Marcus Bishop, J. Matthew Douglass, Götz Pfeiffer, and Gerhard Röhrle,
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 thep th graded piece of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements ofW . Our refined conjecture relates the character ofW on thep th graded piece of its Orlik-Solomon algebra with the descent algebra of~W . A consequence of our conjecture is that both the regular character ofW and the character ofW acting on its Orlik-Solomon algebra have parallel, graded decompositions as sums of characters induced from linear characters of centralizers of elements ofW , one for each conjugacy class of elements ofW .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 typesB_3 ,H_3 ,B_4 ,D_4 ,F_4 , andH_4 as sums of induced representations indexed by the set of conjugacy classes ofW .
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