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Relativistic Toda lattice and equivariant $K$-homology of affine Grassmannian

Takeshi Ikeda, Shinsuke Iwao, Satoshi Naito, Kohei Yamaguchi

TL;DR

This work realizes the $K$-Peterson map in the equivariant setting for $SL_n$ by embedding $QK_T(SL_n(\mathbb{C})/B)$ into $K_*^T(\mathrm{Gr}_{SL_n})$ through an explicit rational substitution. Central to the approach are the Relativistic Toda lattice, the centralizer family, and a symmetric-function realization via the $K$-theoretic double $k$-Schur functions, which correspond to Schubert classes on the affine Grassmannian. The authors establish a precise correspondence $\tilde{\Phi}_n( Q^\xi \mathfrak{G}^Q_w(z|\eta) ) = \tilde{g}_{x^{\omega_k}}^{(k)}(y|b)$, intertwining Demazure actions and enabling determinantal formulas for key classes. As an application, they derive a factorization for the maximal $k$-irreducible partition $\nu_n$, demonstrating concrete structural decompositions in the equivariant $K$-theoretic setting. Overall, the paper provides a concrete, algebraic realization of the quantum affine correspondence in equivariant $K$-theory with new combinatorial tools for studying the $K$-Peterson isomorphism.

Abstract

We investigate the phenomenon known as ``quantum equals affine'' in the setting of $T$-equivariant quantum $K$-theory of the flag variety $G/B$, as established by Kato for any semisimple algebraic group $G$. In particular, we focus on the $K$-Peterson isomorphism between the $T$-equivariant quantum $K$-ring $QK_T(SL_n(\mathbb{C})/B)$ and the $T$-equivariant $K$-homology ring $K_*^T(\mathrm{Gr}_{SL_n})$ of the affine Grassmannian, after suitable localizations on both sides. Building on an earlier work by Ikeda, Iwao, and Maeno, we present an explicit algebraic realization of the $K$-Peterson map via a rational substitution that sends the generators of the quantum $K$-theory ring to explicit rational expressions in the fundamental generators of $K_*^T(\mathrm{Gr}_{SL_n})$, thereby matching the Schubert bases on both sides. Our approach builds on recent developments in the theory of $QK_T(SL_n(\mathbb{C})/B)$ by Maeno, Naito, and Sagaki, as well as the theory of $K$-theoretic double $k$-Schur functions introduced by Ikeda, Shimozono, and Yamaguchi. This concrete formulation provides new insight into the combinatorial structure of the $K$-Peterson isomorphism in the equivariant setting. As an application, we establish a factorization formula for the $K$-theoretic double $k$-Schur function associated with the maximal $k$-irreducible $k$-bounded partition.

Relativistic Toda lattice and equivariant $K$-homology of affine Grassmannian

TL;DR

This work realizes the -Peterson map in the equivariant setting for by embedding into through an explicit rational substitution. Central to the approach are the Relativistic Toda lattice, the centralizer family, and a symmetric-function realization via the -theoretic double -Schur functions, which correspond to Schubert classes on the affine Grassmannian. The authors establish a precise correspondence , intertwining Demazure actions and enabling determinantal formulas for key classes. As an application, they derive a factorization for the maximal -irreducible partition , demonstrating concrete structural decompositions in the equivariant -theoretic setting. Overall, the paper provides a concrete, algebraic realization of the quantum affine correspondence in equivariant -theory with new combinatorial tools for studying the -Peterson isomorphism.

Abstract

We investigate the phenomenon known as ``quantum equals affine'' in the setting of -equivariant quantum -theory of the flag variety , as established by Kato for any semisimple algebraic group . In particular, we focus on the -Peterson isomorphism between the -equivariant quantum -ring and the -equivariant -homology ring of the affine Grassmannian, after suitable localizations on both sides. Building on an earlier work by Ikeda, Iwao, and Maeno, we present an explicit algebraic realization of the -Peterson map via a rational substitution that sends the generators of the quantum -theory ring to explicit rational expressions in the fundamental generators of , thereby matching the Schubert bases on both sides. Our approach builds on recent developments in the theory of by Maeno, Naito, and Sagaki, as well as the theory of -theoretic double -Schur functions introduced by Ikeda, Shimozono, and Yamaguchi. This concrete formulation provides new insight into the combinatorial structure of the -Peterson isomorphism in the equivariant setting. As an application, we establish a factorization formula for the -theoretic double -Schur function associated with the maximal -irreducible -bounded partition.
Paper Structure (45 sections, 51 theorems, 243 equations)

This paper contains 45 sections, 51 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. TEXT-theoretic Peterson map --- abstract form
  3. TEXT-Peterson map from the Relativistic Toda lattice
  4. TEXT-theoretic double TEXT-Schur function
  5. TEXT-rectangles and the TEXT-functions
  6. Quantum double Grothendieck polynomials
  7. Correspondence of Schubert bases
  8. Factorization formula for the maximal TEXT-irreducible partition TEXT
  9. Organization
  10. Relativistic Toda lattice and the centralizer famify
  11. Relativistic Toda lattice
  12. Quantum TEXT-ring of the flag variety
  13. Centralizer family TEXT
  14. TEXT-functions
  15. Construction of TEXT
  16. ...and 30 more sections

Key Result

Theorem 1.1

For $x\in \hat{W}_G^0$, write $x=wt_\xi$ with $w\in W, \xi\in Q^\vee$. Then

Theorems & Definitions (117)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Maeno, Naito, Sagaki MNS1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • ...and 107 more