ACRT Seminar

	This is the webpage for the Algebraic Combinatorics and Representation Theory Seminar (ACRT) at KIAS. The seminar is run by Hyun Kyu Kim, Myungho Kim, Seungjin Lee, and Hwanchul Yoo. If you have questions or if you want to receive emails about upcoming seminars, please contact Hwanchul Yoo (hcyoo@kias.re.kr). Click [ACRT Seminar] to subscribe our seminar calinder in iCal format.

Upcoming seminars

Past Seminars

• August 28 (Friday)
  • UhiRinn Suh (SNU)
  • 10:30 AM - 12:00 PM in KIAS 1424
  • Title: (classical) W-algebras and Drinfeld-Sokolov hierarchies
  • Abstract: In 1980's, Drinfeld and Sokolov discovered a connection between a Lie algebra and a family of integrable systems. A W-algebra associated to a Lie algebra is obtained by so called 'Drinfeld-Sokolov reduction' and there are integrable systems which consist of a D-S hierarchy associated to the W-algebra. In this talk, I will explain how to get W-algebras and D-S hierarchies from Lie algebras.
• August 22 (Friday)
  • Suho Oh (Texas State University)
  • 4:00 PM - 5:00 PM in KIAS 1424
  • Title: Fun with wires
  • Abstract: Wiring diagrams are widely used combinatorial objects that are mainly used to describe reduced words of a permutation. In this talk, I will mention a fun property I recently found about those diagrams, and then introduce other results and problems related to this property.
• July 24 (Thursday)
  • Kyungyong Lee (Wayne State University & KIAS CMC)
  • 10:00 AM - 12:00 PM in KIAS 1423
  • Title: Maximal green sequences and upper cluster algebras
  • Abstract: Maximal green sequences are closely related to quantum dilogarithm identities and Bridgeland stability. They also appear in physics, in particular in the context of the complete spectrum of a BPS particle. Very recently it was observed that the existence of a maximal green sequence on a quiver Q may be equivalent to closing the gap between the upper cluster algebra and the ordinary cluster algebra for Q. Building on works of many mathematicians, we settle this question for a large class of quivers. This is joint work with Ilke Canakci and Ralf Schiffler.
• July 15 (Tuesday)
  • Soojin Cho (Ajou University)
  • 4:00 PM - 6:00 PM in KIAS 1424
  • Title: K-theoretic Schur functions
  • Abstract: A combinatorial model called set-valued tableaux for K-theoretic Schur¡¯s S-functions (Grothendieck polynomials) was introduced by A. Buch. Buch also proved the Littlewood-Richardson rule for Grothendieck polynomials and it is reproved by T. Ikeda and T. Shimazaki. Recently, K-theoretic Schur¡¯s P-functions are defined and a combinatorial model called set-valued shifted tableaux is introduced by T. Ikeda and H. Naruse. Littlewood-Richardson rule for K-theoretic Schur¡¯s P-functions is not known yet. In this talk, basics on K-theoretic Schur functions will be discussed in a combinatorial point of view, then a conjecture on the Littlewood-Richardson rule for K-theoretic Schur¡¯s P-functions and the progress to prove the conjecture will be explained; this is an ongoing work jointly with T. Ikeda and M. Nakasuji.
• July 17 (Thursday)
  • Kyungyong Lee (Wayne State University & KIAS CMC)
  • 10:00 AM - 12:00 PM in CMC 8403
  • Title: Introduction to cluster algebras and quiver Grassmannians
  • Abstract: Cluster algebras were discovered by Fomin and Zelevinsky in 2001. Since then, they have been shown to be related to diverse areas of mathematics such as Total positivity, Quiver representations, String theory, Statistical physics, Non-commutative geometry, Teichm\"uller theory, Hyperbolic geometry, Tropical geometry, KP solitons, Integrable systems, Quantum mechanics, Lie theory, Algebraic combinatorics and Poisson geometry. As an introduction, I'll explain how they occur in such a wide variety of branches of mathematics, especially in geometry. The talk will be totally elementary and will not assume any preliminary knowledge of cluster algebras.
• April 24 (Thursday)
  • Se-jin Oh (Seoul National University)
  • 4:00 PM - 6:00 PM in KIAS 1424
  • Title: Quivers and convex orders on the finite simply laced root system
