Representation theory is a field of mathematics for researching symmetry.
This course deals with the (normal) linear representation of groups. In particular, we will cover representation theory for symmetric groups from the following three perspectives.
- Generalized theory of representation theory for groups and rings
- Symmetric functions
- Combinatorics
Through specific examples of representation theory of symmetric groups, students will gain an understanding of the following three topics.
- Students will deepen their understanding of generalized representation theory.
- Students will realize that apparently separate fields (in this case, representation theory of symmetric groups, and symmetric functions) in mathematics are actually closely related.
- Students will become familiar with the elementary material of algebraic combinatorics, learning to actually calculate quantities associated with representation theory of symmetric groups.
By the end of this course, students will be able to:
(1) Understand that the (ordinary) irreducible representations of symmetric groups
are parametrized by partitions.
(2) Understand various ways of constructing irreducible representations of symmetric groups.
(3) Make use of Young diagrams and related combinatorial algorithms.
(4) Be familiar with the ring of symmetric functions and Schur functions.
(5) Make use of the Littlewood-Richardson rule and Kostka numbers.
(6) Understand that representation theory of symmetric groups are closely related with
the ring of symmetric functions.
Symmetric groups, representation theory, characters, symmetric functions,
Littlewood-Richardson rule, Kostka number, Robinson-Schensted-Knuth correspondence,
Young diagrams, Young tableaux, Specht modules, hook length formula,
Frobenius character formula, Maschke's theorem. plactic monoid, categorification
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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- | - | ✔ | - | - |
This is a standard lecture course.
There will be some assignments.
Course schedule | Required learning | |
---|---|---|
Class 1 | Definition of linear representations of groups and construction of group algebras | Details will be provided during each class session. |
Class 2 | Characters of group representations and their orthogonality | Details will be provided during each class session. |
Class 3 | Maschke's theorem and Wedderburn's theorem | Details will be provided during each class session. |
Class 4 | Induced representations and Frobenius reciprocity | Details will be provided during each class session. |
Class 5 | Introduction to Young diagrams | Details will be provided during each class session. |
Class 6 | A construction of irreducible representations via Young symmetrizers | Details will be provided during each class session. |
Class 7 | Young modules and Specht modules | Details will be provided during each class session. |
Class 8 | Jeu de taquin and plactic monoid | Details will be provided during each class session. |
Class 9 | Robinson-Schendted-Kunuth correspondence and hook length formula | Details will be provided during each class session. |
Class 10 | Symmetries related with Robinson-Schensted-Knuth correspondence | Details will be provided during each class session. |
Class 11 | Littlewood-Richardson rule and Kostka number | Details will be provided during each class session. |
Class 12 | The ring of symmetric functions and Schur functions | Details will be provided during each class session. |
Class 13 | Frobenius character formula and Pieri rule | Details will be provided during each class session. |
Class 14 | Okounkov-Vershik's approach I: the degenerate affine Hecke algebra of type A | Details will be provided during each class session. |
Class 15 | Okounkov-Vershik's approach II: branching rules and Kashiwara's crystal structure | Details will be provided during each class session. |
Btuce E. Sagan, The Symmetric Group, GTM 230,
William Fulton, Young Tableaux, London Mathematical Society, Student Texts 35.
Course materials are provided during class.
Assignments (100 %).
No prerequisites are necessary, but enrollment in the related courses is desirable.