Signatures of Type $A$ Root Systems

TL;DR

We develop a signature for size-$n+1$ subsets of the type $A_n$ positive roots, encoding it as an unordered pair $\{a,b\}$ via cofactors and show the counting formula $s_{a,b}=u_{a,b}$ with $u_{a,b}=(n+1)^{n+1-a-b}\binom{n}{a+b-1}\left\langle {a+b-1 \atop a-1}\right\rangle$, tying signatures to Eulerian numbers. Interpreting $\Phi^+$ as edges of the complete graph $K_{n+1}$ and $S$ as unicyclic subgraphs links signatures to cycle permutations through $\{\widetilde{\rm asc},\widetilde{\rm dsc}\}$ statistics, enabling explicit enumeration. These signatures facilitate exact computations of invariants for cones over deformed arrangements (Shi, Catalan, Linial, Ish), including $T_{\mathcal A}(1,1)$ and $T_{\mathcal A}^{\operatorname{arith}}(1,1)$ and the minimum period of the characteristic quasi-polynomial. The paper also determines the minimum periods for $[\ell,m]$-deformations and Ish, showing $\rho_{\mathcal A_n^{[\ell,m]}}=\operatorname{lcm}\{1,2,\ldots,mn-\ell\}$ and $\rho_{\mathbf{c}\mathrm{Ish}_n}=\operatorname{lcm}\{1,2,\ldots,n\}$, while illustrating that Shi and Ish share the same minimum period but can have different quasi-polynomials.

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 $(β_1, \ldots, β_{n+1})$, 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 $(1, 1)$.

Figures (2)

• Figure 1: A graphical interpretation of the subset $S$ in $A_3$ from Example \ref{['ex:1st']}. The graph corresponding to $S$ is unicyclic, with the cycle $\Gamma = (1,2,4,1)$ of length 3. The permutation $\sigma = 124 \in \mathfrak{S}_\Gamma$ (which is isomorphic to $\mathfrak{S}_3$) has $\widetilde{{\rm asc}}(\sigma) = 2$ and $\widetilde{{\rm dsc}}(\sigma) = 1$. According to Lemma \ref{['lem:uni']}, the signature of $S$ is given by $\operatorname{sign}(S) = \{1,2\}$, which aligns with the signature calculation presented in Example \ref{['ex:1st']}.
• Figure 2: A graphical illustration of the proof of Theorem \ref{['thm:main']}.

Theorems & Definitions (35)

• Remark 1.1
• Lemma 1.2
• Definition 1.3
• Example 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.9
• Theorem 1.10
• Proposition 2.1: Peter05