Sign-twisted Poincaré series and odd inversions in Weyl groups
Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 621-644.

Following recent work of Brenti and Carnevale, 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.

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DOI: https://doi.org/10.5802/alco.62
Classification: 05E15,  05A19,  20F55
Keywords: Weyl group, root system, Poincaré series, inversion
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     author = {Stembridge, John R.},
     title = {Sign-twisted {Poincar\'e} series and odd inversions in {Weyl} groups},
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     publisher = {MathOA foundation},
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     doi = {10.5802/alco.62},
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Stembridge, John R. Sign-twisted Poincaré series and odd inversions in Weyl groups. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 621-644. doi : 10.5802/alco.62. https://alco.centre-mersenne.org/articles/10.5802/alco.62/

[1] Bourbaki, N. Groupes et Algèbres de Lie, Chp. IV–VI, Masson, Paris, 1981 | Zbl 0483.22001

[2] Brenti, F.; Carnevale, A. Odd length for even hyperoctahedral groups and signed generating functions, Discrete Math., Volume 340 (2017) no. 12, pp. 2822-2833 | Article | MR 3698070 | Zbl 1370.05010

[3] Brenti, F.; Carnevale, A. Odd length in Weyl groups (2017) (https://arxiv.org/abs/1709.03320) | Zbl 07140427

[4] Brenti, F.; Carnevale, A. Proof of a conjecture of Klopsch–Voll on Weyl groups of type A, Trans. Amer. Math. Soc., Volume 369 (2017), pp. 7531-7547 | Article | MR 3683117 | Zbl 1368.05007

[5] Humphreys, J. E. Reflection Groups and Coxeter Groups, Cambridge Univ. Press, Cambridge, 1990 | Article | Zbl 0725.20028

[6] Klopsch, B.; Voll, C. Igusa-type functions associated to finite formed spaces and their functional equations, Trans. Amer. Math. Soc., Volume 361 (2009) no. 8, pp. 4405-4436 | Article | MR 2500892 | Zbl 1229.05288

[7] Knapp, Anthony W. Lie groups beyond an introduction, Progress in Mathematics, 140, Birkhäuser, Boston, MA, 1996 | MR 1399083 | Zbl 0862.22006

[8] Landesman, A. Proof of Stasinski and Voll’s hyperoctahedral group conjecture, Australas. J. Combin., Volume 71 (2018) no. 2, pp. 196-240 | MR 3786907 | Zbl 1406.05010

[9] Macdonald, I. G. The Poincaré series of a Coxeter group, Math. Ann., Volume 199 (1972) no. 2, pp. 161-174 | Article | MR 322069 | Zbl 0286.20062

[10] Stasinski, A.; Voll, C. A new statistic on the hyperoctahedral groups, Electron. J. Combin., Volume 20 (2013) no. 3, Paper no. P50, 23 pages | Article | MR 3118958 | 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., Volume 136 (2014) no. 2, pp. 501-550 | Article | MR 3188068 | Zbl 1286.11140

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