Signatures of Type A Root Systems

authored by

Michael Cuntz, Hung Manh Tran, Tan Nhat Tran, Shuhei Tsujie

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).

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

Phenikaa University
Binghamton University
Hokkaido University of Education

Type

Preprint

No. of pages

17

Publication date

07.04.2025

Publication status

E-pub ahead of print

Electronic version(s)

https://arxiv.org/abs/2504.05423 (Access: Open)