# KCL2023.07

KAIST Combinatorics Lab. Workshop

It is a triannual (three-times-a-year) seminar organized by academic genealogy from Professor Dongsu Kim in KAIST Combinatorics Laboratory, which KCL stands for. The aim of this seminar is to bring together active combinatorialists to discuss recent and prospective advances in algebraic and enumerative combinatorics and related areas. The style of talk tends to be less formal.

### Information

Title KAIST Combinatorics Lab. Workshop 2023.07 (KCL2023.07)

Date July 10 (Monday), 2023

Venue Sangyeonjae - Main building (상연재 본관), Seoul

### Program

July 10 (Monday)

13:50 - 14:00 Registration

14:00 - 15:00 Invited Talk

Speaker Ae Ja Yee

Title The Rogers–Ramanujan Identities and Ariki–Koike Algebras

Abstract In 2000, Ariki and Mathas showed that the simple modules of the Ariki–Koike algebras ℋℂ, v; Q1, …, Qm(G(m,1,n)) (when the parameters are roots of unity and v ≠ 1) are labeled by the so-called Kleshchev multipartitions. This together with Ariki’s categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl–Kac character formula. In this talk, I will revisit this generating function for v = Q1 = ⋯ = Qa = − 1, Qa + 1 = ⋯ = Qm = 1. This case is particularly interesting, for the corresponding Kleshchev multipartitions have a very close connection to the Rogers–Ramanujan identities. I will discuss an analytic proof of this generating function.

The second objective of this talk is to count the number of simple modules of the Ariki–Koike algebra in a fixed block. It is known that the simple modules of the Ariki–Koike algebras in a fixed block are labeled by the Kleshchev multiparitions with a fixed partition residue statistic. This partition statistic was also studied in the works of Berkovich, Garvan, and Uncu. Employing their results, we can get two bivariate generating function identities for m = 2. I will also discuss these identities.

This talk is based on joint work with S. Chern, Z. Li, D. Stanton, and T. Xue.

15:00 - 15:20 Coffee Break

15:20 - 17:00 Lightning Talks (Up to 10 minutes each for all participants)

17:00 - 20:00 Dinner (Mad for Garlic, Gwanghwamun D Tower)

### Participants

Dongsu Kim, KAIST

Ae Ja Yee, The Pennsylvania State University

Seunghyun Seo, Kangwon National University

Heesung Shin, Inha University

Jang Soo Kim, Sungkyunkwan University

Jihyeug Jang, Sungkyunkwan University

Taehyun Eom, KAIST

Sangwook Kim, Chonnam National University

Meesue Yoo, Chungbuk National University

Minho Song, Sungkyunkwan University

JiSun Huh, Ajou University

Donghyun Kim, Sungkyunkwan University

Cheolwon Heo, KIAS

### Organizer

Seunghyun Seo, Kangwon National University