Week 3

Program

Farewell Banquet on 6/20 (thu) 18:00-20:00 at Hokubu-Shokudo 2F (campus cafeteria) (info)

Title and Abstract

İlke Çanakçı (Newcastle)

Generalised friezes and the weak Ptolemy map

Frieze patterns, introduced by Coxeter, are infinite arrays of numbers where neighbouring numbers satisfy a local arithmetic rule. Frieze patterns with positive integer values are of a special interest since they are in one-to-one correspondence with triangulations of polygons by Conway--Coxeter. Remarkably, this established a connection to cluster algebras–predating them by 30 years– and to cluster categories. Several generalisations of frieze patterns are known. I will report on joint with Jørgensen in which we associated frieze patterns to dissections of polygons where the entries are over a (commutative) ring. Furthermore, I will discuss an explicit combinatorial formula for the entries of these friezes by generalising the "T-path formula" which was introduced by Schiffler to give explicit formulas for cluster variables for cluster algebras of type A.

Leonid Chekhov (Steklov)

Quantum cluster algebras, SL_N character varieties and the groupoid of bi-linear forms

I will begin with the description of the groupoid of upper-triangular matrices corresponding to dynamics of bi-linear forms admitting a Poisson structure preserved by a braid-group action. The main objective of my presentation is to show how the newly constructed quantum cluster algebra description of the SL_N character manifold on a surface $\Sigma_{g,s,n}$ of arbitrary genus g with $s>0$ holes and with $n>0$ decorated boundary cusps (marked points on hole boundaries) helps to solve a long-standing problems of constructing Darboux coordinates for the upper-triangular matrices and of representing the braid-group action in terms of mutations of cluster variables. All such manifolds can be constructed by amalgamation procedure from elementary blocks which are ideal triangles $\Sigma_{0,1,3}$ endowed with the Fock-Goncharov cluster algebra structure, which therefore plays a pivotal role. We show that for any planar directed (acyclic) network, elements of source-sink matrices satisfy quantum R-matrix relations, and for a special network corresponding to the Fock-Goncharov quiver in the triangle $\Sigma_{0,1,3}$ we can obtain the representation of the quantum groupoid of upper-triangular matrices. We also represent the quantum braid-group action on this groupid as sequences of quantum mutations (Forthcoming joint paper with M.Shapiro).

Ben Davison (Edinburgh)

Positivity for quantum skew symmetric cluster algebras

I will explain a categorification of quantum cluster algebras, and how to use it to prove that all cluster monomials in skew-symmetric quantum cluster algebras are positive Laurent polynomials in the cluster monomials of any given seed, with coefficients polynomials in q^{1/2} enjoying the "hard Lefschetz" property. In particular, this implies positivity for the coefficients of these polynomials.

Anna Felikson (Durham)

Non-integer quivers of finite mutation type

We will discuss how geometry helps to classify non-integer quivers of finite mutation type.

This work is joint with Pavel Tumarkin.

Philippe Di Francesco (Illinois)

Triangular Ice Combinatorics

Alternating Sign Matrices (ASM) are at the confluent of many interesting combinatorial/algebraic problems: Laurent phenomenon for the octahedron equation, configurations of the Square Ice (Six Vertex model), Descending Plane Partitions (DPP), etc. Here we consider the Triangular Lattice version of the Ice model with suitable boundary conditions leading to an integrable 20 Vertex model. Configurations give rise to generalizations of ASM, which we coin Alternating Phase Matrices (APM). We generalize the ASM-DPP correspondence by showing that APM are equinumerous to the quarter-turn symmetric domino tilings of a quasi-Aztec square with a central cross-shaped hole, and obtain a compact determinant formula for their enumeration. We argue that APM may describe the Laurent phenomenon for some yet to bedefined cluster algebra. We present conjectures for triangular Ice with other types of boundary conditions.

Sebastian Franco (CUNY)

Graded Quivers, Generalized Dimer Models and Toric Geometry

The open string sector of the topological B-model model on CY (m+2)-folds is described by m-graded quivers with superpotentials. This connection extends to general m the celebrated correspondence between CY (m+2)-folds and quantum field theories in (6-2m) dimensions. These quivers exhibit new order-(m+1) mutations, which reproduce the recently discovered dualities of the associated quantum field theories for m≤3 and generalize them to m>3. In the first part of this talk we will discuss the general framework of graded quivers, which also involves ideas on higher Ginzburg algebras and higher cluster categories.

