Generating functions of dual $K$-theoretic $P$- and $Q$-functions and boson-fermion correspondence
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866911016640577536 |
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| author | Iwao, Shinsuke |
| author_facet | Iwao, Shinsuke |
| contents | In this paper, we present a new algebraic description of Ikeda-Naruse's $K$-theoretic Schur $P$- and $Q$-functions and their dual functions in terms of neutral fermion operators. We introduce four families of ``$β$-deformed neutral-fermion operators'' depending on a parameter $β$, which reduce to the usual neutral-fermion operators when $β$ is zero. Using these operators, we introduce two families of $β$-deformed vertex operators, power sums, and boson-fermion correspondences. From commutation relations among these operators, we naturally derive the $K$-theoretic Cauchy kernel of Nakagawa-Naruse. Exploiting this fact, we show that the four $K$-theoretic functions can be realized as vacuum expectation values of certain $β$-deformed fermionic operators. This presentation also allows us to derive generating functions for the dual $K$-theoretic $P$-and $Q$-functions, as conjectured by Nakagawa-Naruse. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_12741 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generating functions of dual $K$-theoretic $P$- and $Q$-functions and boson-fermion correspondence Iwao, Shinsuke Combinatorics Algebraic Geometry K-Theory and Homology 05E05, 05E14, 13M10, 14N15 In this paper, we present a new algebraic description of Ikeda-Naruse's $K$-theoretic Schur $P$- and $Q$-functions and their dual functions in terms of neutral fermion operators. We introduce four families of ``$β$-deformed neutral-fermion operators'' depending on a parameter $β$, which reduce to the usual neutral-fermion operators when $β$ is zero. Using these operators, we introduce two families of $β$-deformed vertex operators, power sums, and boson-fermion correspondences. From commutation relations among these operators, we naturally derive the $K$-theoretic Cauchy kernel of Nakagawa-Naruse. Exploiting this fact, we show that the four $K$-theoretic functions can be realized as vacuum expectation values of certain $β$-deformed fermionic operators. This presentation also allows us to derive generating functions for the dual $K$-theoretic $P$-and $Q$-functions, as conjectured by Nakagawa-Naruse. |
| title | Generating functions of dual $K$-theoretic $P$- and $Q$-functions and boson-fermion correspondence |
| topic | Combinatorics Algebraic Geometry K-Theory and Homology 05E05, 05E14, 13M10, 14N15 |
| url | https://arxiv.org/abs/2301.12741 |