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Sign-twisted Poincaré series and odd inversions in Weyl groups. (English) Zbl 1417.05246
Summary: Following recent work of F. Brenti and A. Carnevale [Trans. Am. Math. Soc. 369, No. 10, 7531–7547 (2017; Zbl 1368.05007); “Odd length in Weyl groups”, Preprint, arXiv:1709.03320], we investigate a sign-twisted Poincaré series for finite Weyl groups \(W\) that tracks “odd inversions”; i.e. the number of odd-height positive roots transformed into negative roots by each member of \(W\). We prove that the series is divisible by the corresponding series for any parabolic subgroup \(W_J\), and provide sufficient conditions for when the quotient of the two series equals the restriction of the first series to coset representatives for \(W/W_J\). We also show that the series has an explicit factorization involving the degrees of the free generators of the polynomial invariants of a canonically associated reflection group.

MSC:
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05A15 Exact enumeration problems, generating functions
05A19 Combinatorial identities, bijective combinatorics
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20B30 Symmetric groups
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