A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

Cristian Lenart; Satoshi Naito; Daisuke Sagaki

doi:10.1007/s00029-024-00924-8

A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

Cristian Lenart
, Satoshi Naito
, Daisuke Sagaki

Mathematics and Statistics

University at Albany

Institute of Science Tokyo
University of Tsukuba

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

We give a Chevalley formula for an arbitrary weight for the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory QK_T(G/B) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of QK_T(G/B). In type A_n-1, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory QK(SL_n/B); we also obtain very explicit information about the coefficients in the respective Chevalley formula.

Original language	English
Article number	39
Journal	Selecta Mathematica, New Series
Volume	30
Issue number	3
DOIs	https://doi.org/10.1007/s00029-024-00924-8
State	Published - Jul 2024

Keywords

05E14
14N15
17B37
81R10
Chevalley formula
Primary 14M15
Quantum Bruhat graph
Quantum Grothendieck polynomials
Quantum K-theory
Quantum LS paths
Quantum alcove model
Secondary 14N10
Semi-infinite flag manifold

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10.1007/s00029-024-00924-8

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title = "A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory",

abstract = "We give a Chevalley formula for an arbitrary weight for the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory QKT(G/B) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of QKT(G/B). In type An-1, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory QK(SLn/B); we also obtain very explicit information about the coefficients in the respective Chevalley formula.",

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author = "Cristian Lenart and Satoshi Naito and Daisuke Sagaki",

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doi = "10.1007/s00029-024-00924-8",

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T1 - A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

AU - Lenart, Cristian

AU - Naito, Satoshi

AU - Sagaki, Daisuke

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.

PY - 2024/7

Y1 - 2024/7

N2 - We give a Chevalley formula for an arbitrary weight for the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory QKT(G/B) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of QKT(G/B). In type An-1, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory QK(SLn/B); we also obtain very explicit information about the coefficients in the respective Chevalley formula.

AB - We give a Chevalley formula for an arbitrary weight for the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory QKT(G/B) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of QKT(G/B). In type An-1, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory QK(SLn/B); we also obtain very explicit information about the coefficients in the respective Chevalley formula.

KW - 05E14

KW - 14N15

KW - 17B37

KW - 81R10

KW - Chevalley formula

KW - Primary 14M15

KW - Quantum Bruhat graph

KW - Quantum Grothendieck polynomials

KW - Quantum K-theory

KW - Quantum LS paths

KW - Quantum alcove model

KW - Secondary 14N10

KW - Semi-infinite flag manifold

UR - https://www.scopus.com/pages/publications/85187302871

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DO - 10.1007/s00029-024-00924-8

M3 - Article

SN - 1022-1824

VL - 30

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

IS - 3

M1 - 39

ER -

A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

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Keywords

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