Symmetric groups and symmetric and quasisymmetric functions
Lecturer
Date
9th October, 2025 ~ 8th January, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday,Friday | 15:20 - 16:55 | A3-1-101 | ZOOM B | 462 110 5973 | BIMSA |
Prerequisite
Basic knowledge of representation theory and graph theory will be useful.
Syllabus
【Lectures 1-3】Symmetric group: definitions, presentation and representations. Transpositions and Moore-Coxeter, Hurwitz, Star, ..., presentations.
【Lectures 4-6】Coxeter relations, Jucys-Murphy elements; ring of polynomials, symmetric functions: monomial, elementary and complete, power sums. Quasisymmetric functions.
【Lectures 7-9】Short overview of representation theory of finite groups. Diagrams, partitions, tableaux. Standard, semistandard Young tableaux and classical representations of a symmetric group.
【Lectures 10-12】Schur symmetric functions. Cauchy identities, Robinson-Schensted-Knuth (RSK) correspondence. Divided differences and associative Yang-Baxter relations.
【Lectures 13-15】Irreducible representations of a symmetric group, characters and Schur functions.
【Lectures 16-18】Quasisymmetric functions: monomial, fundamental (a.k.a. Gessel) functions. Compositions: weak and standard. Algebra of QSym. Examples.
【Lectures 19-21】Quasisymmetric action of a symmetric group (F. Hivert). Quasisymmetric invariants and coinvariants. Hopf algebras of symmetric and quasisymmetric groups.
【Lectures 22-25】Part 1. CLassical case: nilcoxeter, o-Hecke plactic and hyperplactic monoids and associated polynomials: Schur, Schubert, Grothendieck, Demazure, Key, plactic (hyperplactic) Grothendieck polynomials. Part 2. Quasisymmetric divivded difference operators, two definitions.
【Lecture 25-27】Summary of the course, conjectures, open problems and potential directions for study of QSym and their generalizations.
【Lectures 4-6】Coxeter relations, Jucys-Murphy elements; ring of polynomials, symmetric functions: monomial, elementary and complete, power sums. Quasisymmetric functions.
【Lectures 7-9】Short overview of representation theory of finite groups. Diagrams, partitions, tableaux. Standard, semistandard Young tableaux and classical representations of a symmetric group.
【Lectures 10-12】Schur symmetric functions. Cauchy identities, Robinson-Schensted-Knuth (RSK) correspondence. Divided differences and associative Yang-Baxter relations.
【Lectures 13-15】Irreducible representations of a symmetric group, characters and Schur functions.
【Lectures 16-18】Quasisymmetric functions: monomial, fundamental (a.k.a. Gessel) functions. Compositions: weak and standard. Algebra of QSym. Examples.
【Lectures 19-21】Quasisymmetric action of a symmetric group (F. Hivert). Quasisymmetric invariants and coinvariants. Hopf algebras of symmetric and quasisymmetric groups.
【Lectures 22-25】Part 1. CLassical case: nilcoxeter, o-Hecke plactic and hyperplactic monoids and associated polynomials: Schur, Schubert, Grothendieck, Demazure, Key, plactic (hyperplactic) Grothendieck polynomials. Part 2. Quasisymmetric divivded difference operators, two definitions.
【Lecture 25-27】Summary of the course, conjectures, open problems and potential directions for study of QSym and their generalizations.
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Anatol Kirillov is a researcher in the area of integrable systems, representation theory, special functions, algebraic combinatorics, and algebraic geometry. He worked as a professor in different universities in Japan for the last 20 years. In 2022 he joined BIMSA as a research fellow.