Computations for Coxeter arrangements and Solomonʼs descent algebra II: Groups of rank five and six

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W acting on the graded components of its Orlik–Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of W. The refined conjecture relates the character above to a decomposition of the regular character of W related to Solomonʼs descent algebra of W. The refined conjecture has been proved for symmetric and dihedral groups, as well as for finite Coxeter groups of rank three and four. In this paper, we prove the conjecture for finite Coxeter groups of rank five and six. The techniques developed and implemented in this paper provide previously unknown decompositions of the regular and Orlik–Solomon characters of the groups considered.