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Relativistic Toda lattice of type B and quantum $K$-theory of type C flag variety

Takeshi Ikeda, Shinsuke Iwao, Takafumi Kouno, Satoshi Naito, Kohei Yamaguchi

Abstract

We introduce a classical integrable system associated with the torus-equivariant quantum $K$-theory of type C flag variety. We prove that its conserved quantities coincide with the generators of the defining ideal of the Borel presentation of the quantum $K$-ring obtained by Kouno and Naito. In particular, the Hamiltonian of the system is naturally regarded as a type B analogue of the relativistic Toda lattice introduced by Ruijsenaars. We also construct Bäcklund transformations describing the discrete time evolution of the system. This construction makes explicit the integrable structure underlying the quantum $K$-theory and provides a framework for further studies of the $K$-theoretic Peterson isomorphism.

Relativistic Toda lattice of type B and quantum $K$-theory of type C flag variety

Abstract

We introduce a classical integrable system associated with the torus-equivariant quantum -theory of type C flag variety. We prove that its conserved quantities coincide with the generators of the defining ideal of the Borel presentation of the quantum -ring obtained by Kouno and Naito. In particular, the Hamiltonian of the system is naturally regarded as a type B analogue of the relativistic Toda lattice introduced by Ruijsenaars. We also construct Bäcklund transformations describing the discrete time evolution of the system. This construction makes explicit the integrable structure underlying the quantum -theory and provides a framework for further studies of the -theoretic Peterson isomorphism.

Paper Structure

This paper contains 10 sections, 5 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The integrable system
  3. Preliminaries: a Lie group decomposition
  4. The Lax matrix via quantum $K$-theory
  5. The Lax equation
  6. The flow of motion
  7. Computation of the Hamiltonian
  8. Bäcklund transformation
  9. Summary and discussions
  10. Proof of Equation \ref{['eq:path_description_of_F']}

Key Result

Proposition 2.1

A generic $2n\times 2n$ matrix $X\in GL_{2n}(\mathord{\mathbb{C}})$ can be uniquely decomposed as $X=KR$, where $K\in G_-$ and $R\in G_+$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: An example of a weighted graph for $n=3$. There are six horizontal lines $L_k$ and $L_{\overline{k}}$ ($k=1,2,3$), four short segments, and three dashed segments. Weights are assigned to some segments as indicated alongside them.

Theorems & Definitions (15)

  • Proposition 2.1
  • proof
  • Example 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 2.6
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • ...and 5 more