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About
President
Governance
Partner Institutions
Visit
People
Management
Faculty
Postdocs
Visiting Scholars
Administration
Academic Support
Research
Research Groups
Courses
Seminars
Join Us
Faculty
Postdocs
Students
Events
Conferences
Workshops
Forum
Life @ BIMSA
Accommodation
Transportation
Facilities
Tour
News
News
Announcement
Downloads
Qiuzhen College, Tsinghua University
Yau Mathematical Sciences Center, Tsinghua University (YMSC)
Tsinghua Sanya International  Mathematics Forum (TSIMF)
Shanghai Institute for Mathematics and  Interdisciplinary Sciences (SIMIS)
BIMSA > Symmetric groups and symmetric and quasisymmetric functions
Symmetric groups and symmetric and quasisymmetric functions
Lecturer
Anatoli Kirillov
Date
9th October ~ 26th December, 2025
Location
Weekday Time Venue Online ID Password
Thursday 13:30 - 15:05 A3-1-103 ZOOM B 462 110 5973 BIMSA
Friday 14:20 - 16:05 A3-1-103 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.
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.
Beijing Institute of Mathematical Sciences and Applications
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