Modular Idempotents for the Descent Algebras of Type A and Higher Lie Powers and Modules

TL;DR

This paper investigates the structure of descent algebras of type A, introducing modular idempotents, analyzing higher Lie powers and modules, and exploring right ideals in symmetric group algebras, both in ordinary and modular contexts.

Contribution

It provides a new construction for modular idempotents and characterizes higher Lie powers and modules within descent algebras of type A.

Findings

Constructed modular idempotents for descent algebras.

Determined dimensions and characters of higher Lie powers.

Analyzed structures of higher Lie modules and right ideals.

Abstract

The article focuses on four aspects related to the descent algebras of type $A$. They are modular idempotents, higher Lie powers, higher Lie modules and the right ideals of the symmetric group algebras generated by the Solomon's descent elements. More precisely, we give a construction for the modular idempotents, describe the dimension and character for higher Lie powers and study the structures of the higher Lie modules and the right ideals both in the ordinary and modular cases.