Tohoku Mathematical Journal
2020
March
SECOND SERIES VOL. 72, NO. 1

Tohoku Math. J.
72 (2020), 87-126

Title TORIC FANO CONTRACTIONS ASSOCIATED TO LONG EXTREMAL RAYS

Author Yasushi Komori, Kohji Matsumoto and Hirofumi Tsumura

(Received November 2, 2017, revised August 28, 2018)
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.

Mathematics Subject Classification. Primary 11M41; Secondary 11B68, 11F27, 11M32, 11M99.
Key words and phrases. Witten's zeta-function, root systems, Weyl groups, Poincaré polynomials.

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