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

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of the representation of a finite Coxeter group W on the pth graded piece 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 of W on the pth graded piece of its Orlik-Solomon algebra with the descent algebra of W . A consequence of our conjecture is that both the regular character of W and the character of W acting on its 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 conjectures have been proved for symmetric and dihedral groups. In this paper we develop algorithmic tools to prove the conjectures computationally for a given W and we use these tools to verify the claim for all finite Coxeter groups of rank three and four. The techniques developed and implemented in this paper provide previously unknown decompositions of the regular characters and the Orlik-Solomon characters of the Coxeter groups of types B3, H3, B4, D4, F4, and H4 as sums of induced representations indexed by the set of conjugacy classes of W .