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

TL;DR

This paper introduces a rank 3 linear q-difference equation with E8^{(1)}-symmetry, utilizing q-middle convolution and comparing it with existing quantum curves, advancing understanding of symmetries in q-difference systems.

Contribution

The paper constructs a new rank 3 q-difference equation with E8^{(1)}-symmetry and links it to q-middle convolution and q-Okubo equations, providing a novel perspective.

Findings

The equation admits affine Weyl group symmetry of type E8^{(1)}.

Comparison with Moriyama-Yamada's quantum curve highlights differences.

Reconstruction of q-middle convolution via q-Okubo type equation enhances symmetry analysis.

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.