Computations for Coxeter arrangements and Solomon's descent algebra: Groups of rank three and four

TL;DR

This paper develops computational tools to verify a refined conjecture relating the characters of finite Coxeter groups, specifically for groups of rank three and four, revealing new character decompositions.

Contribution

The paper introduces algorithms to computationally verify a refined conjecture for Coxeter groups of rank three and four, expanding understanding of their character decompositions.

Findings

Verified the conjecture for all rank three Coxeter groups

Verified the conjecture for all rank four Coxeter 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 characters of a finite Coxeter group $W$ afforded by the homogeneous components of its Orlik-Solomon algebra as sums of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture also relates the Orlik-Solomon characters above to the terms of a decomposition of the regular character of $W$ related to the descent algebra of $W$. A consequence of our conjecture is that both the regular character of $W$ and the character of the Orlik-Solomon algebra have parallel, graded decompositions as sums of characters induced from linear characters of centralizers of elements of $W$, one for each conjugacy class of elements of $W$. The refined conjecture has been proved for symmetric and dihedral groups. In this paper we develop algorithmic tools to prove the conjecture computationally for a given finite Coxeter group. We use these tools to verify the conjecture for all finite Coxeter groups of rank three and four, thus providing previously unknown decompositions of the regular characters and the Orlik-Solomon characters of these groups.