Polynomial representations of the Witt Lie algebra
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909854483873792 |
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| author | Sam, Steven V Snowden, Andrew Tosteson, Philip |
| author_facet | Sam, Steven V Snowden, Andrew Tosteson, Philip |
| contents | The Witt algebra W_n is the Lie algebra of all derivations of the n-variable polynomial ring V_n=C[x_1, ..., x_n] (or of algebraic vector fields on A^n). A representation of W_n is polynomial if it arises as a subquotient of a sum of tensor powers of V_n. Our main theorems assert that finitely generated polynomial representations of W_n are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of Fin^op, where Fin is the category of finite sets. We also show that polynomial representations of W_n are equivalent to polynomial representations of the endomorphism monoid of A^n. These equivalences are a special case of an operadic version of Schur--Weyl duality, which we establish. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_00399 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Polynomial representations of the Witt Lie algebra Sam, Steven V Snowden, Andrew Tosteson, Philip Representation Theory The Witt algebra W_n is the Lie algebra of all derivations of the n-variable polynomial ring V_n=C[x_1, ..., x_n] (or of algebraic vector fields on A^n). A representation of W_n is polynomial if it arises as a subquotient of a sum of tensor powers of V_n. Our main theorems assert that finitely generated polynomial representations of W_n are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of Fin^op, where Fin is the category of finite sets. We also show that polynomial representations of W_n are equivalent to polynomial representations of the endomorphism monoid of A^n. These equivalences are a special case of an operadic version of Schur--Weyl duality, which we establish. |
| title | Polynomial representations of the Witt Lie algebra |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2210.00399 |