We will then introduce m-dimers, which fully encode the m-graded quivers and their superpotentials in the case of toric CY (m+2)-folds. Generalizing the standard m=1 case, m-dimers significantly simplify the map between geometry and m-graded quivers.

Michael Gekhtman (Notre Dame)

Periodic staircase matrices and generalized cluster algebras

As is well-known, cluster transformations in cluster algebras of geometric type are often modeled on determinant identities, such short Plücker relations, Desnanot-Jacobi identities and their generalizations. I will present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in GL(n) compatible with a certain subclass of Belavin-Drinfeld Poisson-Lie backers, in the Drinfeld double of GL(n) and in spaces of periodic difference operators.

(Based on joint work with M. Shapiro and A. Vainshtein).

Rei Inoue (Chiba)

Cluster realizations of Weyl groups and their applications

For symmetrizable Kac-Moody Lie algebra g and an integer m bigger than one, we define a weighted quiver Q, such that the cluster modular group for Q contains the Weyl group of g. It has a several interesting applications, and in this talk we introduce: (1) When g is of finite

type, green sequences and the cluster Donaldson-Thomas transformation for Q are systematically obtained. (2) When g is of classical finite type and m is the Coxeter number of g, the quiver Q is related to the cluster realization of the quantum group studied by Schrader-Shapiro and Ip.

This talk is based on a joint work with Tsukasa Ishibashi and Hironori Oya.

Ivan Ip (HKUST)

Positive Peter-Weyl Theorem

For a compact Lie group G, the classical Peter-Weyl Theorem states that the regular representation of G on L^2(G) decomposes as the direct sum of its irreducible unitary representations. Similar results are being generalized to the case of real reductive groups by Harish-Chandra, as well as compact quantum groups by Woronowicz, but the case for non-compact quantum groups is pretty much unclear.

In this talk, I will explain the Peter-Weyl Theorem for split real quantum groups of type An. I will talk about the necessary ingredients needed to state and proof the theorem, including the GNS representation of C*-algebra, quantum parallel transports, and cluster realization of positive representations. The talk is based on joint work with G. Schrader and A. Shapiro.

Rinat Kedem (Illinois)

Integrable structure of the Q-system cluster algebras

For the classical series g=ABCD, the Q-systems which give the structure of the Grothendieck ring of the untwisted Yangians are mutations in cluster algebras, which do not have a bipartite structure for types BC. The conserved quantities which appear in the linear recursion relations satisfied by cluster variables can be obtained from the coefficients of difference L-operators. Under quantization, the conserved quantities act on a special subset of q-graded characters as q-difference Toda Hamiltonians. I will give some recent results about factorized L-operators for the Q-systems, and discuss the relation with the generating functions of the fundamental q-characters. Joint work with Philippe Di Francesco.

Kyungyong Lee (Nebraska Lincoln)

Geometric description of c-vectors and real Lösungen

We propose a combinatorial/geometric model and formulate several conjectures to describe the c-matrices of an arbitrary skew-symmetrizable matrix. In particular, we introduce real Lösungen as an analogue of real roots and conjecture that c-vectors are real Lösungen given by non-self-crossing curves on a Riemann surface. We show that all our conjectures are true for numerous special cases. This is a joint work with Kyu-Hwan Lee.

Jianrong Li (Graz)

Quantum affine algebras and Grassmannians

I will talk about our recent work (Joint with Wen Chang, Bing Duan, and Chris Fraser) on quantum affine algebras of type A and Grassmannian cluster algebras.

Let g=sln and U_q(^g) the corresponding quantum affine algebra. Hernandez and Leclerc proved that there is an isomorphism Phi from the Grothendieck ring R_l^g of a certain subcategory C_l^g of finite dimensional U_q(^g)-modules to a certain quotient S_{n,n+l+1} of a Grassmannian cluster algebra (certain frozen variables are sent to 1). We proved that this isomorphism induced an isomorphism between the monoid of dominant monomials and the monoid of rectangular semi-standard Young tableaux. Using the isomorphism, we defined ch(T) in S_{n,n+l+1} for every rectangular tableau T.

