Tohoku Mathematical Journal
2018
March
SECOND SERIES VOL. 70, NO. 1

Tohoku Math. J.
70 (2018), 39-63

Title WORPITZKY PARTITIONS FOR ROOT SYSTEMS AND CHARACTERISTIC QUASI-POLYNOMIALS

Author Masahiko Yoshinaga

(Received July 21, 2015, revised December 14, 2015)
Abstract. For a given irreducible root system, we introduce a partition of (coweight) lattice points inside the dilated fundamental parallelepiped into those of partially closed simplices. This partition can be considered as a generalization and a lattice points interpretation of the classical formula of Worpitzky.
 This partition, and the generalized Eulerian polynomial, recently introduced by Lam and Postnikov, can be used to describe the characteristic (quasi)polynomials of Shi and Linial arrangements. As an application, we prove that the characteristic quasi-polynomial of the Shi arrangement turns out to be a polynomial. We also present several results on the location of zeros of characteristic polynomials, related to a conjecture of Postnikov and Stanley. In particular, we verify the “functional equation” of the characteristic polynomial of the Linial arrangement for any root system, and give partial affirmative results on ``Riemann hypothesis'' for the root systems of type $E_6, E_7, E_8$, and $F_4$.

Mathematics Subject Classification. Primary 52C35; Secondary 20F55.
Key words and phrases. Root system, Shi arrangement, Linial arrangement, characteristic quasi-polynomial, Worpitzky identity, Eulerian polynomial.

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