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

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 $k{\otimes }_Z{T}^{\Delta }(B,B)$, for a field k. We show that if the Cartan matrix of B has 1 as an elementary divisor then $\left[B\right]$ is not primitive in $T^{\Delta }(B,B)$. If B has cyclic defect groups we determine a primitive decomposition of $\left[B\right]$ 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 $k{\otimes }_Z{T}^{\Delta }(B,B)$ explicitly as a direct product of a matrix algebra and n group algebras.