Organizers: Sin-Myung Lee (KIAS, sinmyunglee@kias.re.kr), Jaeseong Oh (Sungkyunkwan University, jaeseongoh@skku.edu)
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Upcoming Talks:
Haruto Murata (The University of Tokyo, Japan)
Date: Sep. 18 (Thu), 2025, 10:30am-11:30am (Korea time, UTC+9)
Title: A diagrammatic approach to reflection functors
Abstract: For an arbitrary symmetrizable generalized Cartan matrix, the category of modules over the quiver Hecke algebra provides a categorification of the associated quantum group. In this talk, I will present a detailed construction of reflection functors that categorify Lusztig’s braid group action, from the viewpoint of higher representation theory. Similar functors have recently been constructed independently by Kashiwara-Kim-Oh-Park.
Nicholas Williams (University of Cambridge, United Kingdom)
Date: Oct. 2 (Thu), 2025, 5:00pm-6:00pm (Korea time, UTC+9)
Title: Oriented matroids and maximal green sequences of type A cluster algebras
Abstract: The theory of cluster algebras of Fomin and Zelevinsky is perhaps best illustrated by the type A cluster algebras. These arise as the homogeneous coordinate rings of the Grassmannians of 2-dimensional subspaces of n-dimensional space. They provide an excellent starting point for understanding cluster algebras, since they have a beautiful combinatorial model. In this model, clusters correspond to triangulations of a convex polygon, with cluster variables corresponding to arcs in the polygon and the mutation operation corresponding to flipping an arc in a quadrilateral. In previous work, I showed that triangulations of three-dimensional cyclic polytopes admit a natural interpretation in this model, as equivalence classes of maximal green sequences of the cluster algebra with the initial cluster given by the linear orientation of the type A Dynkin diagram. Maximal green sequences are maximal sequences of mutations from a given initial cluster, and were introduced due to their role in quantum dilogarithm identities and related physics. The obvious question is: if one takes an initial cluster given by a different orientation of the diagram, is there a three-dimensional polytope whose triangulations give equivalence classes of maximal green sequences for that initial cluster? In this talk I will explain recent work which goes a long way to answering this question. Namely, I show that there exists a rank-four oriented matroid whose triangulations give such equivalence classes of maximal green sequences. If time permits, I will explain some of the techniques I use to prove this result, namely the associated cluster category defined by Buan, Marsh, Reineke, Reiten, and Todorov.
Weizhe Zheng (Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China)
Date: Oct. 30 (Thu), 2025, 10:30am-11:30am (Korea time, UTC+9)
Title: TBA
Abstract: TBA
Jihyeug Jang (University of Geneva, Switzerland)
Date: TBD
Title: TBA
Abstract: TBA
Past talks:
2025
Hunter Spink (University of Toronto, Canada)
Date: Sep. 11 (Thu), 2025, 10:30am-11:30am (Korea time, UTC+9)
Title: The Coxeter flag variety: A toric complex on noncrossing partitions paved by clusters
Abstract: (Joint with Nantel Bergeron, Lucas Gagnon, and Vasu Tewari) In this talk I will introduce a new toric complex we call the "Coxeter flag variety" in G/B, defined as the vanishing of Plucker coordinates away from the subset of the Weyl group called the "c-noncrossing partitions" for c a standard Coxeter element. The combinatorics of this variety encodes the geometry of certain Richardson varieties R_{w,wc}, each of which has a distinguished affine chart whose characters correspond to positive clusters.
Donggun Lee (Institute for Basic Science, CCG, Korea)
Date: Jun. 20 (Fri), 2025, 10:30am-11:30am (Korea time, UTC+9)
Title: Geometry of regular semisimple Lusztig varieties
Abstract: Lusztig varieties were introduced in the study of character sheaves and related representation theory. They can be seen as variants of both Schubert varieties and Hessenberg varieties. For example, they admit a Weyl group action on their intersection cohomology through monodromy, and their singularities can be resolved via Bott-Samelson-type resolutions.
I will present results on their geometry, including vanishing theorems for the cohomology of line bundles, their relations to Hessenberg varieties, and diffeomorphism types. Along the way, we also establish that their open cells are affine, and that the same is true for Deligne-Lusztig varieties, settling a question that has been open since the foundational paper of Deligne and Lusztig.
Some of the results were motivated by our forthcoming study of the representations via a trace map of the Hecke algebra. Based on joint works with Patrick Brosnan and Jaehyun Hong.
