basic Morita equivalent to $D \rtimes \mathbb{E}$.