Using the isomorphism and the results of Kashiwara, Kim, Oh, and Park and the results of Qin, we proved that every cluster monomial (resp. cluster variable) in a Grassmannian cluster algebra is of the form ch(T) for some real (resp. prime real) rectangular semi-standard Young tableau T.

We translated a formula of Arakawa–Suzuki and Lapid-Minguez to the setting of q-characters and obtained an explicit q-character formula for a finite-dimensional U_q(^g)-module. These formulas are useful in studying real modules, prime modules, and compatibility of two cluster variables.

We described the mutations of a Grassmannian cluster algebra using semi-standard Young tableaux and described the mutations of modules.

Gregg Musiker (Minnesota)

Dimer Interpretations of toric cluster variables associated to del Pezzo quivers

By applying sequences of toric mutations to vertices of the dP1 (resp. dP2 or dP3) quiver, that is allowing mutation only at vertices with two incoming and two outgoing arrows, we obtain a family of cluster variables that can be parameterized by one (resp. two or three) integers. In most of these cases, such cluster variables have Laurent expansions obtainable via elegant combinatorial interpretations in terms of dimer partition functions. Furthermore, this allows us to study a variety of historical enumerative combinatorics questions all under one roof. We also include a conjectural interpretation, utilizing double-dimers, for the remaining cluster variables, as well as work in progress towards this conjecture. This is based on joint work with Tri Lai.

Alfred Nájera Chávez (UNAM)

Toric degeneration of cluster varieties, cluster duality and mirror symmetry

In this talk we will introduce X-cluster varieties with coefficients. We use this notion to build a flat degeneration of every skew-symmetrizable specially completed X-cluster variety (in the sense of Fock-Goncharov) to the toric variety associated to its g-fan. We further show that our construction is cluster dual to Gross-Hacking-Keel-Kontsevich's toric degeneration of A-cluster varieties. If time permits we will outline the following applications:

1) We can show that the toric degeneration of Grassmannians constructed by Rietsch-Williams coincides with GHKK's toric degeneration.

2) We can use our construction to give a precise relation between cluster duality and Batyrev-Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

This is based on joint work with Lara Bossinger, Man-Wai Cheung, Bosco Frías-Medina and Timothy Magee.

Atsushi Nobe (Chiba)

A family of integrable and non-integrable difference equations arising from cluster algebras

Integrability of a one-parameter family of second order nonlinear difference equations, each of which is arising from seed mutations of a rank 2 cluster algebra, is discussed. Four members of the family posses integrable structure, while the remaining infinitely many members do not. In order to evaluate their dynamics, algebraic entropy of the birational maps equivalent to the difference equations are explicitly computed. Moreover, in integrable cases, generators of the corresponding cluster algebras are explicitly obtained by solving initial value problems of the difference equations.

Yu Qiu (Tsinghua)

Calabi-Yau-X and cluster-X categories

We introduce Calabi-Yau-X categories D_X as q-deformation of topological Fukaya categories TFuk whose Calabi-Yau-N orbit categories are (subcategories of) derived Fukaya categories. Moreover, we show that TFuk is equivalent to the cluster categories associated to D_X. I will also mention q-deformation of stability conditions (and q-quadratic differentials). This is a joint work with Akishi Ikeda.

Gus Schrader (Columbia)

Gelfand-Zeitlin modules under Whittaker transform

A certain class of infinite dimensional quantum group representations, termed Gelfand-Zeitlin modules, were introduced in the 2000’s by Gerasimov, Kharchev, Lebedev and Oblezin. These representations and their generalizations have attracted recent interest due to their role in the theory of quantized K-theoretic Coulomb branch algebras as constructed by Braverman, Finkelberg and Nakajima. I will explain how the Whittaker spectral transform for the q-difference open Toda chain can be used to reveal the cluster structure behind this class of representations and the associated integrable systems.

