A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part I: the defining ideal

Toshiaki Maeno; Satoshi Naito; Daisuke Sagaki

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arXiv · 2302.09485

A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part I: the defining ideal

Toshiaki Maeno·Satoshi Naito·Daisuke Sagaki

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Abstract

We give a presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, as a quotient of a polynomial ring by an explicit ideal. This is the torus-equivariant version of our previous result, which gives a presentation of the non-equivariant quantum $K$-theory ring of flag manifolds of type $A$. However, the method of proof for the torus-equivariant one is completely different from that for the non-equivariant one; our proof is based on the result in the $Q = 0$ limit, and uses Nakayama-type arguments to upgrade it to the quantum situation. Also, in contrast to the non-equivariant case in which we used the Chevalley formula, we make use of the inverse Chevalley formula for the torus-equivariant $K$-group of semi-infinite flag manifolds to obtain a relation which yields our presentation.

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Toshiaki Maeno, Satoshi Naito, Daisuke Sagaki. 2023-02-19. A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part I: the defining ideal. https://arxiv.org/abs/2302.09485

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