Generating series formulas for the structure constants of Solomon's descent algebra

TL;DR

This paper derives explicit series formulas for the structure constants of Solomon's descent algebra, revealing new connections with quasisymmetric functions and extending results to type B Coxeter groups.

Contribution

It introduces series formulas for the structure constants of Solomon's descent algebra and generalizes known relations to hyperoctahedral groups using type B quasisymmetric functions.

Findings

Structure constants relate to Robinson-Schensted-Knuth correspondence

Formulas connect descent algebra to quasisymmetric functions

Results extend to hyperoctahedral groups

Abstract

Introduced by Solomon in his 1976 paper, the descent algebra of a finite Coxeter group received significant attention over the past decades. As proved by Gessel, in the case of the symmetric group its structure constants give the comultiplication table for the fundamental basis of quasisymmetric functions. We show that this property actually implies several well known relations linked to the Robinson-Schensted-Knuth correspondence and some of its generalisations. We further use the theory of type B quasisymmetric functions introduced by Chow to provide analogue results when the Coxeter group is the hyperoctahedral group.