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

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 $\mathbb{E}=E⋊F$, where $E\cong C_{2^n-1}$, $F\le C_n$, 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 $S{L}_2(2^n)⋊F$, $D⋊\mathbb{E}$, or $J_1$ (where $J_1$ occurs only when $n=3$). When $m>1$, $B$ is basic Morita equivalent to $D⋊\mathbb{E}$.

17 pages; typos and references corrected; theorem 4.12 corrected and improved, main result improved by remark 4.9