Symmetric groups and symmetric and quasisymmetric functions
讲师
日期
2025年10月09日 至 12月26日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周四 | 13:30 - 15:05 | A3-1-103 | ZOOM B | 462 110 5973 | BIMSA |
| 周五 | 14:20 - 16:05 | A3-1-103 | ZOOM B | 462 110 5973 | BIMSA |
修课要求
Basic knowledge of representation theory and graph theory will be useful.
课程大纲
【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.
视频公开
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笔记公开
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语言
英文
讲师介绍
Anatol Kirillov的研究领域是可积系统、表示论、特殊函数、代数组合学和代数几何。他在过去的20年里，在日本不同的大学担任教授。2022年，他加入BIMSA任研究员。