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Combinatorics of the \(K\)-theory of affine grassmannians. (English) Zbl 1238.05276
Summary: We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and \(k\)-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and \(k-K\)-Schur functions – Schubert representatives for the \(K\)-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.

MSC:
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
17B65 Infinite-dimensional Lie (super)algebras
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