Morita equivalence classes of blocks with extraspecial defect groups $p_+^{1+2}$

TL;DR

This paper classifies Morita equivalence classes of blocks with extraspecial defect groups of order p^{1+2} for primes p ≥ 5, confirming key conjectures and providing explicit classifications for p=5.

Contribution

It characterizes Morita equivalence classes for blocks with extraspecial defect groups, proving Donovan's and Alperin-McKay conjectures for these groups and reducing the problem for p=3.

Findings

Confirmed Donovan's conjecture for p ≥ 5.

Confirmed Alperin-McKay conjecture for these blocks.

Listed Morita equivalence classes for p=5.

Abstract

We characterise the Morita equivalence classes of blocks with extraspecial defect groups $p_+^{1+2}$ for $p \geq 5$, and so show that Donovan's conjecture and the Alperin-McKay conjecture hold for such $p$-groups. For $p=3$ we reduce Donovan's conjecture for blocks with defect group $3_+^{1+2}$ to bounding the Cartan invariants for such blocks of quasisimple groups. We apply the characterisation to the case $p=5$ as an example, to list the Morita equivalence classes of such blocks.