Peterson Isomorphism in $K$-theory and Relativistic Toda Lattice

TL;DR

This paper establishes a $K$-theory version of the Peterson isomorphism by constructing a birational morphism between the spectra of the affine Grassmannian's $K$-homology ring and the quantum $K$-theory of the flag variety, using relativistic Toda lattice techniques.

Contribution

It explicitly constructs a birational morphism linking two important $K$-theoretic rings and introduces a $K$-theory Peterson isomorphism using integrable systems methods.

Findings

Constructed a birational morphism between the spectra of two $K$-theoretic rings.

Established a $K$-theory analogue of the Peterson isomorphism.

Determined the image of Lenart--Maeno's quantum Grothendieck polynomial for Grassmannian permutations.

Abstract

The $K$-homology ring of the affine Grassmannian of $SL_n(C)$ was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, for the quantum $K$-theory of the flag variety $Fl_n$, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars's relativistic Toda lattice with unipotent initial condition. From this result, we obtain a $K$-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart--Maeno's quantum Grothendieck polynomial associated with a Grassmannian permutation.