MATHS 721 : Representations and Structure of Algebras and Groups

Science

Course Prescription

Representation theory studies properties of abstract groups and algebras by representing their elements as linear transformations of vector spaces or matrices, thus reducing many problems about the structures to linear algebra, a well-understood theory.

Course Overview

This course is intended for students who have enjoyed the introduction to abstract algebra and parts of groups and rings presented in MATHS 255 / 254 / 320 / 326 / 328. The main focus of the course is on structural properties and actions on abelian groups/vector spaces, and illustrating their fundamental role in the study of the structure of algebras and groups:  for example, the character, module and ring approaches. Emphasis is placed on the contexts in which these structures occur, methods by which they can be modelled and analysed, and their diverse applications. We introduce students to some of the major results in the area:  for example, the Wedderburn and Krull-Schmidt theorems. A student completing this course will have a good knowledge of foundations of representation theory,  its connections and applications and approaches in this field of growing importance. This is a core mathematics course for the BSc (Hons) and PGDipSci. It is also important to student wishing to pursue other graduate level courses in pure mathematics, and for those wishing to pursue graduate education in mathematics.

Course Requirements

Prerequisite: MATHS 320

Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity

Learning Outcomes

By the end of this course, students will be able to:
1. Display a high level of knowledge of algebras and groups and their properties. (Capability 1 and 2)
2. Demonstrate, via your written work, a clear understanding of the connections and significant applications of critical results, including those of Schur and Masche, to other areas of pure mathematics. (Capability 1 and 3)
3. Display ability to model and solve problems. (Capability 2 and 3)
4. Demonstrate, via your written work, a clear understanding of the nature of proof and precision in mathematics, present both formally and informally arguments in both verbal and written form. (Capability 2, 4 and 5)

Assessments

Assessment Type Percentage Classification
Assignments 30% Individual Coursework
Final Exam 70% Individual Coursework
1 2 3 4
Assignments
Final Exam

A student's final grade will be calculated as the maximum of E and 0.3A + 0.7E, where A denotes the assignment percentage and E the examination percentage.

Key Topics

Elementary commutative rings (1/2 week)

Modules over PID  (1 and 1/2 weeks)

Projective and injective modules (1 week)

Representations of rings (1/2 week)

Wedderburn's Theorems (2  weeks)

Applications to Modular Representation Theory of Groups ( 1/2 week)

FG-Modules: irreducibility (1 week)

Maschke's Theorem, Schur's Lemma  (1 week)

Irreducible modules and the group algebra (1 and 1/2 weeks)

Character Theory: tables, orthogonality relations (2 and 1/2 weeks)

Special Requirements

No special requirements

This course is a standard 15 point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect 3 hours of lectures and 5 hours of reading and thinking about the content each week. You can expect 5 hours of work on each of the 4 assignments and exam preparation.

Delivery Mode

Campus Experience

Attendance is strongly recommend at scheduled lectures.
The course will not include live online events.
Attendance on campus is required for the exam.
The activities for the course are scheduled as a standard weekly timetable.

Learning Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

The lectures delivered will cover all of the relevant material.
We suggest the following as sensible sources to consult:
S. Maclane & G. Birkhoff,  Algebra
Walter Lederman, Representation Theory
Walter Lederman, it Introduction to Group Characters
M. Aschbacher,  Finite Group Theory
Victor Hill, Groups, Representation, and Characters
T. W. Hungerford, Algebra
S. Lang,   Algebra

Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

Class Representatives

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website

Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page .

This should be done as soon as possible and no later than seven days after the affected test or exam date.

Learning Continuity

In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter .

Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.

Published on 02/11/2021 09:54 p.m.