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Note on Exponents Associated with Y-Systems

Ryo Takenaka

TL;DR

The paper investigates exponents of level-2 Y-systems attached to classical Dynkin diagrams by leveraging cluster-algebraic structures from quiver mutations and Y-seed dynamics. By formulating exponents via the Jacobian of the cluster transformation at its fixed point and exploiting periodicity, it connects to Mizuno's conjectural expressions, $N_{X_n,\ell}(z)$ and $D_{X_n,\ell}(z)$. It proves Mizuno's conjecture for $(B_n,2)$ and $(D_n,2)$, and reformulates the $(C_n,2)$ case under a new conjecture (Conjecture $\text{Csol}$). The results strengthen the bridge between Y-systems, restricted Q-systems, and representation theory, providing explicit spectral data (exponents) tied to root-system combinatorics and offering a conditional path for the $C_n$ case. The methodology combines explicit quiver constructions, Jacobian analyses, and root-theoretic polynomials to realize the conjectured determinant identities.

Abstract

Let $(X_n,\ell)$ be the pair consisting of the Dynkin diagram of finite type $X_n$ and a positive integer $\ell\geq2$, called the level. Then we obtain the Y-system, which is the set of algebraic relations associated with this pair. Related to the Y-system, a sequence of integers called exponents is defined through a quiver derived from the pair $(X_n,\ell)$. Mizuno provided conjectured formulas for the exponents associated with Y-systems in [Mizuno Y., SIGMA 16 (2020), 028, 42 pages, arXiv:1812.05863]. In this paper, we study the exponents associated with level 2 Y-systems for classical Dynkin types. As a result, we present proofs of Mizuno's conjecture for $(B_n,2)$ and $(D_n,2)$, and give a reformulation for $(C_n,2)$.

Note on Exponents Associated with Y-Systems

TL;DR

The paper investigates exponents of level-2 Y-systems attached to classical Dynkin diagrams by leveraging cluster-algebraic structures from quiver mutations and Y-seed dynamics. By formulating exponents via the Jacobian of the cluster transformation at its fixed point and exploiting periodicity, it connects to Mizuno's conjectural expressions, and . It proves Mizuno's conjecture for and , and reformulates the case under a new conjecture (Conjecture ). The results strengthen the bridge between Y-systems, restricted Q-systems, and representation theory, providing explicit spectral data (exponents) tied to root-system combinatorics and offering a conditional path for the case. The methodology combines explicit quiver constructions, Jacobian analyses, and root-theoretic polynomials to realize the conjectured determinant identities.

Abstract

Let be the pair consisting of the Dynkin diagram of finite type and a positive integer , called the level. Then we obtain the Y-system, which is the set of algebraic relations associated with this pair. Related to the Y-system, a sequence of integers called exponents is defined through a quiver derived from the pair . Mizuno provided conjectured formulas for the exponents associated with Y-systems in [Mizuno Y., SIGMA 16 (2020), 028, 42 pages, arXiv:1812.05863]. In this paper, we study the exponents associated with level 2 Y-systems for classical Dynkin types. As a result, we present proofs of Mizuno's conjecture for and , and give a reformulation for .
Paper Structure (12 sections, 11 theorems, 75 equations, 1 figure)

This paper contains 12 sections, 11 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Quiver and Y-seed mutations
  4. Dynkin quiver $\boldsymbol{Q(X_n,\ell)}$ and mutation loop on $\boldsymbol{Q(X_n,\ell)}$
  5. Q-systems and Y-systems
  6. Exponents of $\boldsymbol{Q(X_n,\ell)}$
  7. Exponents
  8. Conjectural formulas
  9. Main results
  10. Type $\boldsymbol{B_n}$
  11. Type $\boldsymbol{D_n}$
  12. Type $\boldsymbol{C_n}$

Key Result

Lemma 2.5

Let $Q=Q(X_n,\ell)$ and $\nu$ be the permutation of vertices in $Q$. Consider the transformation Then $Q^\prime$ and $Q^{\prime\prime}$ are independent on orders of the mutations in $\mu_+$ and $\mu_-$. Furthermore, we have $Q=\nu(Q^{\prime\prime})$.

Figures (1)

  • Figure 1: Dynkin quivers $Q(X_n,2)$.

Theorems & Definitions (21)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Lemma 2.5
  • Theorem 2.6: IIKKNIIKKN2K
  • Proposition 2.7: KNS
  • Theorem 2.8: IIKKNIIKKN2KN
  • Theorem 2.9: LL2
  • Proposition 3.1: M
  • ...and 11 more