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no. 4
Sign-twisted Poincaré series and odd inversions in Weyl groups
Stembridge, John R.1
1 Dept. of Mathematics University of Michigan Ann Arbor Michigan 48109–1043, USA
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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.62
Classification: 05E15, 05A19, 20F55
Keywords: Weyl group, root system, Poincaré series, inversion

Stembridge, John R. 1

1 Dept. of Mathematics University of Michigan Ann Arbor Michigan 48109–1043, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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/

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