Adam Salminen

On the sources of simple modules in nilpotent blocks

Abstract
Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $p$.  If $b$ is a nilpotent block of $kG$ with defect group $P$, then there is a unique isomorphism class of simple $kGb$-modules and Puig proved that the source of this module is an endo-permutation $kP$-module.  It is conjectured that the image of this source is always torsion in the Dade group.

Let $H$ be a finite group and let $P$ be a $p$-subgroup of $\Aut(H)$.  Also let $c$ be a defect zero block of $kH$.  If $c$ is $P$-stable and $Br_P(c)\neq 0$, then $c$ is a nilpotent block of $k(H\rtimes P)$ and $k(H\rtimes P)c$ has $P$ as a defect group.  In this paper, we will investigate the sources of the simple $k(H\rtimes P)c$-modules when $P\cong C_p\times C_p$.  Suppose that we can find an $H$ and $c$ as above such that a source of a simple $k(H\rtimes P)c$-module is not torsion in the Dade group.  Then we can find $H$ and $c$ as above with $H$ a central $p'$-extension of a simple group.  When $p\geq 3$ we show that $H$ can be found in a quite restrictive subset of simple groups.


Status
Published J. Algebra 319 (2008) 4559-4574