Daniel Bump (Stanford University, USA)
Date: May 23 (Fri), 2025, 9:00am-10:00am (Korea time, UTC+9)
Title: Solvable Lattice Models and Yang-Baxter Groupoids
Abstract: We will review some applications of the Yang-Baxter equation to proving that certain partition functions represent functions of interest to combinatorialists such as Schur polynomials, Hall-Littlewood polynomials or Schubert polynomials. The underlying mechanism is the parametrized Yang-Baxter equation. Typically these are regarded as coming from quantum groups, and we will see how for the six-vertex model there is a contrast between the "free-fermionic" case relying on the quantum group U_q(sl(1|1)) and the non-free-fermionic case relying on U_q(sl(2)). A deeper look at the non-free-fermionic case leads to Naprienko's six-vertex groupoid. We will argue that groupoid parametrized Yang-Baxter equations may be common, and give another example of a groupoid parametrized Yang-Baxter equation in the five-vertex model. This talk will rely on joint work with Brubaker, Buciumas and Gustafsson, and with Naprienko.
Soo-Hong Lee (Seoul National University, Korea)
Date: May 2 (Fri), 2025, 10:30am-11:30am (Korea time, UTC+9)
Title: Infinite level Fock space, crystal bases, and tensor products of extremal weight modules of type A_+∞
Abstract: We construct an infinite-level Fock space that admits commuting actions of the quantized enveloping algebra of infinite affine type A_+∞ and the parabolic boson algebra associated with (gl_∞, gl_{∞\0}). This construction extends the classical Howe duality on higher-level Fock spaces. Its study revisits a subtle phenomenon in which the crystal base of a proper submodule coincides with that of the entire space — a behavior previously observed in level-zero representations of affine type by Kashiwara, Beck, and Nakajima. In the A+∞ type, we show that this combinatorial phenomenon is intricately tied to the socle filtration of ambient representations. Utilizing this connection, we construct a saturated crystal base through a limiting process and provide a comprehensive description of the socle filtration for extremal weight modules of type A+∞, recovering coefficients first obtained by Penkov and Styrkas. This is a joint work with Jae-Hoon Kwon.
Sean Griffin (University of Vienna, Austria)
Date: Apr. 18 (Fri), 2025, 04:00pm-05:00pm (Korea time, UTC+9)
Title: On Macdonald expansions of q-chromatic symmetric functions and the Stanley-Stembridge Conjecture
Abstract: The Stanley-Stembridge conjecture asserts that the chromatic symmetric function of a (3+1)-free graph is e-positive, meaning that it expands positively in the elementary symmetric function basis. Motivated by a geometric construction of Kato, recently Hikita proved this conjecture by giving an explicit e-expansion of the Shareshian-Wachs q-chromatic refinement (or chromatic quasisymmetric function) for unit interval graphs. Using the A_{q,t} algebra of Carlsson and Mellit, we give an expansion of these q-chromatic symmetric functions into Macdonald polynomials. Upon setting t=1, we obtain another proof of the Stanley-Stembridge conjecture and rederive Hikita's formula. Upon setting t=0, we obtain an expansion into Hall-Littlewood symmetric functions. This is joint work with Mellit, Romero, Weigl, and Wen.
Serena An (Massachusetts Institute of Technology, USA) and Katherine Tung (Harvard University, USA)
Date: Apr. 3 (Thu), 2025, 10:30am-11:30am (Korea time, UTC+9)
Title: Postnikov–Stanley polynomials are Lorentzian
Abstract: The M-convexity of dual Schubert polynomials was first proven by Huh, Matherne, Mészáros, and St. Dizier in 2022. We give an alternative, elementary proof of this result, explicitly characterize the Newton polytopes of dual Schubert polynomials, and provide a polynomial-time algorithm for determining if a dual Schubert polynomial coefficient vanishes. We furthermore generalize the M-convexity result by showing that Postnikov–Stanley polynomials, also called skew dual Schubert polynomials, are Lorentzian. This work is joint with Yuchong Zhang and is contained in papers arXiv:2411.16654 and arXiv:2412.02051.
