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In recent papers we have refined a conjecture of Lehrer and Solomonexpressing the characters of a finite Coxeter group $W$ afforded by thehomogeneous components of its Orlik-Solomon algebra as sums of charactersinduced from linear characters of centralizers of elements of $W$. Our refinedconjecture also relates the Orlik-Solomon characters above to the terms of adecomposition 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, gradeddecompositions as sums of characters induced from linear characters ofcentralizers 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 conjecturecomputationally for a given finite Coxeter group. We use these tools to verifythe conjecture for all finite Coxeter groups of rank three and four, thusproviding previously unknown decompositions of the regular characters and theOrlik-Solomon characters of these groups.

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