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A degeneration of the $q$-Garnier system of fourth order arises from confluences in quivers

Kazuya Matsugashita, Takao Suzuki, Satoshi Tsuchimi

Abstract

The $q$-Garnier system was first proposed by Sakai and its other directions of discrete time evolutions were given by Nagao and Yamada. Recently, it was shown that all of those directions of discrete time evolutions are derived from a birational representation of an extended affine Weyl group which arises from the cluster algebraic construction established by Masuda, Okubo and Tsuda. In this article, we investigate a degeneration structure of the $q$-Garnier system of fourth order by using confluences in quivers.

A degeneration of the $q$-Garnier system of fourth order arises from confluences in quivers

Abstract

The -Garnier system was first proposed by Sakai and its other directions of discrete time evolutions were given by Nagao and Yamada. Recently, it was shown that all of those directions of discrete time evolutions are derived from a birational representation of an extended affine Weyl group which arises from the cluster algebraic construction established by Masuda, Okubo and Tsuda. In this article, we investigate a degeneration structure of the -Garnier system of fourth order by using confluences in quivers.

Paper Structure

This paper contains 15 sections, 13 theorems, 129 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Cluster mutation and confluence in quiver
  3. Birational representation of affine Weyl group
  4. $q$-Garnier system
  5. Degeneration of the $q$-Garnier system: from 12 vertices to 11 vertices
  6. Degeneration of the $q$-Garnier system: from 11 vertices to 10 vertices
  7. Confluence $4\to5$
  8. Confluence $6\to4$
  9. Confluence $5\to8$
  10. Confluence $11\to2$
  11. Confluence $1\to11$
  12. Degeneration of the particular solution
  13. Quiver $Q_{11}$
  14. Quiver $Q_{101}$
  15. Quiver $Q_{102}$

Key Result

Theorem 5.2

Through the confluence $12\to1$, the simple reflections and the Dynkin diagram automorphisms are reduced as follows. The simple roots are also reduced as follows. $\blacktriangleleft$$\blacktriangleleft$

Figures (8)

  • Figure 1: Confluence in quiver
  • Figure 2: Quiver $Q_{12}$
  • Figure 3: Quiver $Q_{11}$
  • Figure 4: Quiver $Q_{101}$
  • Figure 5: Quiver $Q_{102}$
  • ...and 3 more figures

Theorems & Definitions (20)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 5.2
  • proof
  • Corollary 5.3
  • Theorem 6.2
  • Corollary 6.3
  • Theorem 6.5
  • ...and 10 more