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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.

##### MSC:
 20C30 Representations of finite symmetric groups 20F55 Reflection and Coxeter groups (group-theoretic aspects) 05E15 Combinatorial aspects of groups and algebras (MSC2010)
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##### References:
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