On the Descent Algebra of Type $D$

TL;DR

This paper provides a combinatorial interpretation of Solomon's multiplication rule in the descent algebra of type D Weyl groups and shows its relation as a homomorphic image of the hyperoctahedral group's descent algebra.

Contribution

It introduces a combinatorial interpretation for the multiplication in the descent algebra of type D and establishes a homomorphism from the hyperoctahedral group's descent algebra.

Findings

Combinatorial interpretation of Solomon's rule for type D

Homomorphic relation between $ ext{Σ}D_n$ and $ ext{Σ}B_{n-2}$

Enhanced understanding of descent algebra structures

Abstract

Here we give a combinatorial interpretation of Solomon's rule for multiplication in the descent algebra of Weyl groups of type $D$, $\Sigma D_n$. From here we show that $\Sigma D_n$ is a homomorphic image of the descent algebra of the hyperoctahedral group, $\Sigma B_{n-2}$.