TY - JOUR

T1 - Peterson isomorphism in K-theory and relativistic toda lattice

AU - Ikeda, Takeshi

AU - Iwao, Shinsuke

AU - Maeno, Toshiaki

N1 - Funding Information:
The work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI [grant numbers 15K04832 to T.I., 26800062 to S.I., 16K05083 to T.M.].
Funding Information:
The work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI [grant
Publisher Copyright:
© The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85101333171&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85101333171&partnerID=8YFLogxK

U2 - 10.1093/IMRN/RNY051

DO - 10.1093/IMRN/RNY051

M3 - Article

AN - SCOPUS:85101333171

VL - 2020

SP - 6421

EP - 6462

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 19

ER -