16th International Symposium on Natural Sciences

The international symposium on natural sciences is an annual event of the research institute of basic sciences at Incheon national university (Republic of Korea). It will be held on Oct 10 (Thu) 2024 (Korea time, UTC+9) via Zoom (Zoom Meeting ID: 842 4954 1762, Passcode: 241010). 

Invited Speakers:

Kyu-Hwan Lee (University of Connecticut)

Katsuyuki Naoi (Tokyo University of Agriculture and Technology)

Naoki Fujita (Kumamoto University)

Euiyong Park (University of Seoul)

Titles and Abstracts (Please note that the timetable is based on UTC (+9)):

1. Kyu-Hwan Lee (University of Connecticut)
   Title: Data-scientific study of Kronecker coefficients
   Abstract:  Kronecker coefficients are decomposition multiplicities of the tensor product of two irreducible representations of a symmetric group. They are hard to compute. Determining nonzero Kronecker coefficients is NP-hard, and there are no combinatorial descriptions of these coefficients. In this talk, we take a data-scientific approach to study whether Kronecker coefficients are zero or not. Motivated by principal component analysis and kernel methods, we define loadings of partitions and use them to describe a sufficient condition for Kronecker coefficients to be nonzero. The results provide new methods and perspectives for the study of these coefficients.
   
   

2. Katsuyuki Naoi (Tokyo University of Agriculture and Technology)
   Title: Generalization of the extended T-systems via strong duality data
   Abstract:  It is an interesting (and difficult) problem to study the tensor product of two finite-dimensional simple modules over a quantum affine algebra. The extended T-systems, introduced by Mukhin–Young, are short exact sequences satisfied by the tensor product of simple modules in types A and B (called snake modules), which contains the celebrated T-systems. In this talk, motivated by the generalization of the T-systems by Kashiwara–Kim–Oh–Park, we introduce a generalization of the extended T-systems using a strong duality datum, which is a family of simple modules with some properties. When we take a strong duality datum consisting of fundamental modules,  this recovers the original Mukhin–Young’s extended T-systems.
   
   

3. Naoki Fujita (Kumamoto University)
   Title: Schubert calculus on polytopes and semi-toric degenerations arising from cluster algebras
   Abstract:  One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the Gelfand-Tsetlin polytopes, Kiritchenko-Smirnov-Timorin realized each Schubert class as a sum of reduced (dual) Kogan faces. In this talk, we discuss its generalization to string polytopes in representation theory and to Newton-Okounkov polytopes of flag varieties. Newton-Okounkov polytopes of flag varieties arising from cluster algebras induce semi-toric degenerations of Schubert varieties, which can be expected as combinatorial models of Schubert classes.
   
   

4. Euiyong Park (University of Seoul)
   Title:  PBW theory for Bosonic extensions
   Abstract: In this talk, we will talk about the PBW theory for the bosonic extension $\widehat{A}_{\mathfrak{g}}$ of a quantum group $U_q(\mathfrak{g})$. When $U_q(\mathfrak{g})$ is of finite simply-laced  type, the algebra $\widehat{A}_{\mathfrak{g}}$ is isomorphic to the quantum Grothendieck ring of the Hernandez-Leclerc category over a quantum affine algebra. We introduce PBW vectors and PBW monomials using the braid group actions on $\widehat{A}_{\mathfrak{g}}$, and define a new family of subalgebras, denoted by $\widehat{A}_{\mathfrak{g}}(b)$, for any element $b$ in the (generalized) Braid group corresponding to $\mathfrak{g}$. The algebras $\widehat{A}_{\mathfrak{g}}(b)$ can be understood as a natural extension of quantum unipotent coordinate rings $A_q(\mathfrak{n}(w))$, and the PBW monomials form an orthogonal basis of $\widehat{A}_{\mathfrak{g}}(b)$. This is a joint work with Masaki Kashiwara, Myungho Kim and Se-jin Oh.

last updated: October 7, 2024