Kang Lu (University of Virginia, USA)
Date: Mar. 28 (Fri), 2025, 10:00am-11:00am (Korea time, UTC+9)
Title: A Drinfeld presentation of twisted Yangians and applications
Abstract: Twisted Yangians were first introduced in R-matrix presentation by Olshanski in 1990 in connections with classical Lie algebras and more generally by Molev-Ragoucy and Guay-Regelskis associated with symmetric pairs of classical type. A Drinfeld type current presentation of (quasi-)split twisted Yangians has been recently obtained in two different approaches, and in this talk we will explain the Gauss decomposition approach. Then we shall outline ongoing work on generalizations and applications of the Drinfeld presentation. This is based on joint work of Weiqiang Wang, Weinan Zhang and collaborators.
Christian Gaetz (University of California, Berkeley, USA)
Date: Mar. 14 (Fri), 2025, 10:30am-11:30am (Korea time, UTC+9)
Title: Hypercube decompositions and combinatorial invariance for Kazhdan-Lusztig polynomials
Abstract: Kazhdan-Lusztig polynomials are of foundational importance in geometric representation theory. Yet the Combinatorial Invariance Conjecture, due to Lusztig and to Dyer, suggests that they only depend on the combinatorics of Bruhat order. I'll describe joint work with Grant Barkley in which we adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove this conjecture for Kazhdan-Lusztig R-polynomials in the case of elementary intervals in the symmetric group. This significantly generalizes the main previously known case of the conjecture, that of lower intervals.
2024
Jaewon Min (University of Illinois Urbana-Champaign, USA)
Date: Nov. 21 (Thu), 2024, 09:30am-10:30am (Korea time, UTC+9)
Title: Saturation of Littlewood-Richardson coefficients and generalization
Abstract: By inventing the notion of honeycombs, A. Knutson and T. Tao proved the saturation conjecture for Littlewood-Richardson coefficients in 1999. Combined with earlier work of A. A. Klyachko, this resolved Horn’s conjecture: relations between eigenvalues of Hermitian matrices A, B and A+B. The Newell-Littlewood numbers are a generalization of the Littlewood-Richardson coefficients. According to K. Koike, in stable range, they are also tensor product multiplicities of classical Lie groups: type B, C and D. By introducing honeycombs on a Möbius strip, I will discuss about the saturation theorem for Newell-Littlewood numbers.
Amanda Burcroff (Harvard University, USA)
Date: Nov. 8 (Fri), 2024, 10:00am-11:00am (Korea time, UTC+9)
Title: Scattering Diagram and Generalized Positivity
Abstract: Cluster algebras are celebrated for their intriguing positivity properties. Two distinct proofs of this positivity have emerged: one through the combinatorics of Dyck paths, and another via scattering diagrams, which originate from mirror symmetry and were previously not combinatorially understood. Combining these approaches, we find a directly computable, manifestly positive, and elementary but highly nontrivial formula describing rank 2 scattering diagrams. Using this, we prove the Laurent positivity of generalized cluster algebras of all ranks, resolving a conjecture of Chekhov and Shapiro from 2014. This is joint work with Kyungyong Lee and Lang Mou.
Ryo Fujita (Research Institute for Mathematical Sciences, Japan)
Date: Oct. 31 (Thu), 2024, 9:30am-10:30am (Korea time, UTC+9)
Title: Deformed Cartan matrices and generalized preprojective algebras, with application to quantum affine algebras
Abstract: In their study of the deformed W-algebras associated with complex simple Lie algebras, E. Frenkel--Reshetikhin introduced certain two parameter deformations of the Cartan matrices. They are also important in the representation theory of quantum affine algebras as a key combinatorial ingredient. In this talk, we give a representation-theoretic interpretation of these deformed Cartan matrices (and their inverses) in terms of the generalized preprojective algebras recently introduced by Geiss--Leclerc--Schröer, and discuss its application to the quantum affine algebras in relation with the cluster theory. This talk is based on a joint work with Kota Murakami.
Kento Osuga (University of Tokyo, Japan)
Date: Oct. 11 (Fri), 2024, 10:00am-11:00am (Korea time, UTC+9)
Title: b-Hurwitz numbers from Whittaker vectors for W-algebras
Abstract: G-weighted Hurwitz numbers can be interpreted as sum of ribbon graphs embedded into orientable surfaces with certain weight G, and G-weighted b-Hurwitz numbers are their non-orientable analogues with the notion of measure of non-orientability proposed by Chapuy-Dolega. It was further observed by Bomzon-Chapuy-Dolega that for (very) simple choices of G, their b-Hurwitz numbers satisfy Virasoro constraints. Then, one may ask: what if we choose G to be a more general form? Do they still satisfy Virasoro constraints or are they related to different algebras? In this talk I will show a nontrivial relation between b-Hurwitz numbers and so-called Whittaker vectors for W-algebras --- a generalisation of the Virasoro algebra. If time permits, I will also discuss how they are related to topological recursion as well as refined topological recursion. This talk is based on joint work with Nitin Chidambaram and Maciej Dolega, and the main reference is arXiv:2401.12814.
