arXiv subjects

Joel Kamnitzer

Publications and source records attributed to Joel Kamnitzer.

At least 19 recordsLinked to original sources

arXiv2026

Category $\mathcal{O}$ for truncated shifted Yangians and the bi-infinite Bott-Samelson variety

In this paper, we study the category $\mathcal{O}$ of representations of shifted Yangians associated to a simply-laced simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. In particular, we prove that the (complexified) Grothendieck ring of this category is isomorphic to the Cox ring of the open bi-infinite Bott-Samelson variety, which is a pro-variety we construct from Bott-Samelson varieties for alternating heaps. Using work of Francone-Leclerc, we prove a conjecture of Hernandez-Zhang by identifying the above Grothendieck ring with a cluster algebra defined by Geiss-Hernandez-Leclerc. Our methods also yield an action of the Langlands dual group $G^{\vee}$ on this Grothendieck ring, and show that the shifted coproducts defined in work of the first and fifth authors with collaborators give rise to coproducts for truncated shifted Yangians. This machinery then allows us to prove further conjectures of Frenkel-Hernandez and Geiss-Hernandez-Leclerc on extended $QQ$-systems, and to obtain a generalization of a duality defined by Hernandez-Leclerc.

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arXiv2025

Cactus flower spaces and monodromy of Bethe vectors

We continue the study of cactus flower moduli spaces $\overline{F}_n$ and Gaudin models started in arXiv:2308.06880, arXiv:2407.06424. We show that isomorphism classes of operadic coverings of the real form $\overline{F}_n(\mathbb{R})$ are naturally one-to-one with equivalence classes of concrete coboundary monoidal categories (i.e. coboundary monoidal categories that admit a faithful monoidal functor to sets) with certain semisimplicity and finiteness conditions. Following the strategy of arXiv:1708.05105, for any complex semisimple Lie algebra $\mathfrak{g}$, we recover Kashiwara $\mathfrak{g}$-crystals, as a concrete coboundary category, from the coverings given by Bethe eigenlines for inhomogeneous Gaudin models. Using this, we compute the monodromy of Bethe eigenlines for trigonometric Gaudin models over two different real loci. In the particular case of minuscule highest weights, this can be regarded as combinatorial version of the wall-crossing conjecture of Bezrukavnikov and Okounkov for quantum cohomology of symplectic resolutions in the case of minuscule resolutions of slices in the affine Grassmannian.

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arXiv2024

Fixed points of reverse Hessenberg convolution varieties

Associated to any unit interval graph, Syu Kato introduced a variety which gives (via the geometric Satake correspondence) a graded $GL_m$ representation whose character is the chromatic quasisymmetric polynomial of the graph. In this short note, we reprove Kato's result by analyzing the fixed points of his varieties.

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arXiv2024

Gaudin models and moduli space of flower curves

We introduce and study the family of trigonometric Gaudin subalgebras in $U g^{\otimes n}$ for arbitrary simple Lie algebra $g$. This is the family of commutative subalgebras of maximal possible transcendence degree that serve as a universal source for higher integrals of the trigonometric Gaudin quantum spin chain attached to $g$. We study the parameter space that indexes all possible degenerations of subalgebras from this family. In particular, we show that (rational) inhomogeneous Gaudin subalgebras of $ U g^{\otimes n}$ previously studied in \cite{ffry} arise as certain limits of trigonometric Gaudin subalgebras. Moreover, we show that both families of commutative subalgebras glue together into the one parameterized by the space $\overline{\mathcal F}_n$, which is the total space of degeneration of the Deligne-Mumford space of stable rational curves $ \overline M_{n+2} $ to the moduli space of cactus flower curves $ \overline F_n $ recently introduced in \cite{iklpr}. As an application, we show that trigonometric Gaudin subalgebras act on tensor products of irreducible finite-dimensional $g$-modules without multiplicities, under some explicit assumptions on the parameters in terms of two different real forms of $\overline M_{n+2}$. This gives rise to a monodromy action of the affine cactus group on the set of eigenstates for the trigonometric Gaudin model. We also explain the relation between the trigonometric Gaudin model and the quantum cohomology of affine Grassmannians slices.

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arXiv2024

Heaps, crystals, and preprojective algebra modules

Fix a simply-laced semisimple Lie algebra. We study the crystal $ B(nλ)$, were $λ$ is a dominant minuscule weight and $n$ is a natural number. On one hand, $B(nλ)$ can be realized combinatorially by height $n$ reverse plane partitions on a heap associated to $λ$. On the other hand, we use this heap to define a module over the preprojective algebra of the underlying Dynkin quiver. Using the work of Saito and Savage-Tingley, we realize $B(nλ)$ via irreducible components of the quiver Grassmannian of $n$ copies of this module. In this paper, we describe an explicit bijection between these two models for $B(nλ)$ and prove that our bijection yields an isomorphism of crystals. Our main geometric tool is Nakajima's tensor product quiver varieties.

