Undergraduate Programme and Module Handbook 2010-2011 (archived)

Module MATH4101: REPRESENTATION THEORY & MODULES IV

Department: Mathematical Sciences

MATH4101: REPRESENTATION THEORY & MODULES IV

Type	Open	Level	4	Credits	20	Availability	Available in 2011/12 and alternate years thereafter	Module Cap	None.	Location	Durham

Prerequisites

Mathematics modules to the value of 100 credits in Years 2 and 3, with at least 40 credits at Level 3 and including Linear Algebra II (MATH2021), Algebra & Number Theory II (MATH2061).

Corequisites

None.

Excluded Combination of Modules

Representation Theory and Modules III (MATH3191).

Aims

To develop and illustrate the theory of modules and that of complex characters of finite groups.

Content

Rings and ideals.
Modules and submodules.
Modules over principle ideal domains.
Similarity of matrices over a field.
(Finite dimensional) semisimple algebras.
Representations of Groups over C.
Reading material on a topic related to: Modules of finite length; modules over Dedekind rings, localization as a tool; Grothendieck groups and categories.

Learning Outcomes

Subject-specific Knowledge:

By the end of the module students will: be able to solve complex, unpredictable and specialised problems in Representation Theory and Modules.
have an understanding of specialised and complex theoretical mathematics in the field of Representation Theory and Modules.
have mastered a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Representations of groups.
Character tables.
Frobenius reciprocity.
Modules and submodules including structure of modules over a PID.
Jordan normal form and rational canonical form.
Simple and semi-simple rings.

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: Modules over Dedekind rings, Grothendieck groups.

Key Skills:

Students will have developed independent learning of an advanced topic.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

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.

Teaching Methods and Learning Hours

Activity	Number	Frequency	Duration	Total/Hours
Lectures	40	2 per week	1 Hour	40
Preparation and Reading				160
Total				200

Summative Assessment

Component: Examination		Component Weighting: 100%
Element	Length / duration	Element Weighting	Resit Opportunity
three-hour examination		100%

Formative Assessment:

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

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University