In this talk, we will introduce the (torus-equivariant) quantum K-theory of flag varieties G/P. We will discuss the sum of the Schubert structure coefficients in the equivariant quantum K-theory of flag varieties G/P. We will show that the sheaf Euler characteristic of the equivariant quantum K-product of a Schubert class and an opposite Schubert class is equal to qd, where d is the smallest degree of a rational curve joining the two Schubert varieties. We will also discuss a cyclic group action on the quantum K-theory of complex Grassmannians and its consequences. This is based on my joint works with Anders Buch, Sjuvon Chung, Leonardo Mihalcea, and Mingzhi Yang.

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