The $p$-modular descent algebras

TL;DR

This paper extends the theory of descent algebras to fields of prime characteristic, analyzing their algebraic structure, radical, modules, and decomposition matrices for Coxeter groups, providing new insights into their representation theory.

Contribution

It introduces descent algebras over prime characteristic fields and computes their decomposition and Cartan matrices for various Coxeter groups, advancing understanding of their algebraic properties.

Findings

Determined the radical and irreducible modules of the descent algebras.

Calculated the decomposition matrices for types A, B, and D Coxeter groups.

Derived the Cartan matrix for descent algebras over finite fields.

Abstract

The concept of descent algebras over a field of characteristic zero is extended to define descent algebras over a field of prime characteristic. Some basic algebraic structure of the latter, including its radical and irreducible modules, is then determined. The decomposition matrix of the descent algebras of Coxeter group types $A$, $B$, and $D$ are calculated, and used to derive a description of the decomposition matrix of an arbitrary descent algebra. The Cartan matrix of a variety of descent algebras over a finite field is then obtained.