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Overview

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 structure of finite-dimensional algebras and their representations and Maschke's theorem relating representations of groups and algebras.    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.

Key Topics

  1. Algebras, division algebras, quaternions. Frobenius theorem 
  2. Division algebras and projective planes. Wedderburn's theorem on finite division rings
  3. Modules and representations of algebras 
  4. Jacobson's density theorem. The structure of simple finite-dimensional algebras 
  5. Radical of algebras. The structure of semisimple  algebras 
  6. Group algebra and its semisimplicity.
  7. FG-Modules: irreducibility 
  8. Maschke's Theorem, Schur's Lemma  
  9. Irreducible modules and the group algebra 
  10. Character Theory: tables, orthogonality relations 

Workload Expectations

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.

Course Prerequisites, Corequisites and Restrictions

Prerequisite

Locations and Semesters Offered

LocationSemester
City

Teaching and Learning

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

Taught courses use a learning and collaboration tool called Canvas to provide students with learning materials including 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.

Additional Information on Learning Resources

The lectures delivered will cover all of the relevant material. Handouts on the content of lectures will be provided. We suggest the following as sensible sources to consult:

  • S. Maclane & G. Birkhoff,  Algebra
  • Walter Lederman, Representation Theory
  • Walter Lederman, Introduction to Group Characters
  • M. Aschbacher,  Finite Group Theory
  • Victor Hill, Groups, Representation, and Characters
  • T. W. Hungerford, Algebra

Copyright

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.

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.

Academic Integrity

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 for potential plagiarism or other forms of academic misconduct, using computerised detection mechanisms.

Similarly, research students must meet the University’s expectations of good research practice. This requires:

  • Honesty - in all aspects of research work
  • Accountability - in the conduct of research
  • Professional courtesy and fairness – in working with others
  • Good stewardship – on behalf of others
  • Transparency – of research process and presentation of results
  • Clarity - communication to be understandable, explainable and accessible

For more information on the University’s expectations of academic integrity, please see the Academic Conduct section of the University policy hub.

Disclaimer

Elements of this outline may be subject to change. The latest information about taught courses is made available to enrolled students in Canvas.

Students may be asked to submit assessments digitally. The University reserves the right to conduct scheduled tests and examinations 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.


Assessment and Learning Outcomes

Additional Information on Assessment

A student's final grade will be calculated as the maximum of (100% Exam grade) and (50% Exam grade + 30% Test grade + 20%Assignment grade).

Course Learning Outcomes

CLO #OutcomeProgramme Capability Link
1
2
3
4

Assessments

Assessment TypeAssessment PercentageAssessment Classification

Assessment to CLO Mapping

Assessment Type1234

Student Feedback, Support and Charter

Student Feedback

Feedback on taught courses is gathered from students at the end of each semester through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback and respond with summaries and actions. Your feedback helps teachers to improve the course and its delivery for future students. In addition, class Representatives in each class can take feedback to the department and faculty staff-student consultative committees.

Additional Information on 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. The course is constantly evolving addressing ideas and suggestions from students and faculty.

Class representatives

Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.

Tuākana

Tuākana is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at Tuakana Learning Communities.

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.

Wellbeing

We all go through tough times during the semester, or see our friends struggling. There is lots of help out there - please see the Support Services page for information on support services in the University and the wider community.

Special Circumstances

If your ability to complete assessed work 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.

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.

Student Academic Complaints and Disputes

Students with concerns about teaching including how a course is delivered, the resources provided, or supervision arrangements, have the right to express their concerns and seek resolution. The university encourages informal resolution where possible, as this is quicker and less stressful. For information on the informal and formal complaints processes, please refer to the Student Academic Complaints Statute in the Student Policies and Guidelines section of the Policy Hub.