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

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.