Categorification of DAHA and Macdonald polynomials
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913522880872448 |
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| author | Kato, Syu Khoroshkin, Anton Makedonskyi, Ievgen |
| author_facet | Kato, Syu Khoroshkin, Anton Makedonskyi, Ievgen |
| contents | We describe a categorification of the Double Affine Hecke Algebra (${\mathcal{H}\kern -.4em\mathcal{H}}$) associated with an affine Lie algebra $\widehat{\mathfrak{g}}$, including a categorification of the polynomial representation and Macdonald polynomials.
Our categorification results are presented in the derived setting, focusing on the derived category of graded modules over the Lie superalgebra ${\mathfrak I}[ξ]$, where ${\mathfrak I} \subset \widehat{\mathfrak{g}}$ is the Iwahori subalgebra of the affine Lie algebra and $ξ$ is a formal odd variable.
First, we show that the compositions of induction and restriction functors associated with minimal parabolic subalgebras ${\mathfrak{p}}_{i}$ categorify the Demazure operators $T_i + 1 \in {\mathcal{H}\kern -.4em\mathcal{H}}$, ensuring that all algebraic relations of $T_i$ have categorical interpretations. Second, for each dominant weight $λ$ we introduce a complex ${\mathbb{EM}}_λ$ of ${\mathfrak{I}}[ξ]$-modules and a complex ${\mathbb{PM}}_λ$ of ${\mathfrak{g}}[z,ξ]$-modules, whose Euler characteristics are equal to nonsymmetric $E_λ$ and symmetric $P_λ$ Macdonald polynomials respectively.
We illustrate our theory with the example $\mathfrak{g}=\mathfrak{sl}_2$ where we construct the cyclic representations of Lie superalgebra ${\mathfrak{I}}[ξ]$ such that their supercharacters coincide with certain normalizations of nonsymmetric Macdonald polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_10009 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Categorification of DAHA and Macdonald polynomials Kato, Syu Khoroshkin, Anton Makedonskyi, Ievgen Representation Theory Algebraic Geometry Quantum Algebra We describe a categorification of the Double Affine Hecke Algebra (${\mathcal{H}\kern -.4em\mathcal{H}}$) associated with an affine Lie algebra $\widehat{\mathfrak{g}}$, including a categorification of the polynomial representation and Macdonald polynomials. Our categorification results are presented in the derived setting, focusing on the derived category of graded modules over the Lie superalgebra ${\mathfrak I}[ξ]$, where ${\mathfrak I} \subset \widehat{\mathfrak{g}}$ is the Iwahori subalgebra of the affine Lie algebra and $ξ$ is a formal odd variable. First, we show that the compositions of induction and restriction functors associated with minimal parabolic subalgebras ${\mathfrak{p}}_{i}$ categorify the Demazure operators $T_i + 1 \in {\mathcal{H}\kern -.4em\mathcal{H}}$, ensuring that all algebraic relations of $T_i$ have categorical interpretations. Second, for each dominant weight $λ$ we introduce a complex ${\mathbb{EM}}_λ$ of ${\mathfrak{I}}[ξ]$-modules and a complex ${\mathbb{PM}}_λ$ of ${\mathfrak{g}}[z,ξ]$-modules, whose Euler characteristics are equal to nonsymmetric $E_λ$ and symmetric $P_λ$ Macdonald polynomials respectively. We illustrate our theory with the example $\mathfrak{g}=\mathfrak{sl}_2$ where we construct the cyclic representations of Lie superalgebra ${\mathfrak{I}}[ξ]$ such that their supercharacters coincide with certain normalizations of nonsymmetric Macdonald polynomials. |
| title | Categorification of DAHA and Macdonald polynomials |
| topic | Representation Theory Algebraic Geometry Quantum Algebra |
| url | https://arxiv.org/abs/2103.10009 |