arXiv subjects

Ben Webster

Publications and source records attributed to Ben Webster.

At least 37 records · Page 2Linked to original sources

arXiv2024

Tilting Generator for the $T^*Gr(2,4)$ Coulomb Branch

Remarkable work of Kaledin, based on earlier joint work with Bezrukavnikov, has constructed a tilting generator of the category of coherent sheaves on a very general class of symplectic resolutions of singularities. In this paper, we give a concrete construction of this tilting generator on the cotangent bundle of $Gr(2,4)$, the Grassmannian of 2-planes in $\mathbb{C}^4$. This construction builds on work of the second author describing these tilting bundles in terms of KLRW algebras, but in this low-dimensional case, we are able to describe our tilting generator as a sum of geometrically natural bundles on $T^*Gr(2,4)$: line bundles and their extensions, as well as the tautological bundle and its perpendicular.

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arXiv2024

Gelfand-Tsetlin modules in the Coulomb context

This paper gives a new perspective on the theory of principal Galois orders, as developed by Futorny, Ovsienko, Hartwig, and others. Every principal Galois order can be written as $eFe$ for any idempotent $e$ in an algebra $F$, which we call a flag Galois order; and in most important cases we can assume that these algebras are Morita equivalent. These algebras have the property that the completed algebra controlling the fiber over a maximal ideal has the same form as a subalgebra in a skew group ring, which gives a new perspective to a number of results about these algebras. We also discuss how this approach relates to the study of Coulomb branches in the sense of Braverman-Finkelberg-Nakajima, which are particularly beautiful examples of principal Galois orders. These include most of the interesting examples of principal Galois orders, such as $U(\mathfrak{gl}_n)$. In this case, all the objects discussed have a geometric interpretation, which endows the category of Gelfand-Tsetlin modules with a graded lift and allows us to interpret the classes of simple Gelfand-Tsetlin modules in terms of dual canonical bases for the Grothendieck group. In particular, we classify the Gelfand-Tsetlin modules over $U(\mathfrak{gl}_n)$ and relate their characters to a generalization of Leclerc's shuffle expansion for dual canonical basis vectors. Finally, as an application, we disprove a conjecture of Mazorchuk, showing that the fiber over a maximal ideal of the Gelfand-Tsetlin subalgebra appearing in a finite-dimensional representation has an infinite-dimensional module in its fiber for $n\geq 6$.

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arXiv2024

Homological mirror symmetry for hypertoric varieties II (with an Appendix written jointly with Laurent Côté and Justin Hilburn)

In this paper, we prove a homological mirror symmetry equivalence for pairs of multiplicative hypertoric varieties, and we calculate monodromy autoequivalences of these categories by promoting our result to an equivalence of perverse schobers. We prove our equivalence by matching holomorphic Lagrangian skeleta, on the A-model side, with non-commutative resolutions on the B-model side. The hyperkähler geometry of these spaces provides each category with a natural t-structure, which helps clarify SYZ duality in a hyperkähler context. Our results are a prototype for mirror symmetry statements relating pairs of K-theoretic Coulomb branches.

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arXiv2024

Gelfand-Tsetlin modules: canonicity and calculations

In this paper, we give a more down-to-earth introduction to the connection between Gelfand-Tsetlin modules over $\mathfrak{gl}_n$ and diagrammatic KLRW algebras, and develop some of its consequences. In addition to a new proof of this description of the category Gelfand-Tsetlin modules appearing in earlier work, we show three new results of independent interest: (1) we show that every simple Gelfand-Tsetlin module is a canonical module in the sense of Early, Mazorchuk and Vishnyakova, and characterize when two maximal ideals have isomorphic canonical modules, (2) we show that the dimensions of Gelfand-Tsetlin weight spaces in simple modules can be computed using an appropriate modification of Leclerc's algorithm for computing dual canonical bases, and (3) we construct a basis of the Verma modules of $\mathfrak{sl}_n$ which consists of generalized eigenvectors for the Gelfand-Tsetlin subalgebra. Furthermore, we present computations of multiplicities and Gelfand-Kirillov dimensions for all integral Gelfand-Tsetlin modules in ranks 3 and 4; unfortunately, for ranks $>4$, our computers are not adequate to perform these computations.

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arXiv2024

Stabilization phenomena in Kac-Moody algebras and quiver varieties

Let X be the Dynkin diagram of a symmetrizable Kac-Moody algebra, and X_0 a subgraph with all vertices of degree 1 or 2. Using the crystal structure on the components of quiver varieties for X, we show that if we expand X by extending X_0, the branching multiplicities and tensor product multiplicities stabilize, provided the weights involved satisfy a condition which we call ``depth'' and are supported outside $X_0$. This extends a theorem of Kleber and Viswanath. Furthermore, we show that the weight multiplicities of such representations are polynomial in the length of X_0, generalizing the same result for A_\ell by Benkart, et al.

