### 2016　Special lectures on advanced topics in Mathematics A

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Instructor(s)
Naito Satoshi  Tsuchioka Shunsuke
Course component(s)
Lecture
Day/Period(Room No.)
Intensive (H201)
Group
-
Course number
MTH.E431
Credits
2
2016
Offered quarter
2Q
Syllabus updated
2016/12/14
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

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.

### Student learning outcomes

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 Littlewood-Richardson rule and Kostka numbers.
(6) Understand that representation theory of symmetric groups are closely
related with the ring of symmetric functions.

### Keywords

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

### Competencies that will be developed

Intercultural skills Communication skills Specialist skills Critical thinking skills Practical and/or problem-solving skills
- - - -

### Class flow

This is a standard lecture course. There will be some assignments.

### Course schedule/Required learning

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-Schensted-Knuth 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 Frebenius 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 rule and Kashiwara's crystal structure Details will be provided during each class session.

### Textbook(s)

Bruce E. Sagan, The Symmetric Group, GTM 203,
William Fulton, Young Tableaux, London Mathematical Society, Student Texts 35.

### Reference books, course materials, etc.

Course materials are provided during class.

### Assessment criteria and methods

Assignments (100 %).

### Related courses

• MTH.A302 ： Algebra II
• MTH.A403 ： Advanced topics in Algebra C
• MTH.A404 ： Advanced topics in Algebra D
• MTH.A301 ： Algebra I

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

No prerequisites are necessary, but enrollment in the related courses is desirable.