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Computations for Coxeter arrangements and Solomon’s descent algebra: groups of rank three and four. (English) Zbl 1257.20004
Summary: 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.

MSC:
20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E10 Combinatorial aspects of representation theory
20-04 Software, source code, etc. for problems pertaining to group theory
20C30 Representations of finite symmetric groups
Software:
CHEVIE; GAP; ZigZag
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