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.
- NSF-PAR ID:
- 10325310
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
Abstract -
more » « less
Abstract Let $\textsf {X}$ and $\textsf {X}^{!}$ be a pair of symplectic varieties dual with respect to 3D mirror symmetry. The $K$-theoretic limit of the elliptic duality interface is an equivariant $K$-theory class $\mathfrak {m} \in K(\textsf {X}\times \textsf {X}^{!})$. We show that this class provides correspondences $$ \begin{align*} & \Phi_{\mathfrak{m}}: K(\textsf{X}) \leftrightarrows K(\textsf{X}^{!}) \end{align*}$$mapping the $K$-theoretic stable envelopes to the $K$-theoretic stable envelopes. This construction allows us to relate various representation theoretic objects of $K(\textsf {X})$, such as action of quantum groups, quantum dynamical Weyl groups, $R$-matrices, etc., to those for $K(\textsf {X}^{!})$. In particular, we relate the wall $R$-matrices of $\textsf {X}$ to the $R$-matrices of the dual variety $\textsf {X}^{!}$. As an example, we apply our results to $\textsf {X}=\textrm {Hilb}^{n}({{\mathbb {C}}}^2)$—the Hilbert scheme of $n$ points in the complex plane. In this case, we arrive at the conjectures of Gorsky and Negut from [10].
-
more » « less
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.
-
more » « less
Abstract We use [11] to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}_J\mathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand–Kirillov dimension $\dim (G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_J\mathcal{O}(G)_{J}\cong U({\mathfrak{g}})$ as algebras, where ${\mathfrak{g}}={\textrm{Lie}}(G)$. We also determine the finite dimensional irreducible representations of ${}_J\mathcal{O}(G)_{J}$, by analyzing twisted function algebras on $(H,H)$-double cosets of the support $H\subset G$ of $J$. Finally, we work out several examples to illustrate our results.
-
more » « less
Abstract We introduce and study the continuous Rokhlin property for actions of compact groups on $C^*$-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong $KK$-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the Universal Coefficient Theorem (UCT) is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. Our analysis of the $KK$-theory of the crossed product allows us to prove a ${{\mathbb {T}}}$-equivariant version of Kirchberg–Phillips: two circle actions with the continuous Rokhlin property on Kirchberg algebras are conjugate whenever they are $KK^{{{\mathbb {T}}}}$-equivalent. In the presence of the UCT, this is equivalent to having isomorphic equivariant $K$-theory. We moreover characterize the $KK^{{{\mathbb {T}}}}$-theoretical invariants that arise in this way. Finally, we identify a $KK^{{{\mathbb {T}}}}$-theoretic obstruction to the continuous property, which is shown to be the only obstruction in the setting of Kirchberg algebras. We show by means of explicit examples that the Rokhlin property is strictly weaker than the continuous Rokhlin property.
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnab352
