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| Original language | English |
|---|---|
| Number of pages | 17 |
| Publication status | E-pub ahead of print - 7 Apr 2025 |
Abstract
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2025.
Research output: Working paper/Preprint › Preprint
}
TY - UNPB
T1 - Signatures of Type A Root Systems
AU - Cuntz, Michael
AU - Tran, Hung Manh
AU - Tran, Tan Nhat
AU - Tsujie, Shuhei
PY - 2025/4/7
Y1 - 2025/4/7
N2 - 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).
AB - 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).
U2 - 10.48550/arXiv.2504.05423
DO - 10.48550/arXiv.2504.05423
M3 - Preprint
BT - Signatures of Type A Root Systems
ER -