Outer functors and a general operadic framework

Powell, Geoffrey

doi:10.1016/j.jalgebra.2024.01.015

Journal of Algebra

2024

DOI: 10.1016/j.jalgebra.2024.01.015

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Outer functors and a general operadic framework

Geoffrey Powell

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2024

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Polynomial Representations of the Witt Lie Algebra

Sam,

Snowden,

Tosteson

2024

International Mathematics Research Notices

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The Witt algebra ${\mathfrak{W}}_{n}$ is the Lie algebra of all derivations of the $n$-variable polynomial ring $\textbf{V}_{n}=\textbf{C}[x_{1}, \ldots , x_{n}]$ (or of algebraic vector fields on $\textbf{A}^{n}$). A representation of ${\mathfrak{W}}_{n}$ is polynomial if it arises as a subquotient of a sum of tensor powers of $\textbf{V}_{n}$. Our main theorems assert that finitely generated polynomial representations of ${\mathfrak{W}}_{n}$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $\textbf{Fin}^{\textrm{op}}$, where $\textbf{Fin}$ is the category of finite sets. We also show that polynomial representations of ${\mathfrak{W}}_{n}$ are equivalent to polynomial representations of the endomorphism monoid of $\textbf{A}^{n}$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.

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Polynomial Representations of the Witt Lie Algebra

Sam,

Snowden,

Tosteson

2024

International Mathematics Research Notices

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show abstract

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Outer functors and a general operadic framework

Cited by 1 publication

References 9 publications

Polynomial Representations of the Witt Lie Algebra

Polynomial Representations of the Witt Lie Algebra

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