The main topics of this course are basic concepts and properties of group representation theory. This course first covers irreducible representation of groups and homomorphic rings, and then Schur's lemma. The instructor then introduces characters for the representation of (finite) groups, explaining the direct relationship between them. The course then covers the description of irreducible decomposition for group rings of finite groups. Finally, the instructor introduces tensor products for (finite) group representation, induced representation, and in connection with those, Frobenius reciprocity. This course is a continuation of "Advanced topics in Algebra C" in the third quarter.
Representation theory for (finite) groups is not just a typical example of the general theory for modules over rings. Its results and methods have broad applications outside of mathematics in physics and chemistry. The objective of this course is for students to become familiar with basic methods of the representation theory of (finite) groups, and to be able to use them correctly.
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.
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
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
Standard lecture course.
|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||Irreducible decomposition of the group algebra of a finite group||Details will be provided during each class session|
|Class 6||Tensor products of groups 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|
Toshiyuki Katsura, Algebra II: Modules over a Ring, Toudaishuppan (Japanese)
Based on the reports with answers of exercise problems presented in the class.