A dual basis for the equivariant quantum $K$-theory of cominuscule varieties

TL;DR

This paper introduces a dual basis in equivariant quantum K-theory for cominuscule varieties, providing explicit combinatorial formulas and applications to Schubert structure constants.

Contribution

It defines a quantized ideal sheaf basis dual to the Schubert basis in equivariant quantum K-theory of cominuscule flag varieties, with explicit formulas and applications.

Findings

Explicit combinatorial formulas for quantized ideal sheaves.

Duality between quantized ideal sheaves and Schubert basis.

Application to compute Schubert structure constants.

Abstract

The equivariant quantum $K$-theory ring of a flag variety is a Frobenius algebra equipped with a perfect pairing called the quantum $K$-metric. It is known that in the classical $K$-theory ring for a given flag variety the ideal sheaf basis is dual to the Schubert basis with regard to the sheaf Euler characteristic. We define a quantization of the ideal sheaf basis for the equivariant quantum $K$-theory of cominuscule flag varieties. These quantized ideal sheaves are then dual to the Schubert basis with regard to the quantum $K$-metric. We prove explicit type-uniform combinatorial formulae for the quantized ideal sheaves in terms of the Schubert basis for any cominuscule flag variety. We also provide an application ultilizing the quantized ideal sheaves to calculate the Schubert structure constants associated to multiplication by the top exterior power of the tautological quotient bundle in $QK_T(Gr(k,n))$.