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2020 Zeta-functions of root systems and Poincaré polynomials of Weyl groups
Yasushi Komori, Kohji Matsumoto, Hirofumi Tsumura
Tohoku Math. J. (2) 72(1): 87-126 (2020). DOI: 10.2748/tmj/1585101623

Abstract

We consider a certain linear combination of zeta-functions of root systems for a root system. Showing two different expressions of this linear combination, we find that a certain signed sum of zeta-functions of root systems is equal to a sum involving Bernoulli functions of root systems. This identity gives a non-trivial functional relation among zeta-functions of root systems, if the signed sum does not identically vanish. This is a generalization of the authors' previous result (Proc. London Math. Soc. 100 (2010), 303–347). We present several explicit examples of such functional relations. We give a criterion of the non-vanishing of the signed sum, in terms of Poincaré polynomials of associated Weyl groups. Moreover we prove a certain converse theorem.

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Yasushi Komori. Kohji Matsumoto. Hirofumi Tsumura. "Zeta-functions of root systems and Poincaré polynomials of Weyl groups." Tohoku Math. J. (2) 72 (1) 87 - 126, 2020. https://doi.org/10.2748/tmj/1585101623

Information

Published: 2020
First available in Project Euclid: 25 March 2020

zbMATH: 07199989
MathSciNet: MR4079426
Digital Object Identifier: 10.2748/tmj/1585101623

Subjects:
Primary: 11M41
Secondary: 11B68, 11F27, 11M32, 11M99

Rights: Copyright © 2020 Tohoku University

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Vol.72 • No. 1 • 2020
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