2019 Advanced topics in Algebra B1

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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.A406
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 topics in Algebra A1".
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

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

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