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In recent papers we have refined a conjecture of Lehrer and Solomonexpressing the character of a finite Coxeter group $W$ acting on the $p$thgraded component of its Orlik-Solomon algebra as a sum of characters inducedfrom linear characters of centralizers of elements of $W$. Our refinedconjecture relates the character above to a component of a decomposition of theregular character of $W$ related to Solomon's descent algebra of $W$. Therefined conjecture has been proved for symmetric and dihedral groups, as wellas finite Coxeter groups of rank three and four.  In this paper, the second in a series of three dealing with groups of rank upto eight (and in particular, all exceptional Coxeter groups), we prove theconjecture for finite Coxeter groups of rank five and six, further developingthe algorithmic tools described in the previous article. The techniquesdeveloped and implemented in this paper provide previously unknowndecompositions of the regular and Orlik-Solomon characters of the groupsconsidered.

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