KIAS Workshop on Combinatorial Problems of Algebraic Origin
July 14 - 15, 2020 |

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**June Huh (KIAS/Stanford University)**

Title: Kazhdan-Lusztig polynomials of matroids

Abstract: I will introduce Kazhdan-Lusztig polynomials of matroids and survey combinatorial and geometric theories built around them. The focus will be on the conjecture of Gedeon, Proudfoot, and Young that all zeros of the Kazhdan-Lusztig polynomial of a matroidlie on the negative real axis.

**Dongkwan Kim (University of Minnesota) Lecture Note**

Title: Robinson-Schensted correspondence for unit interval orders

Abstract: Stanley-Stembridge conjecture, currently one of the most famous conjectures in algebraic combinatorics, asks whether a certain generating function with respect to a natural unit interval order is a nonnegative linear combination of complete homogeneous symmetricfunctions. There are many partial progress on this conjecture, including its connection with the geometry of Hessenberg varieties. Here, instead we study its Schur positivity, which is originally proved by Haiman and Gasharov. We define an analogue of Knuthmoves with respect to a natural unit interval order and study its equivalence classes in terms of D graphs introduced by Assaf. Then, we show that if the given order avoids certain two suborders then an analogue of Robinson-Schensted correspondence is well-defined,which proves that the generating function attached to each equivalence class is Schur positive. It is hoped that it proposes a new combinatorial aspect to investigate the Stanley-Stembridge conjectures and cohomology of Hessenberg varieties. This work is jointwith Pavlo Pylyavskyy.

**Myungho Kim (Kyunghee University)**

Title : Polynomial invariants on matrices and directed multigraphs

Abstract: In 2005, J. F Willenbring discovered that the Hilbert series of the conjugation action of the orthogonal group on the set of n x n matrices has a stable limit. Moreover, the d-th coefficient of the stable Hilbert series can be identified with the number of disjoint union of unlabeled directed cycles whose total number of edges are equal to d. When a subgroup G of the general linear group acts on the set of n x n matrices by conjugation, we show that the dimension of the space of homogeneous polynomial invariants of degree d is equal to the dimension of a certain subalgebra of the algebra of linear endomorphisms on the d-th tensor power of the natural representation. It provides another way to explain Willenbring’s stability of Hilbert series, where the centralizer of the symmetric group in the Brauer algebra plays a central role. I will explain how one can relate the directed cycles (resp. multidigraphs) with the Brauer diagrams (resp. partition diagrams) in this picture. Some questions, related but not yet answered, will be presented. This is a joint work with Doyun Koo.

**Woong Kook (Seoul National University)**

Title: Tree numbers of rank selected posets

Abstract: Inspired by high-dimensional matrix-tree theorem, tree numbers of simplicial and cell complexes that arise in combinatorics became an active research topic. One can also find a motivation for this subject in high-dimensional Kirchhoff's laws as mathematical foundations of simplicial network analysis. In this talk, we will suggest computation of tree numbers of rank selected posets as an elementary problem for both pure and practical purposes. Time permitting, previous results and open questions regarding tree numbers of matroid complexes will be outlined as a possible approach

**Kyu-Hwan Lee (University of Connecticut) Lecture Note**

Title: L-matrices of quiver mutations

Abstract: We define a family of reflections along with associated vectors for each mutation sequence of an arbitrary finite quiver and show that these vectors coincide with the c-vectors. This new presentation of c-vectors reveals some essential features in mutations of (non-acyclic) quivers and leads us to define an L-matrix for each mutation sequence. In this talk, some conjectures on L-matrices will be presented. This is a collaboration with Kyungyong Lee and Matthew Mills.

**Li Li (Oakland University) Lecture Note**

Title: Singularities of Schubert varieties and Nakajima's quiver varieties

Abstract: Special varieties including Schubert varieties and Nakajima's singular quiver varieties play important roles in the intersection of algebra, geometry, representation theory and combinatorics. Their singularities are often suitably mild and can bestudied using combinatorial methods. I will talk about some problems on combinatorial objects such as Young tableaux, pipe dreams, and non-intersecting paths that are related to these varieties.

**Dinakar Muthiah (IPMU) Lecture Note**

Title: Double-affine Bruhat order

Abstract: In the course of studying Iwahori-Hecke algebras for p-adic loop groups, Braverman, Kazhdan, and Patnaik defined a certain order, the double affine Bruhat order, that generalizes the Bruhat order and affine Bruhat order (for finite and affine Weyl groups). Conjecturally this order should describe the geometry of double affine Schubert varieties in the double affine flag variety. I will discuss progress that has been made in understanding the combinatorics of this order. However, many fundamental questions remain, which I will explain. This involves joint work with Dan Orr.

**Se-Jin Oh (Ewha Womens University) **

Title: Reduced expressions, their grouping and applications

Abstract: In this talk, I will introduce several ways of grouping for reduced expressions of Weyl groups. The notions themselves are quite simple and easy, but their applications are universe. For instance, special commutation classes tell lots of information on representation theory over quantum affine algebras and quiver Hecke algebra, quantum cluster algebras, interestingly. Also the applications are closely related to root system of finite type. This talk is based on the my recent works with Kashiwara-Kim-Park, Fujita and Scrimshaw.

**Sug Woo Shin (KIAS/University of California Berkeley)**

Title: a combinatorial problem arising from Tunnell’s theorem

Abstract: Let n be an odd integer.

In 1983, Tunnell proved that if *n* is a congruent number, i.e. the area of a right triangle with rational side lengths, then the number of triplets of integers (*x*, *y*, *z*) satisfying 2*x*2 + *y*2 + 8*z*2 = *n* is twice the number of triplets satisfying 2*x*2 + *y*2 + 32*z*2 = *n*. (An interesting problem is whether one can see this combinatorially.) The converse is true if the Birch and Swinnerton-Dyer Conjecture is assumed. He also proved an analogue when n is even. In this talk, we introduce the congruent number problem and the proof of Tunnell’s theorem.

**Hideya Watanabe (RIMS/Kyoto University) Lecture Note**

Title: Combinatorial representation theory arising from quantum symmetric pairs

Abstract: Combinatorics and representation theory are closely related to each other. It often happens that a problem in one side is solved by means of techniques in the other side. A typical example is Kashiwara's crystal bases. Crystal basis theory, which originates from the representation theory of quantum groups, provides us a combinatorial model for representations of complex semisimple Lie algebras. From a view point of combinatorics, one can see many properties of representations just by investigating certain colored directed graph even if he know nothing about the representation theory. In this talk, I will introduce a new combinatorial model for representations of the orthogonal and the symplectic Lie algebras which arises from quantum symmetric pairs of type AI and AII.

**Meesue Yoo (Chungbuk National University) Lecture Note**

Title: Schur expansion of LLT polynomials related to certain graphs

Abstract: LLT polynomials are a family of symmetric functions introduced by Lascoux, Leclerc and Thibon in 1997 which naturally arise in the description of the power-sum plethysm operators on symmetric functions. Grojnowski and Haiman proved that they are Schur positive using Kazhdan-Lusztig theory, but there is no Known combinatorial formula for the Schur coefficients. In this talk, we utilize linear relations introduced by Lee to prove some combinatorial formulas for the Schur coefficients of LLT polynomials, when the LLT polynomials are indexed by certain diagrams related to particular type of graphs. This is joint work with Jisun Huh and Sun-Young Nam.