Recently, Blasiak–Morse–Seelinger introduced symmetric func- tions called Katalan functions, and proved that the
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K
-theoretic
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k
-Schur functions due to Lam–Schilling–Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called closed
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k
-Schur Katalan functions is identified with the Schubert structure sheaves in the
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-homology of the affine Grassmannian. Our main result is a proof of this conjecture.
We also study a
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-theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a nongeometric manner, based on the unipotent solution of the relativistic Toda lattice of Ruijsenaars. We prove that the map sends a Schubert class of the quantum
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-theory ring of the flag variety to a closed
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-
k
k
-Schur Katalan function up to an explicit factor related to a translation element with respect to an antidominant coroot. In fact, we prove this map coincides with a map whose existence was conjectured by Lam, Li, Mihalcea, Shimozono, and proved by Kato, and more recently by Chow and Leung.