  • Abstract: Let $Q$ be a Dynkin quiver whose underlying graph is a finite simply laced Dynkin diagram. Representation of quiver $Q$ is deeply related to the positive root system $\Phi^+$ associated to $Q$ by Gabriel's theorem. More precisely, for the Auslander-Reiten quiver $\Gamma_Q$ associated to $Q$, it is well-known that (i) the vertices of $\Gamma_Q$ can be identified with the set $\Phi^+$ and the set of all isomorphism classes of indecomposable modules over the path algebra $\C Q$, (ii) arrows in $\Gamma_Q$ present the irreducible morphisms between the indecomposables. On the other hand, $\Gamma_Q$ provides {\it the unique convex partial order} $\prec_Q$ on $\Phi^+$ which is compatible with pathes in $\Gamma_Q$. Recall that a reduced expression of the longest element of Weyl group induces the convex total order on $\Phi^+$. In this talk, I will briefly introduce the Auslander-Reiten quiver $\Gamma_Q$ related to $\Phi^+$ and convex orders $\prec_Q$ compatible with pathes in $\Gamma_Q$ by using easy examples. Observing combinatorial features of $\Gamma_Q$, I will give an application of $\Gamma_Q$ to the finite representations which categorifying $U_q^-(\g)$ for $\g=A_n$ or $D_n$.
• April 7 (Monday)
  • Chul-hee Lee (Seoul National University)
  • 4:00 PM - 6:00 PM in KIAS 1424
  • Title: Bethe ansatz for beginners I
  • Abstract: In 1928, Heisenberg proposed a quantum mechanical model of ferromagnets, which can be considered as the prototypical example of a large class of exactly solvable models. Bethe introduced his now famous Bethe ansatz to obtain the spectrum of the Heisenberg Hamiltonian in 1931. The search for algebraic structures behind his solution and its extensions eventually led to the discovery of quantum groups. The purpose of this lecture is to review some basics of this topic. In Part I, the coordinate Bethe ansatz for the Heisenberg spin 1/2 XXX model will be reviewed. It is Bethe's original approach to diagonalize the Hamiltonian of the model. We will obtain the Bethe ansatz equation and see how counting the number of solutions to this equation under the string hypothesis is related to combinatorial representation theory.
• April 11 (Friday)
  • Chul-hee Lee (Seoul National University)
  • 4:00 PM - 6:00 PM in KIAS 1424
  • Title: Bethe ansatz for beginners II
  • Abstract: In Part II, we will study the algebraic Bethe ansatz for the Heisenberg spin 1/2 XXX model, which is based on the Yang-Baxter equation. Here the Bethe ansatz equation will be re-derived in a more algebraic and conceptual way. We will see how a quantum group (Yangian in this case) emerges as the underlying symmetry of the model.
• February 24 (Monday)
  • Kyoungsuk Park (Ajou University)
  • 4:00 PM - 5:00 PM in KIAS 1424
  • Title: A combinatorial proof of a symmetry of $(t,q)$-Eulerian numbers of type $B$ and $D$
  • Abstract: A symmetry of $(t,q)$-Eulerian numbers of type $B$ is combinatorially proved using permutation tableaux of type $B$; we define a bijection preserving many important statistics on the set of permutation tableaux of type $B$. This bijection proves a symmetry of the generating polynomial $\hat{D}_{n, k}(p,q,r)$ of number of crossings, (two types of) alignments, and hence $q$-Eulerian numbers of type $A$ (defined by L. Williams). By considering the restriction of our bijection on permutations of type $D$, we were lead to define a new statistic on the set of permutations of type $D$ and $(t,q)$-Eulerian numbers of type $D$, which is proved to have a nice symmetry also. We conjecture that our new statistic is in the family of Eulerian statistics for permutations of type $D$.
• January 20 (Monday)
  • 4:00 PM - 5:00 PM in KIAS 1424
  • Speaker: Shuhei Kamioka (Kyoto University)
  • Title: Laurent biorthogonal polynomials, $q$-Narayana polynomials and domino tilings of the Aztec diamonds
  • Abstract: A Toeplitz determinant whose entries are described by a $q$-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schr\"oder paths. As an application, a new proof is given to the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp concerning domino tilings of the Aztec diamonds. The proof is based on the correspondence with non-intersecting Schr\"oder paths developed by Johansson.

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