Relativistic Toda lattice and equivariant $K$-homology of affine Grassmannian
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914101983182848 |
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| author | Ikeda, Takeshi Iwao, Shinsuke Naito, Satoshi Yamaguchi, Kohei |
| author_facet | Ikeda, Takeshi Iwao, Shinsuke Naito, Satoshi Yamaguchi, Kohei |
| contents | We investigate the phenomenon known as ``quantum equals affine'' in the setting of $T$-equivariant quantum $K$-theory of the flag variety $G/B$, as established by Kato for any semisimple algebraic group $G$. In particular, we focus on the $K$-Peterson isomorphism between the $T$-equivariant quantum $K$-ring $QK_T(SL_n(\mathbb{C})/B)$ and the $T$-equivariant $K$-homology ring $K_*^T(\mathrm{Gr}_{SL_n})$ of the affine Grassmannian, after suitable localizations on both sides. Building on an earlier work by Ikeda, Iwao, and Maeno, we present an explicit algebraic realization of the $K$-Peterson map via a rational substitution that sends the generators of the quantum $K$-theory ring to explicit rational expressions in the fundamental generators of $K_*^T(\mathrm{Gr}_{SL_n})$, thereby matching the Schubert bases on both sides. Our approach builds on recent developments in the theory of $QK_T(SL_n(\mathbb{C})/B)$ by Maeno, Naito, and Sagaki, as well as the theory of $K$-theoretic double $k$-Schur functions introduced by Ikeda, Shimozono, and Yamaguchi. This concrete formulation provides new insight into the combinatorial structure of the $K$-Peterson isomorphism in the equivariant setting. As an application, we establish a factorization formula for the $K$-theoretic double $k$-Schur function associated with the maximal $k$-irreducible $k$-bounded partition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02941 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relativistic Toda lattice and equivariant $K$-homology of affine Grassmannian Ikeda, Takeshi Iwao, Shinsuke Naito, Satoshi Yamaguchi, Kohei Representation Theory Algebraic Geometry Combinatorics K-Theory and Homology 05E05, 14N15 We investigate the phenomenon known as ``quantum equals affine'' in the setting of $T$-equivariant quantum $K$-theory of the flag variety $G/B$, as established by Kato for any semisimple algebraic group $G$. In particular, we focus on the $K$-Peterson isomorphism between the $T$-equivariant quantum $K$-ring $QK_T(SL_n(\mathbb{C})/B)$ and the $T$-equivariant $K$-homology ring $K_*^T(\mathrm{Gr}_{SL_n})$ of the affine Grassmannian, after suitable localizations on both sides. Building on an earlier work by Ikeda, Iwao, and Maeno, we present an explicit algebraic realization of the $K$-Peterson map via a rational substitution that sends the generators of the quantum $K$-theory ring to explicit rational expressions in the fundamental generators of $K_*^T(\mathrm{Gr}_{SL_n})$, thereby matching the Schubert bases on both sides. Our approach builds on recent developments in the theory of $QK_T(SL_n(\mathbb{C})/B)$ by Maeno, Naito, and Sagaki, as well as the theory of $K$-theoretic double $k$-Schur functions introduced by Ikeda, Shimozono, and Yamaguchi. This concrete formulation provides new insight into the combinatorial structure of the $K$-Peterson isomorphism in the equivariant setting. As an application, we establish a factorization formula for the $K$-theoretic double $k$-Schur function associated with the maximal $k$-irreducible $k$-bounded partition. |
| title | Relativistic Toda lattice and equivariant $K$-homology of affine Grassmannian |
| topic | Representation Theory Algebraic Geometry Combinatorics K-Theory and Homology 05E05, 14N15 |
| url | https://arxiv.org/abs/2505.02941 |