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A $3\times3$ linear $q$-difference system with $E_8^{(1)}$-symmetry

Takahiko Nobukawa

TL;DR

The paper constructs a rank-$3$ linear $q$-difference equation with the affine Weyl group symmetry of type $E_8^{(1)}$, realized via a $q$-middle convolution implemented through a $q$-Okubo reformulation. The main result, Theorem E8mc, provides a birational transformation of the equation’s parameters under $q$-middle convolution, yielding a concrete $W(E_8^{(1)})$-action generated by a distinguished $s_0$ and adjacent transpositions. The rank-$3$ system is shown to reduce to a scalar, non-autonomous version of Moriyama–Yamada’s quantum curve MY, establishing a link to quantum Painlevé-type dynamics and suggesting connections to quantum integrable systems. The work thus integrates geometric Weyl-group symmetries, $q$-convolution techniques, and scalar reductions to illuminate higher-symmetry $q$-difference equations and their potential Lax structures and degenerations.

Abstract

We present a linear $q$-difference equation of rank $3$, which admits the affine Weyl group symmetry of type $E_8^{(1)}$. We further compare this equation with Moriyama-Yamada's quantum curve which has $W(E_8^{(1)})$-symmetry. The symmetry of our equation is provided by the $q$-middle convolution, defined by Sakai-Yamaguchi and reformulated by Arai-Takemura. In this paper, we provide a reconstruction of the $q$-middle convolution via a $q$-Okubo type equation.

A $3\times3$ linear $q$-difference system with $E_8^{(1)}$-symmetry

TL;DR

The paper constructs a rank- linear -difference equation with the affine Weyl group symmetry of type , realized via a -middle convolution implemented through a -Okubo reformulation. The main result, Theorem E8mc, provides a birational transformation of the equation’s parameters under -middle convolution, yielding a concrete -action generated by a distinguished and adjacent transpositions. The rank- system is shown to reduce to a scalar, non-autonomous version of Moriyama–Yamada’s quantum curve MY, establishing a link to quantum Painlevé-type dynamics and suggesting connections to quantum integrable systems. The work thus integrates geometric Weyl-group symmetries, -convolution techniques, and scalar reductions to illuminate higher-symmetry -difference equations and their potential Lax structures and degenerations.

Abstract

We present a linear -difference equation of rank , which admits the affine Weyl group symmetry of type . We further compare this equation with Moriyama-Yamada's quantum curve which has -symmetry. The symmetry of our equation is provided by the -middle convolution, defined by Sakai-Yamaguchi and reformulated by Arai-Takemura. In this paper, we provide a reconstruction of the -middle convolution via a -Okubo type equation.
Paper Structure (9 sections, 16 theorems, 165 equations)

This paper contains 9 sections, 16 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. $q$-middle convolution
  4. A $3\times3$ system
  5. Scalar equation
  6. Summary and discussions
  7. Proof of Theorem \ref{['E8mc']}
  8. Proof of Proposition \ref{['propbraid']}
  9. Proof of Proposition \ref{['propsc']}

Key Result

Lemma 2.1

The $q$-Leibniz rule holds:

Theorems & Definitions (54)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 44 more