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Linhui Shen

Publications and source records attributed to Linhui Shen.

16 recordsLinked to original sources
arXiv2025

Comparing cluster algebras on braid varieties

Braid varieties parametrize linear configurations of flags with transversality conditions dictated by positive braids. They include and generalize reduced double Bruhat cells, positroid varieties, open Bott-Samelson varieties, and Richardson varieties, among others. Recently, two cluster algebra structures were independently constructed in the coordinate rings of braid varieties: one using weaves and the other using Deodhar geometry. The main result of the article is that these two cluster algebras coincide. More generally, our comparative study matches the different concepts and results from each approach to the other, both on the combinatorial and algebraic geometric aspects.

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arXiv2023

Intersections of Dual $SL_3$-Webs

We introduce a topological intersection number for an ordered pair of $\operatorname{SL}_3$-webs on a decorated surface. Using this intersection pairing between reduced $(\operatorname{SL}_3,\mathcal{A})$-webs and a collection of $(\operatorname{SL}_3,\mathcal{X})$-webs associated with the Fock--Goncharov cluster coordinates, we provide a natural combinatorial interpretation of the bijection from the set of reduced $(\operatorname{SL}_3,\mathcal{A})$-webs to the tropical set $\mathcal{A}^+_{\operatorname{PGL}_3,\hat{S}}(\mathbb{Z}^t)$, as established by Douglas and Sun in \cite{DS20a, DS20b}. We provide a new proof of the flip equivariance of the above bijection, which is crucial for proving the Fock--Goncharov duality conjecture of higher Teichm\"uller spaces for $\operatorname{SL}_3$.

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arXiv2023

The Chromatic Lagrangian: Wavefunctions and Open Gromov-Witten Conjectures

Inside a symplectic leaf of the cluster Poisson variety of Borel-decorated $PGL_2$ local systems on a punctured surface is an isotropic subvariety we will call the chromatic Lagrangian. Local charts for the quantized cluster variety are quantum tori defined by cubic planar graphs, and can be put in standard form after some additional markings giving the notion of a framed seed. The mutation structure is encoded as a groupoid. The local description of the chromatic Lagrangian defines a wavefunction which, we conjecture, encodes open Gromov-Witten invariants of a Lagrangian threefold in threespace defined by the cubic graph and the other data of the framed seed. We also find a relationship we call framing duality: for a family of "canoe" graphs, wavefunctions for different framings encode DT invariants of symmetric quivers.

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arXiv2022

Cluster Nature of Quantum Groups

We present a rigid cluster model to realize the quantum group ${\bf U}_q(\mathfrak{g})$ for $\mathfrak{g}$ of type ADE. That is, we prove that there is a natural Hopf algebra isomorphism from the quantum group ${\bf U}_q(\mathfrak{g})$ to a quotient algebra of the Weyl group invariants of the Fock-Goncharov quantum cluster algebra $\mathcal{O}_q(\mathscr{P}_{{\rm G},\odot})$. By applying the quantum duality of cluster algebras, we show that ${\bf U}_q(\mathfrak{g})$ admits a natural basis $\bar{\bf \Theta}$ whose structural coefficients are in $\mathbb{N}[q^{\frac{1}{2}}, q^{-\frac{1}{2}}]$. The basis $\bar{\bf \Theta}$ satisfies an invariance property under Lusztig's braid group action, the Dynkin automorphisms, and the star anti-involution.

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arXiv2022

Cluster structures on braid varieties

We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig's coordinates. Several explicit seeds are provided and the quiver and cluster variables are readily computable. We prove that these upper cluster algebras equal their cluster algebras, show local acyclicity, and explicitly determine their DT-transformations as the twist automorphisms of braid varieties. The main result also resolves the conjecture of B. Leclerc on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties.

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arXiv2022

$\mathscr{A}=\mathscr{U}$ for cluster algebras from moduli spaces of $G$-local systems

For a finite-dimensional simple Lie algebra $\mathfrak{g}$ admitting a non-trivial minuscule representation and a connected marked surface $\Sigma$ with at least two marked points and no punctures, we prove that the cluster algebra $\mathscr{A}_{\mathfrak{g},\Sigma}$ associated with the pair $(\mathfrak{g},\Sigma)$ coincides with the upper cluster algebra $\mathscr{U}_{\mathfrak{g},\Sigma}$. The proof is based on the fact that the function ring $\mathcal{O}(\mathcal{A}^\times_{G,\Sigma})$ of the moduli space of decorated twisted $G$-local systems on $\Sigma$ is generated by matrix coefficients of Wilson lines introduced in [IO20]. As an application, we prove that the Muller-type skein algebras $\mathscr{S}_{\mathfrak{g}, \Sigma}[\partial^{-1}]$ [Muller,IY23,IY22] for $\mathfrak{g}=\mathfrak{sl}_2, \mathfrak{sl}_3,$ or $\mathfrak{sp}_4$ are isomorphic to the cluster algebras $\mathscr{A}_{\mathfrak{g}, \Sigma}$.

