2019 Advanced topics in Algebra A1

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Naito Satoshi 
Course component(s)
Lecture
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
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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.

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.

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