Alexander Shaprio (Tronto)

Coulomb branches and integrable systems

Recently, Braverman, Finkelberg, and Nakajima gave a mathematical definition of Coulomb branches of 3d N=4 gauge theories of cotangent type. It was then conjectured by Gaiotto that these Coulomb branches bear structure of cluster varieties. I will discuss cluster-algebraic tools needed to prove Gaiotto's conjecture for quiver gauge theories. These include a "bifundamental Baxter operator", a special cluster modular group element associated to a pair of Coxeter-Toda integrable systems, as well as the construction of generalized nil-Macdonald operators from a quantum measurement matrix associated to the Coxeter-Toda cluster algebra.

Michael Shapiro (Michigan State)

Cluster algebras with Grassmann variables (joint with V. Ovsienko)

We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of “extended quivers” which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in different contexts. This project is a step towards understanding the notion of cluster superalgebra.

Salvatore Stella (Haifa)

Acyclic cluster algebras via Coxeter double Bruhat cells and generalized minors

Cluster algebras come with a canonical partial basis: the cluster monomials. Extending this partial basis to a full basis has been one of the central problems in the theory giving raise to a plethora of constructions.

In this talk we will explain how Lie theory can be used to relate them. Specifically, after recalling the basic definitions, we will explain how any acyclic cluster algebra can be seen as the ring of coordinates of a suitable double Bruhat cell in the associated Kac-Moody group. Under this identification we will interpret cluster monomials as generalized minors - certain functions on a Kac-Moody group defined in terms of its representations - and explain how one can use generalized minors to extend cluster monomials to a continuous family of bases of the cluster algebra in the affine cases.

This talk is based on joint works with D. Rupel and H. Williams.

Pavel Tumarkin (Durham)

Mutation-finite cluster algebras and extended affine Weyl groups

With every unpunctured surface we can associate two groups: an

extended affine Weyl group of type A and a certain group which behaves

nicely with respect to mutations. I will discuss a connection between

these groups and a (conjectural) characterization of mutation-finite

quivers in terms of positive semidefinite symmetric matrices. The talk

is based on joint works with Anna Felikson, John Lawson and Michael

Shapiro.

Alek Vainshtein (Haifa)

Plethora of cluster structures on GL_n

I will explain our recent results in the study of multiple cluster structures in the rings of regular functions on GL_n, SL_n and Mat_n that are compatible with Poisson-Lie and Poisson-homogeneous structures. All possible A_n type Belavin-Drinfeld (BD) data will be subdivided into oriented and non-oriented kinds. In the oriented case, one can single out BD data satisfying a certain combinatorial condition called aperiodicity. Our main result claims that for any BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson–Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on SL_n compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of SL_n equipped with two different Poisson-Lie brackets. In all the cases, the corresponding upper cluster algebra is isomorphic to the rihg of regular functions. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. (Based on a joint work with M. Gekhtman and M. Shapiro).

Harold Williams (UC Davis)

Canonical Bases for Coulomb Branches

Following work of Kapustin-Saulina and Gaiotto-Moore-Neitzke it is anticipated that the expectation values of irreducible half-BPS line defects define a canonical basis in the quantized Coulomb branch of a 4d N=2 field theory. In this talk we propose a mathematical definition of the category of such defects, hence of the associated canonical basis, in the case of N=2 gauge theories of cotangent type. The definition takes the form of a finite length t-structure on the DG category of coherent sheaves on the space of triples introduced by Braverman-Finkelberg-Nakajima. This t-structure is a non-Noetherian generalization of the perverse coherent t-structure, to which it specializes in the case of pure N=2 gauge theory. It is anticipated that these categories provide a large new class of monoidal cluster categorifications, which we confirm in various examples. This is work in progress with Sabin Cautis.

Lauren Williams (UC Berkeley)

Cluster structures and superpotentials for Schubert varieties

I will begin my describing the combinatorics of cluster structures in Schubert varieties in the Grassmannian, based on joint work with Khrystyna Serhiyenko and Melissa Sherman-Bennett. I'll then describe Newton Okounkov bodies for Schubert varieties and a conjectural superpotential function, based on joint work with Konstanze Rietsch.

(last updated: June 17, 2019)