KAIST Discrete Math Seminar

In this talk, the diagrams of affine permutations and their balanced labellings will be introduced. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the column strict balanced labellings is the affine Stanley symmetric function defined by Lam. The affine Stanley symmetric function is the object of active research in the field of Schubert calculus. It is the affine counterpart of the Stanley symmetric function which is the limit of Schubert polynomials. Our construction is a natural tableau-theoretic realization of this function. We also give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we will give a necessary and sufficient condition for a diagram to be an affine permutation diagram. If time allows, we will introduce some conjectures about when the affine Stanley symmetric functions coincide. This talk is based on the joint work with Taedong Yun.

Weakly separated set families were first studied by Leclerc and Zelevinsky in the context of quantum flag variety. Two quantum Plücker coordinates quasi-commute whenever their indexing sets are weakly separated. It was conjectured that maximal such families always have the same size. Similar question was asked by Scott when she studied quantum Grassmannian. These conjectures were independently proved by Danilov-Karzanov-Koshevoy and Oh-Postnikov-Speyer using some planar graphs and by the author using truncation. In this talk, definitions and motivations for the weakly separated set families will be explained, including Oh-Postnikov-Speyer’s point of view on the subject. The proof of the purity conjecture using truncation will be provided, and related questions will be discussed.

List of speakers

• Tuesday 10:30AM-12PM Seunghyun Seo (서승현), Kangwon National University, Refined enumeration of trees by the size of maximal decreasing trees
• Tuesday 1:30PM-3PM HwanChul Yoo (유환철), KIAS, Specht modules of general diagrams and their Hecke counterparts
• Tuesday 4PM-5:30PM Heesung Shin (신희성), Inha University, q-Hermite 다항식을 포함하는 두 항등식에 관하여
• Wednesday 10:30AM-12PM Soojin Cho (조수진), Ajou University, Skew Schur P-functions
• Wednesday 1:30PM-3PM Sangwook Kim (김상욱), Chonnam National University, Flag vectors of polytopes

In 2006 at MSRI, nine tropical geometers and combinatorialists met and announced the list of ten key open problems in (algebraic and combinatorial side of) tropical geometry. Axiomatization of tropical oriented matroids was one of them. After the work of Develin and Sturmfels, tropical oriented matroids were conjectured to be in bijection with subdivisions of product of simplices as well as with tropical pseudohyperplane arrangements. Ardila and Develin defined tropical oriented matroid, and showed one direction that tropical oriented matroids encode subdivision of product of simplices. Recently, in joint work with Oh, we proved that every triangulation of product of simplices encodes a tropical oriented matroid.

In this talk, I will give a survey on this topic, and discuss this well known conjecture. I will also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes.