Symplectic Fourier-Deligne transforms on G/U and the algebra of braids and ties
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911947302109184 |
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| author | Morton-Ferguson, Calder |
| author_facet | Morton-Ferguson, Calder |
| contents | We explicitly identify the algebra generated by symplectic Fourier-Deligne transforms (i.e. convolution with Kazhdan-Laumon sheaves) acting on the Grothendieck group of perverse sheaves on the basic affine space $G/U$, answering a question originally raised by A. Polishchuk. We show it is isomorphic to a distinguished subalgebra, studied by I. Marin, of the generalized algebra of braids and ties (defined in Type $A$ by F. Aicardi and J. Juyumaya and generalized to all types by Marin), providing a connection between geometric representation theory and an algebra defined in the context of knot theory. Our geometric interpretation of this algebra entails some algebraic consequences: we obtain a short and type-independent geometric proof of the braid relations for Juyumaya's generators of the Yokonuma-Hecke algebra (previously proved case-by-case in types $A, D, E$ by Juyumaya and separately for types $B, C, F_4, G_2$ by Juyumaya and S. S. Kannan), a natural candidate for an analogue of a Kazhdan-Lusztig basis, and finally an explicit formula for the dimension of Marin's algebra in Type $A_n$ (previously only known for $n \leq 4$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_01998 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Symplectic Fourier-Deligne transforms on G/U and the algebra of braids and ties Morton-Ferguson, Calder Representation Theory Algebraic Geometry Rings and Algebras We explicitly identify the algebra generated by symplectic Fourier-Deligne transforms (i.e. convolution with Kazhdan-Laumon sheaves) acting on the Grothendieck group of perverse sheaves on the basic affine space $G/U$, answering a question originally raised by A. Polishchuk. We show it is isomorphic to a distinguished subalgebra, studied by I. Marin, of the generalized algebra of braids and ties (defined in Type $A$ by F. Aicardi and J. Juyumaya and generalized to all types by Marin), providing a connection between geometric representation theory and an algebra defined in the context of knot theory. Our geometric interpretation of this algebra entails some algebraic consequences: we obtain a short and type-independent geometric proof of the braid relations for Juyumaya's generators of the Yokonuma-Hecke algebra (previously proved case-by-case in types $A, D, E$ by Juyumaya and separately for types $B, C, F_4, G_2$ by Juyumaya and S. S. Kannan), a natural candidate for an analogue of a Kazhdan-Lusztig basis, and finally an explicit formula for the dimension of Marin's algebra in Type $A_n$ (previously only known for $n \leq 4$). |
| title | Symplectic Fourier-Deligne transforms on G/U and the algebra of braids and ties |
| topic | Representation Theory Algebraic Geometry Rings and Algebras |
| url | https://arxiv.org/abs/2304.01998 |