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.

Revised : 2019-01-17

Accepted : 2019-01-21

Published online : 2019-08-01

DOI : https://doi.org/10.5802/alco.62

Classification: 05E15, 05A19, 20F55

Keywords: Weyl group, root system, Poincaré series, inversion

@article{ALCO_2019__2_4_621_0, author = {Stembridge, John R.}, title = {Sign-twisted Poincar\'e series and odd inversions in Weyl groups}, journal = {Algebraic Combinatorics}, publisher = {MathOA foundation}, volume = {2}, number = {4}, year = {2019}, pages = {621-644}, doi = {10.5802/alco.62}, language = {en}, url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_4_621_0} }

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. alco.centre-mersenne.org/item/ALCO_2019__2_4_621_0/

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