Closed $k$-Schur Katalan functions as $K$-homology Schubert representatives of the affine Grassmannian
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866929302561357824 |
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| author | Ikeda, Takeshi Iwao, Shinsuke Naito, Satoshi |
| author_facet | Ikeda, Takeshi Iwao, Shinsuke Naito, Satoshi |
| contents | Recently, Blasiak-Morse-Seelinger introduced symmetric functions called Katalan functions, and proved that the $K$-theoretic $k$-Schur functions due to Lam-Schilling-Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called the closed $k$-Schur Katalan functions are identified with the Schubert structure sheaves in the $K$-homology of the affine Grassmannian. The main result is a proof of the conjecture.
We also study a $K$-theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a non-geometric 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 $K$-theory ring of the flag variety to a closed $K$-$k$-Schur Katalan function up to an explicit factor related to a translation element with respect to an anti-dominant coroot. In fact, we prove the above 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_14483 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Closed $k$-Schur Katalan functions as $K$-homology Schubert representatives of the affine Grassmannian Ikeda, Takeshi Iwao, Shinsuke Naito, Satoshi Combinatorics Mathematical Physics Algebraic Geometry K-Theory and Homology Representation Theory 05E05, 14N15 Recently, Blasiak-Morse-Seelinger introduced symmetric functions called Katalan functions, and proved that the $K$-theoretic $k$-Schur functions due to Lam-Schilling-Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called the closed $k$-Schur Katalan functions are identified with the Schubert structure sheaves in the $K$-homology of the affine Grassmannian. The main result is a proof of the conjecture. We also study a $K$-theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a non-geometric 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 $K$-theory ring of the flag variety to a closed $K$-$k$-Schur Katalan function up to an explicit factor related to a translation element with respect to an anti-dominant coroot. In fact, we prove the above 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. |
| title | Closed $k$-Schur Katalan functions as $K$-homology Schubert representatives of the affine Grassmannian |
| topic | Combinatorics Mathematical Physics Algebraic Geometry K-Theory and Homology Representation Theory 05E05, 14N15 |
| url | https://arxiv.org/abs/2203.14483 |