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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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References:
 [1] Bourbaki, N., Groupes et Algèbres de Lie, Chp. IV-VI, (1981), Masson: Masson, Paris · Zbl 0483.22001 [2] Brenti, F.; Carnevale, A., Odd length for even hyperoctahedral groups and signed generating functions, Discrete Math., 340, 12, 2822-2833, (2017) · Zbl 1370.05010 [3] Brenti, F.; Carnevale, A., Odd length in Weyl groups, (2017) [4] Brenti, F.; Carnevale, A., Proof of a conjecture of Klopsch-Voll on Weyl groups of type $${A}$$, Trans. Amer. Math. Soc., 369, 7531-7547, (2017) · Zbl 1368.05007 [5] Humphreys, J. E., Reflection Groups and Coxeter Groups, (1990), Cambridge Univ. Press: Cambridge Univ. Press, Cambridge · Zbl 0725.20028 [6] Klopsch, B.; Voll, C., Igusa-type functions associated to finite formed spaces and their functional equations, Trans. Amer. Math. Soc., 361, 8, 4405-4436, (2009) · Zbl 1229.05288 [7] Knapp, Anthony W., Lie groups beyond an introduction, 140, (1996), Birkhäuser: Birkhäuser, Boston, MA · Zbl 0862.22006 [8] Landesman, A., Proof of Stasinski and Voll’s hyperoctahedral group conjecture, Australas. J. Combin., 71, 2, 196-240, (2018) · Zbl 1406.05010 [9] Macdonald, I. G., The Poincaré series of a Coxeter group, Math. Ann., 199, 2, 161-174, (1972) · Zbl 0286.20062 [10] Stasinski, A.; Voll, C., A new statistic on the hyperoctahedral groups, Electron. J. Combin., 20, 3, 23 p. pp., (2013) · Zbl 1295.05038 [11] Stasinski, A.; Voll, C., Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type $${B}$$, Amer. J. Math., 136, 2, 501-550, (2014) · Zbl 1286.11140
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