96730 - Algebraic Combinatorics

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. 

The course does not need special prerequisites. Standard notions of groups theory and linear algebra will be sufficient. 

Course programme.

1. Introduction to the representation theory of finite groups.
2. Character theory.
3. Construction of the irreducible representations of the symmetric group.
4. Combinatorics of Young tableaux.
5. The Robinson-Schensted algorithm. Knuth relations and jeu de Taquin.
6. Introduction to generating functions.
7. The Frobenious characteristic.
8. The Littlewood-Richardson rule.
9. Combinatorics of permutations.

Students interested to study in deep detail other aspects of representation theory are invited to follow, this or the next year, the course Representation Theory (96759) that will treat Lie algebras and their representations. Algebraic combinatorics (96730), proposed every two years, and Representation Theory (96759) proposed every year, have compatible timetable this semester.

Readings/Bibliography

1. Bruce E. Sagan. The Symmetric Group. Representations, Combinatorial Algorithms, and Symmetric Functions. Graduate Texts in Mathematics 203. Springer.
2. Richard. P. Stanley. Enumerative Combinatorics 2. Cambridge University Press.
   
3. 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