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- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Naito Satoshi
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Thr5-6(H137)
- Group
- -
- Course number
- MTH.A405
- Credits
- 1
- Academic year
- 2019
- Offered quarter
- 1Q
- Syllabus updated
- 2019/3/18
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
In representation theory, one of the most important problems is to give a (good) basis for each irreducible representation, which enables us to obtain an explicit formula for its character.
In this course, we explain a combinatorial model of finite-dimensional, irreducible (highest weight) reprersentations of complex finite-dimensional semi-simple Lie algebras; this model is called Littelmann's path model.
The aim of this course is to give an explicit combinatorial parametrization of a certain good basis of each finite-dimensional, irreducible representation of a complex finite-dimensional semi-simple Lie algebra.
Student learning outcomes
There exists a one-to-one correspondence between the set of equivalence classes of finite-dimensional irreducible highest weight representations of a complex finite-dimensional semi-simple Lie algebra and the set of dominant integral weights.
The goal of this course is become able to write down explicitly all the Lakshmibai-Seshadri (LS) paths of an arbitrary fixed shape (or, dominant integral weight), which indexes a certain good basis of the finite-dimensional irreducible representation with the given highest weight; here an LS path is a certain combinatorial object, which is described in terms of root systems and Weyl groups of semi-simple Lie algebras.
Keywords
complex semi-simple Lie algebra, irreducible highest weight representation, crystal basis, Lakshmibai-Seshadri path, Littelmann's path model
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 | Complex semi-simple Lie algebras and their root systems | Details will be provided during each class session. |
| Class 2 | Weyl groups and the Bruhat order | Details will be provided during each class session. |
| Class 3 | Action of root operators (Kashiwara operators) on paths | Details will be provided during each class session. |
| Class 4 | Properties of root (Kashiwara operators) operators | Details will be provided during each class session. |
| Class 5 | Lakshmibai-Seshadri (LS) paths | Details will be provided during each class session. |
| Class 6 | Properties of LS paths | Details will be provided during each class session. |
| Class 7 | Action of root operators on LS paths | Details will be provided during each class session. |
| Class 8 | Littelmann's path model | 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.
Related courses
- MTH.A203 ： Introduction to Algebra III
- MTH.A204 ： Introduction to Algebra IV
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None.
Other
Do not hesitate to ask any questions.
