On the Left Connected Subalgebra of the Descent Algebra of a Coxeter Group of Classical Type

TL;DR

This paper investigates the structure of the descent algebra in classical Coxeter groups, revealing properties of subgroup intersections and providing explicit formulas for a commutative subalgebra using Stirling numbers.

Contribution

It characterizes the intersection properties of subgroups in classical Coxeter groups and derives explicit multiplication formulas for a specific commutative subalgebra of the descent algebra.

Findings

Intersections of conjugate subgroups are of the same type in classical Coxeter groups.

Explicit polynomial expressions for basis elements using Stirling numbers.

Derived multiplication formulas for a commutative subalgebra of the descent algebra.

Abstract

A Coxeter group of classical type $A_n$, $B_n$ or $D_n$ contains a chain of subgroups of the same type. We show that intersections of conjugates of these subgroups are again of the same type, and make precise in which sense and to what extent this property is exclusive to the classical types of Coxeter groups. As the main tool for the proof we use Solomon's descent algebra. Using Stirling numbers, we express certain basis elements of the descent algebra as polynomials and derive explicit multiplication formulas for a commutative subalgebra of the descent algebra.