The ring of perfect $p$-permutation bimodules for blocks with cyclic defect groups

TL;DR

This paper investigates the algebraic structure of the ring of perfect p-permutation bimodules for blocks with cyclic defect groups, revealing conditions for primitivity and explicit decompositions in various algebraic settings.

Contribution

It provides a detailed analysis of the ring structure of perfect p-permutation bimodules for blocks with cyclic defect groups, including primitive decompositions and explicit algebraic descriptions.

Findings

If the Cartan matrix has 1 as an elementary divisor, the class is not primitive.

For blocks with cyclic defect groups, a primitive decomposition of the class is determined.

Explicit descriptions of the tensor product algebra are given for fields of characteristic different from p.

Abstract

Let $B$ be a block algebra of a group algebra $FG$ of a finite group $G$ over a field $F$ of characteristic $p>0$. This paper studies ring theoretic properties of the representation ring $T^\Delta(B,B)$ of perfect $p$-permutation $(B,B)$-bimodules and properties of the $k$-algebra $k\otimes_\mathbb{Z} T^\Delta(B,B)$, for a field $k$. We show that if the Cartan matrix of $B$ has $1$ as an elementary divisor then $[B]$ is not primitive in $T^\Delta(B,B)$. If $B$ has cyclic defect groups we determine a primitive decomposition of $[B]$ in $T^\Delta(B,B)$. Moreover, if $k$ is a field of characteristic different from $p$ and $B$ has cyclic defect groups of order $p^n$ we describe $k\otimes_\mathbb{Z} T^\Delta(B,B)$ explicitly as a direct product of a matrix algebra and $n$ group algebras.