Trivial source characters in blocks of domestic representation type
Bernhard Böhmler
TL;DR
The paper analyzes trivial source modules in blocks of domestic representation type over a field of characteristic $2$, showing that such blocks fall into Puig-equivalence classes with $kV_4$, $k\mathfrak{A}_4$, or the principal block of $k\mathfrak{A}_5$. It develops the trivial source character theory by computing full trivial source character tables for $V_4$, $\mathfrak{A}_4$, and $\mathfrak{A}_5$, and then transfers this information across splendid Morita equivalences to blocks with Klein four defect groups. The results yield explicit lists of trivial source $B$-modules and their ordinary characters, including the associated decomposition matrices, in the three Puig-equivalent cases, and extend to the $D_{4v}$-family via Morita transport. Two detailed examples illustrate the method: the dihedral family $D_{4v}$ and a concrete domestic-type group of order $972$, where the full trivial source character data are obtained. Overall, the work provides a structured, Morita-invariant approach to determining trivial source characters in small blocks, contributing to a growing TSCT database and enabling systematic analysis of trivial source modules across related block theories.
Abstract
Let $G$ be a finite group of even order, let $k$ be an algebraically closed field of characteristic $2$, and let $B$ be a block of the group algebra $kG$ which is of domestic representation type. Up to splendid Morita equivalence, precisely three cases can occur: $kV_4$, $k\mathfrak{A}_4$ and the principal block of $k\mathfrak{A}_5$. In each case, given the character values of the ordinary irreducible characters of $B$, we determine the ordinary characters of all trivial source $B$-modules.