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arXiv2024

The moduli space of cactus flower curves and the virtual cactus group

The space $ \ft_n = \C^n/\C $ of $n$ points on the line modulo translation has a natural compactification $ \overline \ft_n $ as a matroid Schubert variety. In this space, pairwise distances between points can be infinite; it is natural to imagine points at infinite distance from each other as living on different projective lines. We call such a configuration of points a ``flower curve'', since we picture the projective lines joined into a flower. Within $ \ft_n $, we have the space $ F_n = \C^n \setminus Δ/ \C $ of $ n$ distinct points. We introduce a natural compatification $ \overline F_n $ along with a map $ \overline F_n \rightarrow \overline \ft_n $, whose fibres are products of genus 0 Deligne-Mumford spaces. We show that both $\overline \ft_n$ and $\overline F_n$, are special fibers of $1$-parameter families whose generic fibers are, respectively, Losev-Manin and Deligne-Mumford moduli spaces of stable genus $0$ curves with $n+2$ marked points. We find combinatorial models for the real loci $ \overline \ft_n(\BR) $ and $ \overline F_n(\BR) $. Using these models, we prove that these spaces are aspherical and that their equivariant fundamental groups are the virtual symmetric group and the virtual cactus groups, respectively. The degeneration of a twisted real form of the Deligne-Mumford space to $\overline F_n(\mathbb{R})$ gives rise to a natural homomorphism from the affine cactus group to the virtual cactus group.

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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

BFN Springer Theory

Given a representation N of a reductive group G, Braverman-Finkelberg-Nakajima have defined a remarkable Poisson variety called the Coulomb branch. Their construction of this space was motivated by considerations from 3d gauge theories and symplectic duality. The coordinate ring of this Coulomb branch is defined as a convolution algebra, using a vector bundle over the affine Grassmannian of G. This vector bundle over the affine Grassmannian maps to the space of loops in the representation N. We study the fibres of this maps, which live in the affine Grassmannian. We use these BFN Springer fibres to construct modules for (quantized) Coulomb branch algebras. These modules naturally correspond to boundary conditions for the corresponding gauge theory. We use our construction to partially prove a conjecture of Baumann-Kamnitzer-Knutson and give evidence for conjectures of Hikita, Nakajima, and Kamnitzer-McBreen-Proudfoot. We also prove a relation between BFN Springer fibres and quasimap spaces.

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arXiv2022

Perfect bases in representation theory: three mountains and their springs

In order to give a combinatorial descriptions of tensor product multiplicites for semisimple groups, it is useful to find bases for representations which are compatible with the actions of Chevalley generators of the Lie algebra. There are three known examples of such bases, each of which flows from geometric or algebraic mountain. Remarkably, each mountain gives the same combinatorial shadow: the crystal B(infty) and the Mirkovic-Vilonen polytopes. In order to distinguish between the three bases, we introduce measures supported on these polytopes. We also report on the interaction of these bases with the cluster structure on the coordinate ring of the maximal unipotent subgroup.

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arXiv2022

Symplectic resolutions, symplectic duality, and Coulomb branches

Symplectic resolutions are an exciting new frontier of research in representation theory. One of the most fascinating aspects of this study is symplectic duality: the observation that these resolutions come in pairs with matching properties. The Coulomb branch construction allows us to produce and study many of these dual pairs. These notes survey much recent work in this area including quantization, categorification, and enumerative geometry. We particularly focus on ADE quiver varieties and affine Grassmannian slices.

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arXiv2022

Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians

Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper, we prove that neighbouring generalized affine Grassmannian slices are related by Hamiltonian reduction by the action of the additive group. We also prove a weaker version of the same result for their quantizations, algebras known as truncated shifted Yangians.

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arXiv2021

Computing fusion products of MV cycles using the Mirkovic-Vybornov isomorphism

The fusion of two Mirkovic-Vilonen cycles is a degeneration of their product, defined using the Beilinson-Drinfeld Grassmannian. In this paper, we put in place a conceptually elementary approach to computing this product in type $A$. We do so by transferring the problem to a fusion of generalized orbital varieties using the Mirkovic-Vybornov isomorphism. As an application, we explicitly compute all cluster exchange relations in the coordinate ring of the upper-triangular subgroup of $GL_4$, confirming that all the cluster variables are contained in the Mirkovic-Vilonen basis.