Eugene Gorsky (University of California, Davis, USA)
Date: Sep. 12 (Thu), 2024, 10:00am-11:00am (Korea time, UTC+9)
Title: Delta Conjecture and affine Springer fibers
Abstract: Delta Conjecture of Haglund, Remmel and Wilson is the identity describing the action of Macdonald operators on elementary symmetric functions. The conjecture was proved independently by Blasiak-Haiman-Morse-Pun-Seelinger, and D'Adderio-Mellit. In this talk, I will give a geometric model for Delta conjecture using affine Springer fibers. This is a joint work with Sean Griffin and Maria Gillespie.
Brendon Rhoades (University of California, San Diego, USA)
Date: Sep. 5 (Thu), 2024, 10:30am-11:30am (Korea time, UTC+9)
Title: Harmonics and graded Ehrhart theory
Abstract: Classical Ehrhart theory gives a family of results on counting lattice points in a lattice polytope P together with its integer dilations. The enumerative results of Ehrhart theory applied to P may be understood algebraically using the affine semigroup ring A_P attached to P. The orbit harmonics method gives a way to attach a graded ring to any finite point locus; we use this idea to define a canonical q-refinement of the Ehrhart series of P and make a number of conjectures about this object analogous to those in the classical setting. We attach a non-obvious bigraded ring H_P to P called the {\em harmonic algebra} of P. We conjecture that the harmonic algebra plays a role in the q-Ehrhart theory of P analogous to the role played by the affine semigroup ring in the classical Ehrhart theory of P. Joint with Vic Reiner.
Hideya Watanabe (Rikkyo University, Japan)
Date: Jun. 21 (Fri), 2024, 10:00am-11:00am (Korea time, UTC+9)
Title: Combinatorial $K$-matrices arising from affine quantum symmetric pairs of type $A$
Abstract: A combinatorial $R$-matrix is a solution to the combinatorial Yang-Baxter equation, which plays a vital role in discrete dynamical systems such as box-ball systems. It has been known that the theory of crystal bases arising from quantum groups provides us with a framework to construct combinatorial $R$-matrices systematically. In a dynamical system which has a boundary, the role of the combinatorial Yang-Baxter equation is replaced by the combinatorial reflection equation, whose solutions are called combinatorial $K$-matrices. In this talk, based on a joint work with Hiroto Kusano and Masato Okado, I will explain how to construct combinatorial $K$-matrices systematically by means of the theory of crystal bases arising from affine quantum symmetric pairs of type $A$.
Theo Douvropoulos (Brandeis University, USA)
Date: May 24 (Fri), 2024, 10:00am-11:00am (Korea time, UTC+9)
Title: Parking functions on the cluster complex and Cyclic Sieving Phenomena
Abstract: The associahedron K_n and its dual, the type-A cluster complex Y(S_n), with their many realizations and generalizations are central objects in algebraic combinatorics. Their faces are labeled by subdivisions of an (n+2)-gon, they have Catalan-many vertices/facets, and their f-vectors are given by the Kirkman numbers.
In past joint work with Josuat-Verges we gave a refinement of the f-vector of Y(n) that keeps track of a cycle type associated to each polygonal subdivision, and is expressed in terms of structure coefficients of Haiman's parking space of diagonal coinvariants. We will present in this talk two research works built on these structures.
In the first work, joint with Josuat-Verges, we give a topological, equivariant version of the combinatorial f-to-h reciprocity for Y(n). We label each polygonal subdivision with a set partition and get as a result a simplicial complex that carries a parking space representation in its top homology. This presents the result of the recent preprint arxiv:2402.03052.
In the second still in progress work, joint with Josuat-Verges and Sommers, we give q-versions of these refined f-vectors (obtained naturally via the parking space interpretation) and prove cyclic sieving phenomena (CSPs) for the natural cyclic rotation of the polygon. In particular, we show that the q-Kirkman numbers satisfy a new, weighted type of CSP. We present further a version that keeps track of dihedral symmetries via the qt-Schroder numbers, utilizing the full grading of the parking space.
