- Docente: Riccardo Biagioli
- Credits: 6
- SSD: MAT/02
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Mathematics (cod. 5827)
-
from Feb 22, 2024 to May 31, 2024
Learning outcomes
At the end of the course, the student knows the representation theory of the symmetric group and the associated combinatorial objects as Young tableaux and symmetric functions. He is able to apply this knowledge for studying the general linear group and Schubert varieties.
Course contents
The representation theory of the symmetric group can be examined from different points of view: by using the general representation theory of finite groups, by applying combinatorial methods, or by employing symmetric functions. In this course, important results coming from these three directions will be examined. In particular, we will stress the combinatorial aspects, giving importance to explicit constructions and algorithmic proofs.
Course programme.
- Introduction to the representation theory of finite groups.
- Character theory.
- Construction of the irreducible representations of the symmetric group.
- Combinatorics of Young tableaux.
- The Robinson-Schensted algorithm. Knuth relations and jeu de Taquin.
- Introduction to generating functions.
- The Frobenious characteristic.
- The Littlewood-Richardson rule.
- Combinatorics of permutations.
Complementary information
The course does not need prerequisites. Elementary notions of group theory and linear algebra will be enough.
The program of this course is independent but related to that of Representation Theory (96759) and Group Theory (09346).
The courses Algebraic Combinatorics and Group Theory are organized on alternate years. The order in which they are attended is not important.
Readings/Bibliography
- Bruce E. Sagan. The Symmetric Group. Representations, Combinatorial Algorithms, and Symmetric Functions. Graduate Texts in Mathematics 203. Springer.
- Richard. P. Stanley. Enumerative Combinatorics 2. Cambridge University Press.
- William Fulton. Young Tableaux. With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts 35. Cambridge University Press.
Teaching methods
Front lectures.
Assessment methods
Oral exam and evaluation of exercises given during the course. The student may replace the oral exam with a presentation of a research paper on a subject related to the course.
Office hours
See the website of Riccardo Biagioli