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Computations for Coxeter arrangements and Solomon’s descent algebra. III: Groups of rank seven and eight. (English) Zbl 1306.20004
Summary: In this paper we extend the computations in parts I and II of this series of papers [J. Symb. Comput. 50, 139-158 (2013; Zbl 1257.20004); J. Algebra 377, 320-332 (2013; Zbl 1277.20007)] and complete the proof of a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group \(W\) acting on the \(p\)-th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of \(W\) for groups of rank seven and eight. For classical Coxeter groups, these characters are given using a formula that is expected to hold in all ranks.

MSC:
20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E10 Combinatorial aspects of representation theory
20C15 Ordinary representations and characters
20C40 Computational methods (representations of groups) (MSC2010)
Software:
CHEVIE; GAP; ZigZag
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References:
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