Schedule for: 22w5173 - Bases for Cluster Algebras

We will present the blue vs. red game where mutations at frozen vertices are counterbalanced by mutations at non frozen vertices. As an application, we will obtain group actions (e.g. braid group actions) on cluster categories and cluster algebras.

I will present a construction that ties together of several diverse notions including spaces of periodic difference operators, Poisson sub manifolds of a Drinfeld double of GL(n) and subsets of Grassmannians stable under the action of powers of a cyclic shift. The theory of generalized cluster algebras serves as a unifying theme. Time permitting, I will discuss potential applica- tions to representation theory of quantum affine algebras at roots of unity. Based on a joint work with M. Shapiro and A. Vainshtein and an ongoing project with C. Fraser and K. Trampel.

The quest to prove the positivity conjecture for cluster algebras resulted in many interesting proofs for special families. One such proof was given by Musiker, Schiffler, and Williams for cluster algebras from surfaces. The main idea of the proof is to realize each cluster variable as a generating function of statistics on a certain labeled graph, called a snake graph. We take this idea and use it to give a proof of positivity for generalized cluster algebras from an orbifold, as defined by Chekhov and Shapiro. In the unpunctured case, this is joint work with Elizabeth Kelley; recently, along with Wonwoo Kang, we have extended the construction to allow punctures. In order to deal with punctures, we use loop graphs, a construction from Jon Wilson.

LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to exhibit the Laurent phenomenon, but positivity remains conjectural in general. Graph LP algebras are finite LP algebras that can be encoded by a graph. For the subclass of graph LP algebras defined by trees, we define a family of clusters called rooted clusters. We give combinatorial interpretations for expansions in terms of these rooted clusters using generalizations of T-paths and snake graphs. (This is joint work with Esther Banaian, Sunita Chepuri, and Sylvester W. Zhang)

In cluster theory, establishing explicit formulae for cluster variables is an important problem. In the setting of surface cluster algebras this may be given combinatorially via snake graphs and homologically by the CC-map. Recently a combinatorial formula was introduced by Musiker--Ovenhouse--Zhang in an attempt to introduce super cluster algebras of type A (which computes super lambda-lengths in Penner-Zeitlin's super-Teichmüller spaces). This formula is given in terms of double dimer covers on snake graphs. Motivated by this construction, we propose a representation theoretic interpretation of super lambda-lengths and introduce a super-CC formula which recovers the combinatorial model. This is joint work in progress with Fedele, Garcia Elsener and Serhiyenko.

Triangular bases are Kazhdan-Lusztig type bases for quantum cluster algebras. They include the dual canonical bases for the quantized coordinate rings of unipotent subgroups. In this talk, we will construct the (common) triangular bases for the (quantized) coordinate rings of algebraic groups and of their double Bruhat cells, generalizing the results for unipotent subgroups. It is worth mentioning that their structure constants are positive when the Cartan datums are symmetric.

When Fomin and Zelevinsky invented cluster algebras, one motivation was to understand canonical bases in representation theory. Finite-type cluster algebras come with a natural basis given by cluster monomials. For infinite-type cluster algebras, the construction of bases containing the cluster monomials has been the subject of intense work in recent years, producing three families of bases: the``generic basis'', ``common triangular basis'', and ``theta basis''. All these bases are parametrized by the tropical points in the dual cluster variety, in line with the Fock-Goncharov conjecture. On the other hand, in representation theory, we also have three families of bases: the ``semicanonical basis'', ``canonical basis'', and ``Mirkovic-Vilonen basis''. Of course, it is very tempting to match up these families with the ones coming from cluster algebras. I will explain known results and open problems in this direction.

Semistandard Young tableaux and irreducible components of Springer fibers model highest weight crystals for $sl_n$ in a compatible way. We present a generalization of these correspondences to ADE Demazure crystals having minuscule weight. Our generalization uses reverse plane partitions in place of tableaux and preprojective algebra modules in place of flags. Do reverse plane partitions play with good bases or clusters? We end by sharing some open questions in this direction. This talk is based on joint work with Elek, Kamnitzer and Morton-Ferguson.

We discuss a generalization in type $A_n$ of the well--known KR-modules for quantum affine algebras. We shall see that our generalization has many of the properties of the KR-modules. Moreover, they allow us to classify all prime representation of the quantum affine algebra which are supported on only one node of the Dynkin diagram of $A_n$. We give a necessary and sufficient for a tensor product of such modules for a fixed node to be irreducible. We discuss an analog of the theory of monoidal categorification of cluster algebras developed by Hernandez and Leclerc for KR-modules. In the final part of our talk we discuss imaginary modules for quantum affine algebras. The first example of such modules appeared in the work of Leclerc. In terms of the infinite rank cluster algebras, coming from monodial categorification, our examples show the existence of an infinite number of pars of cluster variables whose product is not in the span of cluster monomials. The talk is based on joint work with Matheus Brito.

Quantum cluster algebras are quantizations of cluster algebras, which are a class of algebras interpolating between integrable systems and combinatorics. These algebras were originally introduced to study positivity phenomena arising in the study of quantum groups, and so one of the key questions regarding them (and their quantum analogues) is whether they admit a basis for which the structure constants are positive. The classical version of this question was settled in the affirmative by Gross, Hacking, Keel and Kontsevich. I will present a proof of the quantum version of this positivity for skew-symmetric quantum cluster algebras, due to joint work with Travis Mandel, based on results in categorified Donaldson-Thomas theory and using scattering diagrams.

