Decompositions of Group Algebras as a Direct Sum of Projective Indecomposable Modules and of Blocks in Positive Characteristic

TL;DR

This dissertation explicitly decomposes group algebras over fields of positive characteristic into projective indecomposable modules, calculates their radical series, and analyzes block structures for specific groups like $A_4$ and $A_5$, extending the Artin--Wedderburn theorem.

Contribution

It provides explicit decompositions of group algebras into projective indecomposable modules and analyzes their block structures in characteristic 2.

Findings

Decomposition of Klein four-group algebra into projective indecomposables

Calculation of Cartan matrices for specific group algebras

Identification of blocks in $kA_4$ and $kA_5$

Abstract

The dissertation focuses on decomposing a group algebra $kG$ over a field of positive characteristic into a direct sum of projective indecomposable modules. Such a decomposition is obtained together with the Artin--Wedderburn Theorem. The main goal of the dissertation is to explicitly decompose given group algebras as a direct sum of their projective indecomposable modules. To achieve this, we determine the radical series of each projective indecomposable module of the given group algebras. For a group algebra over characteristic $p$, each projective indecomposable module has a simple head that is isomorphic to its socle. Projective covers and injective envelopes are used to construct these modules. A cyclic group algebra is uniserial, and a $p$-group algebra over characteristic $p$ is itself a projective indecomposable module. Using these properties, we explicitly find all projective indecomposable modules for the following group algebras over characteristic $2$: the Klein four-group, the alternating group $A_4$, and the alternating group $A_5$. Their relationships play an important role in this process. Since $p$-group algebras have trivial head and trivial socle, the Klein four-group algebra has a corresponding radical series. Its decomposition into a direct sum of projective indecomposable modules is described explicitly, and the Cartan matrix of a group algebra is obtained by calculating the multiplicities of simples in its projective indecomposable modules. The topic is then extended slightly by considering the unique decomposition of a group algebra into a direct sum of particular modules known as blocks. For $kA_4$, the primitive orthogonal idempotents are calculated, and since $kA_4$ has one block, it is equal to its block decomposition. For $kA_5$, we show that there are two blocks, determined by checking the nonzero entries in its Cartan matrix.