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1. Set Partitions, Fermions, and Skein Relations

   https://doi.org/10.1093/imrn/rnac110  

   Kim, Jesse; Rhoades, Brendon (May 2022, International Mathematics Research Notices)

   Abstract Let $$\Theta _n = (\theta _1, \dots , \theta _n)$$ and $$\Xi _n = (\xi _1, \dots , \xi _n)$$ be two lists of $$n$$ variables, and consider the diagonal action of $${{\mathfrak {S}}}_n$$ on the exterior algebra $$\wedge \{ \Theta _n, \Xi _n \}$$ generated by these variables. Jongwon Kim and the 2nd author defined and studied the fermionic diagonal coinvariant ring$$FDR_n$$ obtained from $$\wedge \{ \Theta _n, \Xi _n \}$$ by modding out by the ideal generated by the $${{\mathfrak {S}}}_n$$-invariants with vanishing constant term. On the other hand, the 2nd author described an action of $${{\mathfrak {S}}}_n$$ on the vector space with basis given by noncrossing set partitions of $$\{1,\dots ,n\}$$ using a novel family of skein relations that resolve crossings in set partitions. We give an isomorphism between a natural Catalan-dimensional submodule of $$FDR_n$$ and the skein representation. To do this, we show that set partition skein relations arise naturally in the context of exterior algebras. Our approach yields an $${{\mathfrak {S}}}_n$$-equivariant way to resolve crossings in set partitions. We use fermions to clarify, sharpen, and extend the theory of set partition crossing resolution. 

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2. Set superpartitions and superspace duality modules

   https://doi.org/10.1017/fms.2022.90  

   Rhoades, Brendon; Wilson, Andrew Timothy (January 2022, Forum of Mathematics, Sigma)

   Abstract The superspace ring $$\Omega _n$$ is a rank n polynomial ring tensored with a rank n exterior algebra. Using an extension of the Vandermonde determinant to $$\Omega _n$$ , the authors previously defined a family of doubly graded quotients $${\mathbb {W}}_{n,k}$$ of $$\Omega _n$$ , which carry an action of the symmetric group $${\mathfrak {S}}_n$$ and satisfy a bigraded version of Poincaré Duality. In this paper, we examine the duality modules $${\mathbb {W}}_{n,k}$$ in greater detail. We describe a monomial basis of $${\mathbb {W}}_{n,k}$$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called ordered set superpartitions . These are ordered set partitions $$(B_1 \mid \cdots \mid B_k)$$ of $$\{1,\dots ,n\}$$ in which the nonminimal elements of any block $$B_i$$ may be barred or unbarred. 

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3. The LLV Algebra for Primitive Symplectic Varieties with Isolated Singularities

   https://doi.org/10.46298/epiga.2025.12186  

   Tighe, Benjamin (March 2025, Épijournal de Géométrie Algébrique)

   We extend results of Looijenga--Lunts and Verbitsky and show that the total Lie algebra $$\mathfrak g$$ for the intersection cohomology of a primitive symplectic variety $$X$$ with isolated singularities is isomorphic to $$\mathfrak g \cong \mathfrak{so}\left(\left(IH^2(X, \mathbb Q), Q_X\right)\oplus \mathfrak h\right),$$ where $$Q_X$$ is the intersection Beauville--Bogomolov--Fujiki form and $$\mathfrak h$$ is a hyperbolic plane. This gives a new, algebraic proof for irreducible holomorphic symplectic manifolds which does not rely on the hyperk\"ahler metric. Along the way, we study the structure of $$IH^*(X, \mathbb Q)$$ as a $$\mathfrak{g}$$-representation -- with particular emphasis on the Verbitsky component, multidimensional Kuga--Satake constructions, and Mumford--Tate algebras -- and give some immediate applications concerning the $P = W$ conjecture for primitive symplectic varieties. Comment: 41 pages; Final journal version; new subsection on LLV algebra for symplectic orbifolds 

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4. Cosets of Free Field Algebras via Arc Spaces

   https://doi.org/10.1093/imrn/rnac367  

   Linshaw, Andrew_R; Song, Bailin (January 2023, International Mathematics Research Notices)

   Abstract Using the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra $${{\mathcal {V}}}$$, we have a surjective homomorphism of differential algebras $$\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$$; equivalently, the singular support of $${{\mathcal {V}}}$$ is a closed subscheme of the arc space of the associated scheme $$X_{{{\mathcal {V}}}}$$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $$L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$$ for all positive integers $$n$$ and $$k$$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular $${{\mathcal {W}}}$$-algebra of $${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$$ at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras. 

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5. Polynomial Representations of the Witt Lie Algebra

   https://doi.org/10.1093/imrn/rnae139  

   Sam, Steven_V; Snowden, Andrew; Tosteson, Philip (June 2024, International Mathematics Research Notices)

   Abstract 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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