Central elements of the Jennings basis and certain Morita invariants

Abstract

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the $n$th (right) socle $Z{S}^n(A):=Z(A)\cap {\mathrm{Soc}}^n(A)$ of a finite-dimensional algebra $A$ is a Morita invariant; This is a generalization of important Morita invariants --- the center $Z(A)$ and the Reynolds ideal $Z{S}^1(A)$. As an example, we also studied $Z{S}^n(FG)$ for the group algebra $FG$ of a finite $p$-group $G$ over a field $F$ of positive characteristic $p$. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent set of $Z{S}^n(FG)$. In fact, such elements form a basis of $Z{S}^n(FG)$ for every integer $1\le n\le p$ if $G$ is powerful. As a corollary we have ${\mathrm{Soc}}^p(FG)\subseteq Z(FG)$ if $G$ is powerful.

11 pages, 1 table

References in corpus (1)

• On the use of Külshammer type invariants in representation theory

Cited by in corpus (2)

• Further results on the structure of (co)ends in finite tensor categories
• On theorems of Brauer-Nesbitt and Brandt for characterizations of small block algebras