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arXiv2023

RoCK blocks for affine categorical representations

Given a categorical action of a Lie algebra, a celebrated theorem of Chuang and Rouquier proves that the blocks corresponding to weight spaces in the same orbit of the Weyl group are derived equivalent, proving an even more celebrated conjecture of Broué for the case of the symmetric group. In many cases, these derived equivalences are $t$-exact, and thus induce equivalences of abelian categories between different blocks. We call two such blocks ``Scopes equivalent.'' In this paper, we describe how Scopes equivalence classes for any affine categorification can be classified by the chambers of a finite hyperplane arrangement, which can be found through simple Lie theoretic calculations. We pay special attention to the largest equivalence classes, which we call RoCK, and show how this matches with recent work of Lyle on Rouquier blocks for Ariki-Koike algebras. We also provide Sage code that tests whether blocks are RoCK and finds RoCK blocks for Ariki-Koike algebras.

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arXiv2023

On the representation theory of Schur algebras in type $B$

We study the representation theory of the type B Schur algebra $\mathcal{L}^n(m)$ with unequal parameters introduced in work of Lai, Nakano and Xiang. For generic values of $q,Q$, this algebra is semi-simple and Morita equivalent to the Hecke algebra, but for special values, its category of modules is more complicated. We study this representation theory by comparison with the cyclotomic $q$-Schur algebra of Dipper, James and Mathas, and use this to construct a cellular algebra structure on $\mathcal{L}^n(m)$. This allows us to index the simple $\mathcal{L}^n(m)$-modules as a subset of the set of bipartitions of $n$. For $m$ large, this will be all bipartitions of $n$ if and only if $\mathcal{L}^n(m)$ is quasi-hereditary, in which case, $\mathcal{L}^n(m)$ is Morita equivalent to the cyclotomic $q$-Schur algebra. We prove a modified version of a conjecture of Lai, Nakano and Xiang giving the values of $(q,Q)$ where this holds: if $m$ is large and odd, $Q\neq -q^k$ for all $k$ satisfying $\frac{4-n}{2}\leq k<n$; if $m$ is large and even, $Q\neq -q^k$ for all $k$ satisfying $-n<k<n.$ We also prove two strengthenings of this result: an indexing of the simple modules when $q$ is not a root of unity, and a characterization of the quasi-hereditary blocks of $\mathcal{L}^n(m)$.

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arXiv2023

Lie algebra actions on module categories for truncated shifted Yangians

We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's induction and restriction functors for Cherednik algebras, but their definition uses different tools. After this general definition, we focus on quiver gauge theories attached to a quiver $Γ$. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak{g}_Γ$ on category $ \mathcal O $ for these Coulomb branch algebras. When $ Γ$ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence. To establish this categorical action, we define a new class of "flavoured" KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.

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arXiv2023

Foundations of Frobenius Heisenberg categories

We describe bases for the morphism spaces of the Frobenius Heisenberg categories associated to a symmetric graded Frobenius algebra, proving several open conjectures. Our proof uses a categorical comultiplication and generalized cyclotomic quotients of the category. We use our basis theorem to prove that the Grothendieck ring of the Karoubi envelope of the Frobenius Heisenberg category recovers the lattice Heisenberg algebra associated to the Frobenius algebra.

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arXiv2023

On the definition of quantum Heisenberg category

We introduce a diagrammatic monoidal category $\mathcal{H}eis_k(z,t)$ which we call the quantum Heisenberg category, here, $k \in \mathbb{Z}$ is "central charge" and $z$ and $t$ are invertible parameters. Special cases were known before: for central charge $k=-1$ and parameters $z = q-q^{-1}$ and $t = -z^{-1}$ our quantum Heisenberg category may be obtained from the deformed version of Khovanov's Heisenberg category introduced by Licata and the second author by inverting its polynomial generator, while $\mathcal{H}eis_0(z,t)$ is the affinization of the HOMFLY-PT skein category. We also prove a basis theorem for the morphism spaces in $\mathcal{H}eis_k(z,t)$.

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arXiv2023

3-dimensional mirror symmetry

This expository article discusses recent advances in understanding 3-dimensional mirror symmetry and the mathematical definitions of the Higgs and Coulomb branches. This is a slightly expanded version of an article appearing in the Notices of the AMS.

