Standard monomials and invariant theory of arc spaces II: Symplectic group

Linshaw, Andrew; Song, Bailin

doi:10.1090/jag/834

J. Algebraic Geom.

2024

DOI: 10.1090/jag/834

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Standard monomials and invariant theory of arc spaces II: Symplectic group

Andrew Linshaw,

Bailin Song

Abstract: This is the second in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field K K , we construct a standard monomial basis for the arc space of the Pfaffian variety over K K . As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the symplectic group.

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2024

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Cited by 2 publications

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Classical freeness of orthosymplectic affine vertex superalgebras

Creutzig¹,

Linshaw²,

Song³

2024

Proc. Amer. Math. Soc.

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The question of when a vertex algebra is a quantization of the arc space of its associated scheme has recently received a lot of attention in both the mathematics and physics literature. This property was first studied by Tomoyuki Arakawa and Anne Moreau (see their paper in the references), and was given the name \lq\lq classical freeness" by Jethro van Ekeren and Reimundo Heluani [Comm. Math. Phys. 386 (2021), no. 1, pp. 495-550] in their work on chiral homology. Later, it was extended to vertex superalgebras by Hao Li [Eur. J. Math. 7 (2021), pp. 1689–1728]. In this note, we prove the classical freeness of the simple affine vertex superalgebra L n ( o s p m | 2 r ) L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{m|2r}) for all positive integers m , n , r m,n,r satisfying − m 2 + r + n + 1 > 0 -\frac {m}{2} + r +n+1 > 0 . In particular, it holds for the rational vertex superalgebras L n ( o s p 1 | 2 r ) L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{1|2r}) for all positive integers r , n r,n .

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Classical freeness of orthosymplectic affine vertex superalgebras

Creutzig¹,

Linshaw²,

Song³

2024

Proc. Amer. Math. Soc.

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Cosets of Free Field Algebras via Arc Spaces

Linshaw

Song

2023

International Mathematics Research Notices

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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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Standard monomials and invariant theory of arc spaces II: Symplectic group

Cited by 2 publications

References 12 publications

Classical freeness of orthosymplectic affine vertex superalgebras

Classical freeness of orthosymplectic affine vertex superalgebras

Cosets of Free Field Algebras via Arc Spaces

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