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- Academic unit or major
- Mathematics
- Instructor(s)
- Naito Satoshi
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Thr5-6(H137)
- Group
- -
- Course number
- ZUA.A332
- Credits
- 1
- Academic year
- 2019
- Offered quarter
- 2Q
- Syllabus updated
- 2019/3/18
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
This course is the continuation of "Advanced courses in Algebra A".
In representation theory, one of the most important problems is to give an explicit formula for characters of irreducible representations.
The aim of this course is to give some important applications of Littelmann's path model to the representation theory of finite-dimensional complex semi-simple Lie algebras; in particular, we explain a character formula for finite-dimensional irreducible (highest weight) representations, which is described in terms of Lakshmibai-Seshadri (LS) paths.
Student learning outcomes
The goal of this course is become able to write down explicitly characters of finite-dimensional irreducible (highest weight) representations of complex finite-dimensinal semi-simple Lie algebras, by using Lakshmibai-Seshadri (LS) paths.
Keywords
semi-simple Lie algebra, irreducible highest weight representation, crystal basis, Lakshmibai-Seshadri path, character formula
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture course.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Abstract crystals | Details will be provided during each class session. |
| Class 2 | Tensor product rule for crystals | Details will be provided during each class session. |
| Class 3 | Action of the Weyl group on LS paths | Details will be provided during each class session. |
| Class 4 | Weyl's character formula | Details will be provided during each class session. |
| Class 5 | Combinatorial character formula in terms of LS paths | Details will be provided during each class session. |
| Class 6 | Proof of the combinatorial character formula | Details will be provided during each class session. |
| Class 7 | Littlewood-Richardson rule | Details will be provided during each class session. |
| Class 8 | PRV conjecture and its proof | Details will be provided during each class session. |
Textbook(s)
None.
Reference books, course materials, etc.
M. Kashiwara, Bases cristallines des groupes quantiques, Cours Specialises, Vol. 9, SMF.
Assessment criteria and methods
Based on evaluation of assignments. Details will be announced during each class session.
Related courses
- MTH.A203 ： Introduction to Algebra III
- MTH.A204 ： Introduction to Algebra IV
- MTH.A301 ： Algebra I
- MTH.A204 ： Introduction to Algebra IV
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None
Other
Do not hesitate to ask any questions.
