Computations for Coxeter arrangements and Solomon's descent algebra II: Groups of rank five and six

TL;DR

This paper proves a refined conjecture relating characters of finite Coxeter groups to Solomon's descent algebra for groups of rank five and six, expanding the understanding of their algebraic structure.

Contribution

It extends the proof of a refined character conjecture to Coxeter groups of rank five and six, developing new algorithmic tools for character decomposition.

Findings

Confirmed the conjecture for rank five and six Coxeter groups.

Developed algorithms for character decomposition of these groups.

Provided new decompositions of regular and Orlik-Solomon characters.

Abstract

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture relates the character above to a component of a decomposition of the regular character of $W$ related to Solomon's descent algebra of $W$. The refined conjecture has been proved for symmetric and dihedral groups, as well as finite Coxeter groups of rank three and four. In this paper, the second in a series of three dealing with groups of rank up to eight (and in particular, all exceptional Coxeter groups), we prove the conjecture for finite Coxeter groups of rank five and six, further developing the algorithmic tools described in the previous article. The techniques developed and implemented in this paper provide previously unknown decompositions of the regular and Orlik-Solomon characters of the groups considered.