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arXiv2020

Augmentations, Fillings, and Clusters

We investigate positive braid Legendrian links via a Floer-theoretic approach and prove that their augmentation varieties are cluster K2 (aka. A-) varieties. Using the exact Lagrangian cobordisms of Legendrian links in [EHK16], we prove that a large family of exact Lagrangian fillings of positive braid Legendrian links correspond to cluster seeds of their augmentation varieties. We solve the infinite-filling problem for positive braid Legendrian links; i.e., whenever a positive braid Legendrian link is not of type ADE, it admits infinitely many exact Lagrangian fillings up to Hamiltonian isotopy.

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arXiv2020

Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems

We study the dual ${\rm G}^\ast$ of a standard semisimple Poisson-Lie group ${\rm G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}({\rm G}^\ast)$ can be naturally embedded into a cluster Poisson algebra with a Weyl group action. We prove that $\mathcal{O}({\rm G}^\ast)$ admits a natural basis which has positive integer structure coefficients and satisfies an invariance property with respect to a braid group action. We continue the study of the moduli space $\mathscr{P}_{{\rm G},\mathbb{S}}$ of ${\rm G}$-local systems introduced in \cite{GS3}, and prove that the coordinate ring of $\mathscr{P}_{{\rm G}, \mathbb{S}}$ coincides with its underlying cluster Poisson algebra.

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arXiv2019

Quantum geometry of moduli spaces of local systems and representation theory

Let G be a split semi-simple adjoint group, and S a colored decorated surface, given by an oriented surface with punctures, special boundary points, and a specified collection of boundary intervals. We introduce a moduli space P(G,S) parametrizing G-local system on S with some boundary data, and prove that it carries a cluster Poisson structure, equivariant under the action of the cluster modular group M(G,S), containing the mapping class group of S, the group of outer automorphisms of G, and the product of Weyl / braid groups over punctures / boundary components. We prove that the dual moduli space A(G,S) carries a M(G,S)-equivariant cluster structure, and the pair (A(G,S), P(G,S)) is a cluster ensemble. These results generalize the works of V. Fock & the first author, and of I. Le. We quantize cluster Poisson varieties X for any Planck constant h s.t. h>0 or |h|=1. First, we define a *-algebra structure on the Langlands modular double A(h; X) of the algebra of functions on X. We construct a principal series of representations of the *-algebra A(h; X), equivariant under a unitary projective representation of the cluster modular group M(X). This extends works of V. Fock and the first author when h>0. Combining this, we get a M(G,S)-equivariant quantization of the moduli space P(G,S), given by the *-algebra A(h; P(G,S)) and its principal series representations. We construct realizations of the principal series *-representations. In particular, when S is punctured disc with two special points, we get a principal series *-representations of the Langlands modular double of the quantum group Uq(g). We conjecture that there is a nondegenerate pairing between the local system of coinvariants of oscillatory representations of the W-algebra and the one provided by the projective representation of the mapping class group of S.

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arXiv2019

Cluster Structures on Double Bott-Samelson Cells

Let $C$ be a symmetrizable generalized Cartan matrix. We introduce four different versions of double Bott-Samelson cells for every pair of positive braids in the generalized braid group associated to $C$. We prove that the decorated double Bott-Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras. We explicitly describe the Donaldson-Thomas transformations on double Bott-Samelson cells and prove that they are cluster transformations. As an application, we complete the proof of the Fock-Goncharov duality conjecture in these cases. We discover a periodicity phenomenon of the Donaldson-Thomas transformations on a family of double Bott-Samelson cells. We give a (rather simple) geometric proof of Zamolodchikov's periodicity conjecture in the cases of $\Delta\square \mathrm{A}_r$. When $C$ is of type $\mathrm{A}$, the double Bott-Samelson cells are isomorphic to Shende-Treumann-Zaslow's moduli spaces of microlocal rank-1 constructible sheaves associated to Legendrian links. By counting their $\mathbb{F}_q$-points we obtain rational functions which are Legendrian link invariants.