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arXiv2021

Associated graded of Hodge modules and categorical sl_2 actions

One of the most mysterious aspects of Saito's theory of Hodge modules are the Hodge and weight filtrations that accompany the pushforward of a Hodge module under an open embedding. In this paper we consider the open embedding in a product of complementary Grassmannians given by pairs of transverse subspaces. The push-forward of the structure sheaf under this open embedding is an important Hodge module from the viewpoint of geometric representation theory and homological knot invariants. We compute the associated graded of this push-forward with respect to the induced Hodge filtration as well as the resulting weight filtration. The main tool is a categorical $\sl_2$ action on the category of $\D_h$-modules on Grassmannians. Along the way we also clarify the interaction of kernels for $\D_h$-modules with the associated graded functor. Both of these results may be of independent interest.

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arXiv2020

The Mirkovic-Vilonen basis and Duistermaat-Heckman measures

Using the geometric Satake correspondence, the Mirkovic-Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group G . We prove that these bases are "perfect", i.e. compatible with the action of the Chevelley generators of the positive half of the Lie algebra g. We compute this action in terms of intersection multiplicities in the affine Grassmannian. We prove that these bases stitch together to a basis for the algebra C[N] of regular functions on the unipotent subgroup. We compute the multiplication in this MV basis using intersection multiplicities in the Beilinson-Drinfeld Grassmannian, thus proving a conjecture of Anderson. In the third part of the paper, we define a map from C[N] to a convolution algebra of measures on the dual of the Cartan subalgebra of g. We characterize this map using the universal centralizer space of G. We prove that the measure associated to an MV basis element equals the Duistermaat-Heckman measure of the corresponding MV cycle. This leads to a proof of a conjecture of Muthiah. Finally, we use the map to measures to compare the MV basis and Lusztig's dual semicanonical basis. We formulate conjectures relating the algebraic invariants of preprojective algebra modules (which underlie the dual semicanonical basis) and geometric invariants of MV cycles. In the appendix, we use these ideas to prove that the MV basis and the dual semicanonical basis do not coincide in SL_6.

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arXiv2020

The quantum Hikita conjecture

The Hikita conjecture relates the coordinate ring of a conical symplectic singularity to the cohomology ring of a symplectic resolution of the dual conical symplectic singularity. We formulate a quantum version of this conjecture, which relates the quantized coordinate ring of the first variety to the quantum cohomology of a symplectic resolution of the dual variety. We prove this conjecture for hypertoric varieties and for the Springer resolution. Our paper includes an appendix, written by Ben Webster, which studies highest weights for quantizations of symplectic resolutions with isolated torus actions.

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arXiv2020

Crystals and monodromy of Bethe vectors

Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for g-representations. These algebras depend on a parameter which is a point in the Deligne-Mumford moduli space of marked stable genus 0 curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the tensor product multiplicity space and gives us a basis of eigenvectors. In this paper, we study the monodromy of these eigenvectors as the parameter varies within the real locus; this gives an action of the fundamental group of this moduli space, which is called the cactus group. We prove a conjecture of Etingof which states that the monodromy of eigenvectors for Gaudin algebras agrees with the action of the cactus group on tensor products of g-crystals. In fact, we prove that the coboundary category of normal g-crystals can be reconstructed using the coverings of the moduli spaces. Our main tool is the construction of a crystal structure on the set of eigenvectors for shift of argument algebras, another family of commutative algebras which act on any irreducible g-representation. We also prove that the monodromy of such eigenvectors is given by the internal cactus group action on g-crystals.

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arXiv2019

On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products

Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights $λ$ and $μ$ for a Lie algebra $\mathfrak{g}$, which we will assume is simply-laced. In this paper, we relate the category $\mathcal{O}$ over truncated shifted Yangians to categorified tensor products: for a generic integral choice of parameters, category $\mathcal{O}$ is equivalent to a weight space in the categorification of a tensor product of fundamental representations defined by the third author using KLRW algebras. We also give a precise description of category $\mathcal{O}$ for arbitrary parameters using a new algebra which we call the parity KLRW algebra. In particular, we confirm the conjecture of the authors that the highest weights of category $\mathcal{O}$ are in canonical bijection with a product monomial crystal depending on the choice of parameters. This work also has interesting applications to classical representation theory. In particular, it allows us to give a classification of simple Gelfand-Tsetlin modules of $U(\mathfrak{gl}_n)$ and its associated W-algebras.

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arXiv2019

Highest weights for truncated shifted Yangians and product monomial crystals

Truncated shifted Yangians are a family of algebras which are natural quantizations of slices in the affine Grassmannian. We study the highest weight representations of these algebras. In particular, we conjecture that the possible highest weights for these algebras are described by product monomial crystals, certain natural subcrystals of Nakajima's monomials. We prove this conjecture in type A. We also place our results in the context of symplectic duality and prove a conjecture of Hikita in this situation.

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