Peterson isomorphism in K-theory and relativistic toda lattice

Takeshi Ikeda, Shinsuke Iwao, Toshiaki Maeno

Research output: Contribution to journalArticlepeer-review

Abstract

The K-homology ring of the affine Grassmannian of SLn(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 Fln, 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.

Original languageEnglish
Pages (from-to)6421-6462
Number of pages42
JournalInternational Mathematics Research Notices
Volume2020
Issue number19
DOIs
Publication statusPublished - 2021
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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