The Gross-Siebert program associates a theta function on X to each boundary valuation on Y, where X and Y are a pair of mirror dual affine log Calabi-Yau varieties with maximal boundary (such as cluster varieties). Since mirror duality is a symmetric relation, this provides two ways to associate an integer to a pair m and n of boundary valuations on X and Y (respectively). 1) Apply the valuation m to the theta function associated to n. 2) Apply the valuation n to the theta function associated to m. Resolving a conjecture of Gross-Hacking-Keel-Kontsevich, we show that these two numbers are equal in a generality which covers all cluster algebras (specifically, when the theta functions are given by enumerating broken lines in a scattering diagram generated by finitely-many elementary incoming walls). Time permitting, I will discuss applications to the theta basis and its localizations. This work is joint with Man-Wai Cheung, Tim Magee, and Travis Mandel.

The main objective of this talk is to provide a description of the cluster complex for cluster $\mathcal{X}$-varieties of type $A$ in terms of $c$-vectors. We present partial results for other $\mathcal{X}$-varieties of finite cluster type. This is based on my ongoing Ph.D. project.

In this talk I will relate two types of tropicalization that are available when dealing with a (partially compactified) cluster variety: (1) the Fock-Goncharov tropicalization of a scheme with a positive atlas, (2) the tropicalization of an ideal associated with an embedding of the variety. We discuss how both types of tropicalzation encode toric degenerations of the cluster variety. We present the construction of a piece wise linear map from (1) to (2). Given certain natural assumptions the piece wise linear map not only identifies the fan structures on either side, but also the associated toric degenerations.

To every positive braid b, one can associate a smooth, affine, irreducible variety X(b), called the braid variety. Braid varieties are a natural generalization of positroid varieties, Richardson varieties, double Bruhat cells, and double Bott-Samelson cells in type A. I'll discuss joint work with P. Galashin, T. Lam, and D. Speyer in which we show the coordinate ring of X(b) is a locally acyclic cluster algebra. We construct seeds for this cluster algebra from "3D plabic graphs", which generalize Postnikov's plabic graphs for positroid varieties. Time permitting, I'll also discuss related joint work with K. Serhiyenko, where we prove that in type A, Leclerc's conjectural cluster structure on Richardsons is indeed a cluster structure.

We explore the polyhedral and tropical geometry of flag positroids, particularly flag positroids whose set of ranks is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety equals the nonnegative flag Dressian, and that points of these spaces give rise to coherent subdivisions of flag positroid polytopes into flag positroid polytopes. Our results have applications to Bruhat interval polytopes and to realizability questions. In particular, we prove that every positively oriented flag matroid of consecutive ranks is realizable. Joint work with Jon Boretsky and Chris Eur.

Given a simple algebraic group $G$ and an element $\beta$ of its positive braid monoid we define a smooth, affine algebraic variety $X(\beta)$, the braid variety of $\beta$, which generalizes well-known varieties in Lie theory, including open Richardson varieties and double Bott-Samelson cells. In joint work with R. Casals, E. Gorsky, M. Gorsky, I. Le and L. Shen we show that the coordinate algebra of $X(\beta)$ admits the structure of a Fomin-Zelevinsky cluster algebra and explicitly construct several initial seeds, using combinatorial objects called weaves and tropicalization of Lusztig's coordinates. I will explain this construction (with several examples) and give properties of the corresponding cluster structure, including local acyclicity and the existence of reddening sequences.

This is a survey talk on the roles of the dilogarithm and the pentagon relation in cluster algebras and cluster scattering diagrams. The topics include the Fock-Goncharov decomposition of mutations, the Hamiltonian formalism for mutations, the algebraic formulation of the dilogarithm and the pentagon relation, and the positive realization of cluster scattering diagrams.

This talk is based on joint work with Véronique Bazier-Matte. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra is 2n, where n is the number of crossing points in the knot diagram. We then construct 2n indecomposable modules T(i) over the Jacobian algebra of the quiver with potential. For each T(i), we show that the submodule lattice of T(i) is isomorphic to the corresponding lattice of Kauffman states of the knot. Furthermore, the Alexander polynomial of the knot is a specialization of the F-polynomial of T(i), for every i. If time permits, I will sketch the current state of our conjecture that the collection of the T(i) forms a cluster in the cluster algebra.

We will show that all root of unity quantum cluster algebras have canonical structures of Cayley-Hamilton algebras (in the sense of Procesi) and Poisson orders (in the sense of De Concini-Kac-Procesi and Brown-Gordon). The first result allows the transfer of finiteness properties between the quantum and classical situations. The second result relates the representation theory of these algebras to the Poisson geometry of the Gekhtman-Shapiro-Vainshtein brackets. We will then prove that the spectrum of each upper cluster algebra equipped with the GSV Poisson structures has an explicit Zariski open torus orbit of symplectic leaves, which is a far-reaching generalization of the Richardson divisor of a Schubert cell in Lie theory. At the end we will combine the above results to describe explicitly the fully Azumaya loci of all (strict) root of unity quantum cluster algebras. This classifies their irreducible representations of maximal dimension. This is a joint work with Shengnan Huang, Thang Le, Greg Muller, Bach Nguyen and Kurt Trampel.