Signatures of Type $A$ Root Systems
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908306087346176 |
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| author | Cuntz, Michael Tran, Hung Manh Tran, Tan Nhat Tsujie, Shuhei |
| author_facet | Cuntz, Michael Tran, Hung Manh Tran, Tan Nhat Tsujie, Shuhei |
| contents | 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 $(β_1, \ldots, β_{n+1})$, the signature of $S$ is defined as an unordered pair $\{a, b\}$, where $a$ and $b$ denote the numbers of $1$s and $-1$s, respectively, among the cofactors $(-1)^k \det(S \setminus \{β_k\})$ for $1 \le k \le n+1$. 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 $(1, 1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05423 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Signatures of Type $A$ Root Systems Cuntz, Michael Tran, Hung Manh Tran, Tan Nhat Tsujie, Shuhei Combinatorics Primary 05C50, Secondary 52C35 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 $(β_1, \ldots, β_{n+1})$, the signature of $S$ is defined as an unordered pair $\{a, b\}$, where $a$ and $b$ denote the numbers of $1$s and $-1$s, respectively, among the cofactors $(-1)^k \det(S \setminus \{β_k\})$ for $1 \le k \le n+1$. 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 $(1, 1)$. |
| title | Signatures of Type $A$ Root Systems |
| topic | Combinatorics Primary 05C50, Secondary 52C35 |
| url | https://arxiv.org/abs/2504.05423 |