In quantum cohomology of the complete flag variety, the special Schubert class acts as a cyclic permutation operator with a quantum monomial weight, and the same is conjectured for quantum K-theory.
Transformation formulas in quantum cohomology , Compos Math, 2004, 140: 778-792
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Toward quantum Pieri rule for $F\ell_n$ via Seidel representation
In quantum cohomology of the complete flag variety, the special Schubert class acts as a cyclic permutation operator with a quantum monomial weight, and the same is conjectured for quantum K-theory.