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arXiv2018

Cyclic Sieving and Cluster Duality of Grassmannian

We introduce a decorated configuration space $\mathscr{C}\!{\rm onf}_n^\times(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $\big(\mathscr{C}\!{\rm onf}_n^\times(a), \mathcal{W}\big)$ canonically parametrizes a linear basis of the homogeneous coordinate ring of the Grassmannian $\operatorname{Gr}_a(n)$ with respect to the Pl\"ucker embedding. We prove that $\big(\mathscr{C}\!{\rm onf}_n^\times(a), \mathcal{W}\big)$ is equivalent to the mirror Landau-Ginzburg model of the Grassmannian considered by Eguchi-Hori-Xiong, Marsh-Rietsch and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.

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arXiv2016

Donaldson-Thomas trasnsformations of moduli spaces of G-local systems

Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category equipped with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in a single formal automorphism of the cluster variety, called the DT-transformation. Let S be an oriented surface with punctures, and a finite number of special points on the boundary considered modulo isotopy. It give rise to a moduli space X(m, S), closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. For each puncture of S, there is a birational Weyl group action on the space X(m, S). We prove that it is given by cluster Poisson transformations. We prove a similar result for the involution * of the space X(m,S) provided by dualising a local system on S. We calculate the DT-transformation of the moduli space X(m,S), with few exceptions. Namely, let C(m,S) be the transformation of the space X(m,S) given by the product of three commuting maps: the involution *, the product, over all punctures of S, of the longest element of the Weyl group action corresponding to the puncture, and the "shift of the special points on the boundary by one" map. Using a characterisation of a class of DT-transformations due to Keller, we prove that C(m,S) = DT. We prove that, burring few exceptions, the Weyl group and the involution * act by cluster transformations of the dual moduli space A(m, S). So the formula C(m,S) = DT is valid for the space A(m,S). Our main result, combined with the work of Gross, Hacking, Keel and Kontsevich, deliver a canonical basis in the space of regular functions on the cluster variety X(m,S), and in the upper cluster algebra with principal coefficients related to the pair (SL(m), S), with few exceptions.

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arXiv2014

Tensor invariants, Saturation problems, and Dynkin automorphisms

Let G be a connected almost simple algebraic group with a Dynkin automorphism {\sigma}. Let G_{\sigma} be the connected almost simple algebraic group associated to G and {\sigma}. We prove that the dimension of the tensor invariant space of G_{\sigma} is equal to the trace of {\sigma} on the corresponding tensor invariant space of G. We prove that if G has the saturation property then so does G{\sigma}. As a consequence, we show that the spin group Spin(2n + 1) is of saturation property with factor 2, which strengthens the results of Belkale-Kumar and Sam in the case of type B_n.

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arXiv2013

Geometry of canonical bases and mirror symmetry

A decorated surface S is a surface with a finite set of special points on the boundary, considered modulo isotopy. Let G be a split reductive group. A pair (G, S) gives rise to a moduli space A(G, S), closely related to the space of G-local systems on S. It has a positive structure. So the set of its integral tropical points is defined. We introduce a rational positive function W on A(G, S), the potential. The condition that its tropicalisation is non-negative determines its subset. For SL(2), we recover the set of integral laminations on S. We prove that when S is a disc with n special points on the boundary, this set parametrises top dimensional components of the convolution varieties. Thus, via geometric Satake correspondence, they provide a canonical basis in tensor product invariants of irreducible modules for the Langlands dual group. When G=GL(m), n=3, there is a special coordinate system on A(G,S). We show that it identifies our set with the set of with Knutson-Tao's hives. Our result generalises a theorem of Kamnitzer, who used hives to parametrise top components of convolution varieties for GL(m), n=3. For n>3, we prove Kamnitzer's conjecture. We define canonical bases in tensor products, generalizing the Mirkovic-Vilonen basis in a single representation. We prove that for any S, the set of positive integral tropical points of A(G, S) parametrise top components in a new space, surface affine Grasmannian. We view W as a potential for Landau-Ginzburg model on A(G,S). We conjecture that the pair (A(G,S), W) is the mirror dual to the moduli space of local systems on S for the Langlands dual group. In a special case, we recover Givental's description of the quantum cohomology connection for flag varieties.

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