MATH42220: Representation Theory

• Prior knowledge of Algebra at undergraduate level.

• None

• None

• To develop and illustrate representation theory for finite groups and Lie groups.

• Representations of finite groups.
• Character theory.
• Modules over group algebra.
• Lie groups and Lie algebras and their representations.
• Representations of SL(2,C), SU(2), SO(3).
• Reading material on a topic related to: Representations of the symmetric group, representations over finite fields.

Subject-specific Knowledge:

• By the end of the module students will: be able to solve novel and/or complex problems in Representation Theory.
• have a systematic and coherent understanding of theoretical mathematics in the field of Representation Theory.
• have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Representations of finite groups.
• Character tables.
• Induced representations, Frobenius reciprocity.
• Representations of abelian groups.
• Lie groups and algebras, exponential map.
• Examples of representations of Lie groups and algebras.

Subject-specific Skills:

• In addition students will have highly specialised and advanced mathematical skills in the following areas: Abstract Reasoning.
• Students will have an advanced understanding in one of the following areas: Representations of the symmetric group, representations over finite fields.

Key Skills:

• Students will have developed independent learning of an advanced topic.

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Eight written or electronic assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

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