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, Tome 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 : 10.5802/alco.62
Classification : 05E15, 05A19, 20F55
Mots clés : Weyl group, root system, Poincaré series, inversion
Stembridge, John R. 1

1 Dept. of Mathematics University of Michigan Ann Arbor Michigan 48109–1043, USA 
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Stembridge, John R. Sign-twisted Poincaré series and odd inversions in Weyl groups. Algebraic Combinatorics, Tome 2 (2019) no. 4, pp. 621-644. doi : 10.5802/alco.62. http://www.numdam.org/articles/10.5802/alco.62/

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