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arXiv2023

Current algebras and categorified quantum groups

We identify the trace, or 0th Hochschild homology, of type ADE categorified quantum groups with the corresponding current algebra of the same type. To prove this, we show that 2-representations defined using categories of modules over cyclotomic (or deformed cyclotomic) quotients of KLR-algebras correspond to local (or global) Weyl modules. We also investigate the implications for centers of categories in 2-representations of categorified quantum groups.

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arXiv2022

Quantizations of conical symplectic resolutions I: local and global structure

We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a version of the Beilinson-Bernstein localization theorem, a theory of Harish-Chandra bimodules and their relationship to convolution operators on cohomology, and a discrete group action on the derived category of representations, generalizing the braid group action on category O via twisting functors. Our primary goal is to apply these results to other quantized symplectic resolutions, including quiver varieties and hypertoric varieties. This provides a new context for known results about Lie algebras, Cherednik algebras, finite W-algebras, and hypertoric enveloping algebras, while also pointing to the study of new algebras arising from more general resolutions.

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arXiv2022

Quantizations of conical symplectic resolutions II: category $\mathcal O$ and symplectic duality

We define and study category $\mathcal O$ for a symplectic resolution, generalizing the classical BGG category $\mathcal O$, which is associated with the Springer resolution. This includes the development of intrinsic properties parallelling the BGG case, such as a highest weight structure and analogues of twisting and shuffling functors, along with an extensive discussion of individual examples. We observe that category $\mathcal O$ is often Koszul, and its Koszul dual is often equivalent to category $\mathcal O$ for a different symplectic resolution. This leads us to define the notion of a symplectic duality between symplectic resolutions, which is a collection of isomorphisms between representation theoretic and geometric structures, including a Koszul duality between the two categories. This duality has various cohomological consequences, including (conjecturally) an identification of two geometric realizations, due to Nakajima and Ginzburg/Mirković-Vilonen, of weight spaces of simple representations of simply-laced simple algebraic groups. An appendix by Ivan Losev establishes a key step in the proof that $\mathcal O$ is highest weight.

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arXiv2022

Hypertoric category O

We study the representation theory of the invariant subalgebra of the Weyl algebra under a torus action, which we call a "hypertoric enveloping algebra." We define an analogue of BGG category O for this algebra, and identify it with a certain category of sheaves on a hypertoric variety. We prove that a regular block of this category is highest weight and Koszul, identify its Koszul dual, compute its center, and study its cell structure. We also consider a collection of derived auto-equivalences analogous to the shuffling and twisting functors for BGG category O.

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arXiv2021

The degenerate Heisenberg category and its Grothendieck ring

The degenerate Heisenberg category $\mathcal{H}eis_k$ is a strict monoidal category which was originally introduced in the special case $k=-1$ by Khovanov in 2010. Khovanov conjectured that the Grothendieck ring of the additive Karoubi envelope of his category is isomorphic to a certain $\mathbb{Z}$-form for the universal enveloping algebra of the infinite-dimensional Heisenberg Lie algebra specialized at central charge $-1$. We prove this conjecture and extend it to arbitrary central charge $k \in \mathbb{Z}$. We also explain how to categorify the comultiplication (generically).

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arXiv2021

Intersection cohomology of hypertoric varieties

A hypertoric variety is a quaternionic analogue of a toric variety. Just as the topology of toric varieties is closely related to the combinatorics of polytopes, the topology of hypertoric varieties interacts richly with the combinatorics of hyperplane arrangements and matroids. Using finite field methods, we obtain combinatorial descriptions of the Betti numbers of hypertoric varieties, both for ordinary cohomology in the smooth case and intersection cohomology in the singular case. We also introduce a conjectural ring structure on the intersection cohomology of a hypertoric variety.

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arXiv2020

Rational Cherednik algebras of $G(\ell,p,n)$ from the Coulomb perspective

We prove a number of results on the structure and representation theory of the rational Cherednik algebra of the imprimitive reflection group $G(\ell,p,n)$. In particular, we: (1) show a relationship to the Coulomb branch construction of Braverman, Finkelberg and Nakajima, and 3-dimensional quantum field theory; (2) show that the spherical Cherednik algebra carries the structure of a principal Galois order; (3) construct a graded lift of category $\mathcal{O}$ and the larger category of Dunkl-Opdam modules, whose simple modules have the properties of a dual canonical basis and (4) give the first classification of simple Dunkl-Opdam modules for the rational Cherednik algebra of the imprimitive reflection group $G(\ell,p,n)$.

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