  In recent papers we have refined a conjecture of Lehrer and Solomon
expressing the characters of a finite Coxeter group $W$ afforded by the
homogeneous components of its Orlik-Solomon algebra as sums of characters
induced from linear characters of centralizers of elements of $W$. Our refined
conjecture also relates the Orlik-Solomon characters above to the terms of a
decomposition 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, 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 conjecture has been proved for symmetric and dihedral groups.
In this paper we develop algorithmic tools to prove the conjecture
computationally for a given finite Coxeter group. We use these tools to verify
the conjecture for all finite Coxeter groups of rank three and four, thus
providing previously unknown decompositions of the regular characters and the
Orlik-Solomon characters of these groups.
