Signatures of Type A Root Systems

Publikation: Arbeitspapier/PreprintPreprint

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  • Phenikaa University
  • Binghamton University
  • Hokkaido University of Education
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OriginalspracheEnglisch
Seitenumfang17
PublikationsstatusElektronisch veröffentlicht (E-Pub) - 7 Apr. 2025

Abstract

Given a type A root system Φ of rank n, we introduce the concept of a signature for each subset S of Φ consisting of n+1 positive roots. For a subset S represented by a tuple (β1,…,βn+1), the signature of S is defined as an unordered pair {a,b}, where a and b denote the numbers of 1s and −1s, respectively, among the cofactors (−1)kdet(S∖{βk}) for 1≤k≤n+1. We prove that the number of tuples with a given signature can be expressed in terms of classical Eulerian numbers. The study of these signatures is motivated by their connections to the arithmetic and combinatorial properties of cones over deformed arrangements defined by Φ, including the Shi, Catalan, Linial, and Ish arrangements. We apply our main result to compute two important invariants of these arrangements: The minimum period of the characteristic quasi-polynomial, and the evaluation of the classical and arithmetic Tutte polynomials at (1,1).

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Signatures of Type A Root Systems. / Cuntz, Michael; Tran, Hung Manh; Tran, Tan Nhat et al.
2025.

Publikation: Arbeitspapier/PreprintPreprint

Cuntz, M., Tran, H. M., Tran, T. N., & Tsujie, S. (2025). Signatures of Type A Root Systems. Vorabveröffentlichung online. https://arxiv.org/abs/2504.05423
Cuntz M, Tran HM, Tran TN, Tsujie S. Signatures of Type A Root Systems. 2025 Apr 7. Epub 2025 Apr 7.
Cuntz, Michael ; Tran, Hung Manh ; Tran, Tan Nhat et al. / Signatures of Type A Root Systems. 2025.
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