授業の目的 【日本語】
Goals of the Course(JPN) | | | 有限群の表現論の基礎を紹介し、主に複素数体上の場合。特に、基本な指標理論の理解が含まれます。後半では、モジュラー表現論に移行するか、有限対称群とその関連する組み合わせ論に特化します。 |
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授業の目的 【英語】
Goals of the Course | | | Introduce basics of the representation theory of finite groups, focusing mainly on the 'ordinary' case where the underlying field is the complex numbers. The first main theme of the course is to explain why this ordinary case is particularly nice to work with. This includes understanding of ordinary character theory. In the latter part, we will move into the modular setting, or specialise in the case of finite symmetric groups. |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | -Understand basic properties of group algebras.
-Be able to translate between representations, characters, and modules.
-Learn to calculate ordinary characters of finite groups of small orders.
-Learn how modular setting is different from the ordinary case, or, how to work with combinatorics related to representations of the symmetric groups. |
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到達目標 【英語】
Objectives of the Course | | -Understand basic properties of group algebras.
-Be able to translate between representations, characters, and modules.
-Learn to calculate ordinary characters of finite groups of small orders.
-Learn how modular setting is different from the ordinary case, or, how to work with combinatorics related to representations of the symmetric groups. |
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授業の内容や構成
Course Content / Plan | | Part I - Basic:
Representations of groups and their homomorphisms, comparison with modules.
Maschke's theorem, tensor and dual, induction and restriction, permutation representations
Part II - Character theory
Characters of group, the space they form, orthogonalities, hands-on calculation
Part III - Modular group representations, or, representations of the symmetric groups
(a) Modular case: Frobenius algebra, symmetric algebra. Non-semisimplicity. Basic block theory
(b) Representations of symmetric groups: Partitions, Young tableaux, tabloid, Specht modules, Schur-Weyl duality |
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履修条件
Course Prerequisites | | | Linear algebra is necessary. Basic knowledge of groups, rings, and modules. |
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関連する科目
Related Courses | | |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | |
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教科書・テキスト
Textbook | | | Will be prepared as the course goes on. |
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参考書
Reference Book | | G. James & M. Liebeck: Representations and characters of groups 2nd ed, Cambridge University Press 2001
A. Zimmermann: Representation Theory: A Homological Algebra Point of View, Algebra and Applications 19, Springer 2014
K. Erdmann, T. Holm: Algebras and representation theory. Springer 2018
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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 | | | Any student is welcome to attend the course |
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他学科聴講の条件
Conditions of Other department student's attendance | | | See necessary material (履修条件) |
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レベル
Level | | |
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キーワード
Keyword | | | representation theory, finite group, ordinary representations, ordinary character theory, rings and modules, symmetric groups |
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履修の際のアドバイス
Advice | | The course is a good opportunity to practice listening to and doing mathematics in English.
この科目は英語での数学を使う機会があって，学生さんが挑戦してみてほしいです． |
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授業開講形態等
Lecture format, etc. | | | The course will be in-person. If necessary, online and remote methods will also be implemented. |
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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) | | | If need arises, lectures will be recorded or held online. |
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