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

We give a presentation of the torus-equivariant (small) quantum $K$-theory ring of flag manifolds of type $A$, as the quotient of a polynomial ring by an explicit ideal. This result is the torus-equivariant version of our previous one, which gives a presentation of the nonequivariant quantum $K$-theory ring of flag manifolds of type $A$. However, the method of proof for the torus-equivariant one is entirely different from that for the nonequivariant 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 nonequivariant 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 relations that yield our presentation.