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Descents and one-dimensional characters for classical Weyl groups. (English) Zbl 0830.20025
This paper examines all sums of the form \(\sum_{\pi\in W}\chi(\pi)t^{d({\pi})}\) where \(W\) is a classical Weyl group, \(\chi\) is a character of degree 1 of \(W\) and \(d(\pi)\) is the descent statistic. The case when \(W\) is of type \(A_n\) (symmetric groups) has already been considered and involves the Eulerian and signed Eulerian polynomials [see J. Désarménien and D. Foata, Discrete Math. 99, 49-58 (1992; Zbl 0769.05094) and M. Wachs, ibid. 99, 59-62 (1992; Zbl 0769.05098)].
This paper extends these results to the other classical Weyl groups of types \(B_n\) and \(D_n\). In fact, the answers turn out to be simpler and generalise further in these cases. In type \(B_n\) the proof emulates that of M. Wachs. For types \(B_n\) and \(D_n\), the sum factors completely into a product of simple linear terms when \(\chi\) is a non-trivial character.

20C30 Representations of finite symmetric groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E15 Combinatorial aspects of groups and algebras (MSC2010)
Full Text: DOI
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