Algebra II (ΑΛ2)

General

School School of Science Department of Mathematics Graduate ΑΛ2 2 Algebra II Lectures (Weekly Teaching Hours: 3, Credits: 7.5) General Background - Greek Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes The main purpose of the course is to introduce the student to the basic concepts, results, tools and methods of the Representation Theory of Finite Groups and its applications to other areas of Mathematics, mainly in Group Theory, and other related sciences, e.g. in Physics. At the end of the course, we expect the student to understand the basic concepts and the main theorems that are analysed in the course, to understand how these are applied to concrete examples arising from different thematic areas of Mathematics and related sciences, to be able to apply them to derive new elementary consequences in various fields, and finally to be able to perform some (not so obvious) calculations related to several problems arising in Group Theory. The course aims at enabling the graduate student to acquire the ability to analyse and synthesize basic knowledge of the basic Representation Theory of Finite Groups, which is an important part of modern Mathematics with numerous applications to other sciences, for instance in Physics. When the graduate student comes in for the first time in connection with the basic notions of representation theory and its applications to group theory, (s)he strengthens her/his creative, analytical and inductive thinking, and her/his ability to apply abstract knowledge in different areas of central interest in Mathematics and related sciences.

Syllabus

• Representations and characters of groups.
• Groups and homomorphisms.
• FG-modules και group-algebras.
• Schur’s Lemma and Maschke’s Theorem.
• Group-algebras and irreducible modules.
• Conjugacy classes and characters.
• Character tables and orthogonality relations.
• Normal subgroups and lifted characters.
• Elementary examples of characters tables.
• Tensor products. Restricting representations to subgroups.
• Applications.

Teaching and Learning Methods - Evaluation

Delivery

Face to face

Use of Information and Communications Technology -
Teaching Methods
Lectures 39
Student's study Hours 78
Exercises: Problem Solving 70.5
Course total 187.5
Student Performance Evaluation

The evaluation is based on the combined performance of the graduate student in:

• Weekly homeworks,
• Presentations during the semester,
• Major Homework at the end of the course,
• Written examination at the end of the courses in Greek with questions and problems of development of theoretical topics and problem solving.

Attached Bibliography

• J.P. Serre: “Linear Representations of Finite Groups”, Springer-Verlag, (1977).
• B. Steinberg: “Representation Theory of Finite Groups: An Introductory Approach”, Springer, (2012).
• C.W. Curtis and V. Reiner: “Methods of Representation Theory: With Applications to Finite Groups and Orders”, Wiley, (1981).
• P. Etingof et al: “Introduction to Representation Theory”, Student Mathematical Library 59, AMS, (2011).
• J.L. Alperin and R.B. Bell: “Groups and Representations”, Springer (1995).
• M. Burrow: “Representation Theory of Finite Groups”, Academic Press, (1965).
• M. Liebeck and G. James: “Representations and Characters of Groups”, CUP, (2001).