Most of these results make sense for all real reflection groups but we will focus our presentation to the symmetric group case. We will end with open questions in each approach or possible common extensions.
Oleksandr Tsymbaliuk (Purdue University, USA)
Date: May 16 (Thu), 2024, 7:30pm-8:30pm (Korea time, UTC+9) (Note the unusual day and time!)
Title: Lyndon words and fused currents in shuffle algebra
Abstract: Classical q-shuffle algebras provide combinatorial models for the positive half U_q(n) of a finite quantum group. We define a loop version of this construction, yielding a combinatorial model for the positive half U_q(Ln) of a quantum loop group. In particular, we construct a PBW basis of U_q(Ln) indexed by standard Lyndon words, generalizing the work of Lalonde-Ram, Leclerc and Rosso in the U_q(n) case. We also connect this to Enriquez' degeneration A of the elliptic algebras of Feigin-Odesskii, proving a conjecture that describes the image of the embedding U_q(Ln) -> A in terms of pole and wheel conditions. The talk shall conclude with the shuffle interpretations of fused currents proposed by Ding-Khoroshkin. This is based on joint works with Andrei Negut.
Arun Ram (University of Melbourne, Australia)
Date: Apr. 26 (Fri), 2024, 11:30am-12:30pm (Korea time, UTC+9)
Title: A new definition of integral form Macdonald polynomials
Abstract: Mellit explained the Modified Macdonald polynomials can be viewed as a generating function for the number of points in parabolic affine Springer fibers. In this talk we explain that the plethystic transformation that relates integral form Macdonald polynomials and modified Macdonald polynomials has an interpretation in terms of traces on the Hecke algebra. This leads to an interpretation of the integral form Macdonald polynomials as generating functions for the number of points Lusztig varieties.
Joshua Wen (Northeastern University, USA)
Date: Apr. 12 (Fri), 2024, 10:00am-11:00am (Korea time, UTC+9)
Title: Wreath Macdonald polynomials
Abstract: Defined by Haiman, wreath Macdonald polynomials are generalizations of the well-known Macdonald polynomials to wreath products of cyclic groups with symmetric groups. For a fixed cylic group Z/rZ, these can be viewed as partially-symmetric polynomials, where there are r families of symmetric variables. Many results for the usual Macdonald polynomials should have analogues in the wreath setting: e.g. Macdonald operators, bispectral duality, evaluation formulas, and norm formulas. Precise conjectures for these analogues can be tricky to write down and even more difficult to prove. A guiding principle is that various quantum algebraic methods in the classical Macdonald theory should have generalizations in the wreath setting. I will present work, joint with Daniel Orr and Mark Shimozono, that studies these polynomials via the rank r quantum toroidal algebra.
Tianyi Yu (University of California, San Diego, USA)
Date: Apr. 5 (Fri), 2024, 2:00pm-3:00pm (Korea time, UTC+9)
Title: Analogue of Fomin-Stanley algebra on bumpless pipedreams
Abstract: Schubert polynomials are distinguished representatives of Schubert cells in the cohomology of the flag variety. Pipedreams (PD) and bumpless pipedreams (BPD) are two combinatorial models of Schubert polynomials. There are many classical perspectives to view PDs: Fomin and Stanley represented each PD as an element in the NilCoexter algebra; Lenart and Sottile converted each PD into a labeled chain in the Bruhat order. In this talk, we unravel the BPD analogues of both viewpoints. One application of our results is a simple bijection between PDs and BPDs via Lenart’s growth diagram.
Huanchen Bao (National University of Singapore, Singapore)
Date: Mar. 15 (Fri), 2024, 3:00pm-4:00pm (Korea time, UTC+9)
Title: Acyclic matchings on Bruhat intervals
Abstract: Discrete Morse theory, developed by Forman, is an efficient tool to determine the homotopy type of a regular CW complex. The theory has been reformulated by Chari in purely combinatorial terms of acyclic matchings on the face poset. In this talk, I will discuss explicit constructions of such acyclic matchings on Bruhat intervals using reflection orders. As an application, we show the totally nonnegative Springer fibres are conctractible, verifying a conjecture of Lusztig. This is based on joint work with Xuhua He.