### 2016　Advanced courses in Algebra D

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Mathematics
Instructor(s)
Naito Satoshi
Course component(s)
Lecture
Day/Period(Room No.)
Tue3-4(H137)
Group
-
Course number
ZUA.A334
Credits
1
2016
Offered quarter
4Q
Syllabus updated
2016/4/27
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

The main subjects of this course are the notion and some of the basic properties of group representations. In this course, we first introduce the notions of irreducible representations of groups and endomorphism algebras of group representations, and then explain Schur's lemma, which is an important result closely related to these notions. Next, we introduce group characters, and then explain the orthogonality relations for them. Also, we explain the irreducible decomposition of the group algebra of a finite group. Finally, we introduce tensor products and induced representations for group representations, and then explai n the Frobenius reciprocity, which is an important result closely related to these notions. This course is a continuation of "Advanced courses in Algebra C" in the third quarter.
The therory of group representations is not just a typical example of the general theory of modules over a ring, but also has a wide variety of applications in physics and chemistry other than mathematics. The aim of this course is to make students familiar with basic methods in group representations and enable them to make use of these correctly.

### Student learning outcomes

By the end of this course, students will be able to:
1) Understand the notions of irreducible representations of groups and endomorphism algebras of group representations, and make use of Schur's lemma.
2) Explain the definition of group characters, and make use of the orthogonality relations for them correctly.
3) Understand the irreducible decomposition of the group algebra of a finite group.
4) Understand the notions of tensor products and induced representations for group representations, and make use of the Frobenius reciprocity.

### Keywords

Schur's lemma, Maschke's theorem, group characters, orthogonality relations for group characters, irreducible decomposition of the group algebra, tensor products of group representations, induced representations, Frobenius reciprocity

### 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 Schur's lemma and Maschke's theorem Details will be provided during each class session.
Class 2 Commutant and endomorphism algebras Details will be provided during each class session.
Class 3 Characters of group representations Details will be provided during each class session.
Class 4 Inner products of characters and orthogonality relations Details will be provided during each class session.
Class 5 Decomposition of the group algebra of a finite group Details will be provided during each class session.
Class 6 Tensor products of group representations Details will be provided during each class session.
Class 7 Induced representations Details will be provided during each class session.
Class 8 Frobenius reciprocity Details will be provided during each class session.

### Textbook(s)

Toshiyuki Katsura, Algebra II: Modules over a ring, Toudaishuppan (Japanese)

Unspecified.

### Assessment criteria and methods

Judging from the performance level of the exercises given during the class.

### Related courses

• MTH.A403 ： Advanced topics in Algebra C
• MTH.A404 ： Advanced topics in Algebra D
• ZUA.A333 ： Advanced courses in Algebra C
• MTH.A301 ： Algebra I
• MTH.A302 ： Algebra II

None required.