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.
Classical aspects of quantum cohomology of generalized flag varieties , Int Math Res Not, 2012, 16: 3706-3722
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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.