# Chevalley formula for anti-dominant weights in the equivariant K-theory of semi-infinite flag manifolds

@article{Naito2018ChevalleyFF,
title={Chevalley formula for anti-dominant weights in the equivariant K-theory of semi-infinite flag manifolds},
author={Satoshi Naito and Daniel Orr and Daisuke Sagaki},
journal={arXiv: Quantum Algebra},
year={2018}
}
• Published 4 August 2018
• Mathematics
• arXiv: Quantum Algebra
We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk formula in the torus-equivariant $K$-group of the formal power series model of semi-infinite flag manifolds, both of which are described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or, equivalently, quantum Lakshmibai-Seshadri paths). In view of recent results of Kato, these formulas give an explicit description of the structure constants for the Pontryagin product in the torus-equivariant $K$-group… Expand
12 Citations
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#### References

SHOWING 1-10 OF 53 REFERENCES
Equivariant K -theory of semi-infinite flag manifolds and the Pieri–Chevalley formula
• Mathematics
• 2017
We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $\mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-ChevalleyExpand
A Chevalley formula for semi-infinite flag manifolds and quantum K-theory (Extended abstract)
• Mathematics
• 2019
We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant K-theory of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As anExpand
Chevalley formula for anti-dominant minuscule fundamental weights in the equivariant quantum $K$-group of partial flag manifolds
• Mathematics
• 2020
In this paper, we give an explicit formula of Chevalley type, in terms of the Bruhat graph, for the quantum multiplication with the class of the line bundle associated to the anti-dominant minusculeExpand
Equivariant $K$-theory of the semi-infinite flag manifold as a nil-DAHA module
The equivariant $K$-theory of the semi-infinite flag manifold, as developed recently by Kato, Naito, and Sagaki, carries commuting actions of the nil-double affine Hecke algebra (nil-DAHA) and aExpand
Affine Weyl Groups in K-Theory and Representation Theory
• Mathematics
• 2003
We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley formula. OurExpand
Equivariant K-Chevalley rules for Kac-Moody flag manifolds
• Mathematics
• 2012
Explicit combinatorial cancellation-free rules are given for the product of an equivariant line bundle class with a Schubert class in the torus-equivariant $K$-theory of a Kac-Moody flag manifold.Expand
Loop structure on equivariant $K$-theory of semi-infinite flag manifolds
We explain that the Pontryagin product structure on the equivariant $K$-group of an affine Grassmannian considered in [Lam-Schilling-Shimozono, Compos. Math. {\bf 146} (2010)] coincides with theExpand
Quantum cohomology of G/P and homology of affine Grassmannian
• Mathematics
• 2007
Let G be a simple and simply-connected complex algebraic group, P ⊂ G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH*(G/P) of a flagExpand
A Uniform Model for Kirillov–Reshetikhin Crystals I: Lifting the Parabolic Quantum Bruhat Graph
• Mathematics, Physics
• 2012
A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties
We prove a type-uniform Chevalley formula for multiplication with divisor classes in the equivariant quantum $K$-theory ring of any cominuscule flag variety $G/P$. We also prove that multiplicationExpand