授業の目的 【日本語】
Goals of the Course(JPN) | | | The purpose of the course is to give an introduction to representation theory, in general, and to complex representations of compact Lie groups, in particular. |
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授業の目的 【英語】
Goals of the Course | | |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | | The goal is to develop a good understanding of basic concepts and results in representation theory. Highlights include the structure of finite dimensional complex representations of symmetric groups and of the compact Lie groups SU(2) and SO(3), as well as the Peter-Weyl theorem and basic notions related to Lie groups. |
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到達目標 【英語】
Objectives of the Course | | |
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授業の内容や構成
Course Content / Plan | | Here is a preliminary list of the content of each of the lectures in the course:
Lecture 1: Introduction and examples.
Lecture 2: Complete reducibility and semisimplicity.
Lecture 3: Unitarity of finite dimensional complex representations. Lecture 4: Dual representation, tensor product of representations. Lecture 5: Extension and restriction of scalars.
Lecture 6: Schur's lemma and its applications.
Lecture 7: Character theory for finite groups.
Lecture 8: Transitive group actions. Schur orthogonality.
Lecture 9: Six-functor formalism for QCoh([G\X]).
Lecture 10: Induction and restriction.
Lecture 11: Representations of symmetric groups.
Lecture 12: The classical groups.
Lecture 13: The Peter-Weyl theorem.
Lecture 14: Smooth manifolds.
Lecture 15: Lie groups. |
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履修条件
Course Prerequisites | | | A good knowledge of linear algebra is essential. Some knowledge of abstract algebra and point-set topology is helpful. |
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関連する科目
Related Courses | | | Representation theory is a mathematical way of encoding symmetries. It is used broadly in mathematics, physics, chemistry, and other fields. |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Grading will based on weekly problem sets. |
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不可(F)と欠席(W)の基準
Criteria for "Fail (F)" & "Absent (W)" grades | | |
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参考書
Reference Book | | The following book is both more specific and more advanced. However, it is very well-written and comprehensive.
Jean-Pierre Serre. Complex semisimple Lie algebras. Translated from the French by G. A. Jones. Reprint of the 1987 edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2001. x+74 pp. ISBN: 3-540-67827-1. |
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教科書・テキスト
Textbook | | | Ernest B. Vinberg. Linear representations of groups. Translated from the 1985 Russian original by A. Iacob. Reprint of the 1989 translation. Modern Birkh?user Classics. Birkh?user/Springer, New York, 2010. vii+146 pp. ISBN: 978-3-0348-0062-4. |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | |
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注意事項
Notice for Students | | |
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他学科聴講の可否
Propriety of Other department student's attendance | | |
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他学科聴講の条件
Conditions for Other department student's attendance | | |
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レベル
Level | | |
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キーワード
Keyword | | | Representations, six-functor formalism, the Peter-Weyl theorem, Lie groups, SU(2). |
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履修の際のアドバイス
Advice | | | The English version of Wikipedia is a great source. It is both quite reliable and often gives good examples that are particularly useful for non-expert readers. |
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授業開講形態等
Lecture format, etc. | | |
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遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | |
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