### 2019　Advanced courses in Algebra A

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Mathematics
Instructor(s)
Naito Satoshi
Course component(s)
Lecture
Day/Period(Room No.)
Thr5-6(H137)
Group
-
Course number
ZUA.A331
Credits
1
2019
Offered quarter
1Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
English
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### 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

Intercultural skills Communication skills Specialist 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.

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

None.

### Other

Do not hesitate to ask any questions.