Unit of study_

# MATH4314: Representation Theory

## Overview

Representation theory is the abstract study of the possible types of symmetry in all dimensions. It is a fundamental area of algebra with applications throughout mathematics and physics: the methods of representation theory lead to conceptual and practical simplification of any problem in linear algebra where symmetry is present. This unit will introduce you to the basic notions of modules over associative algebras and representations of groups, and the ways in which these objects can be classified. You will learn the special properties that distinguish the representation theory of finite groups over the complex numbers, and also the unifying principles which are common to the representation theory of a wider range of algebraic structures. By learning the key concepts of representation theory you will also start to appreciate the power of category-theoretic approaches to mathematics. The mental framework you will acquire from this unit of study will enable you both to solve computational problems in linear algebra and to create new mathematical theory.

### Details

Academic unit Mathematics and Statistics Academic Operations MATH4314 Representation Theory Semester 1, 2021 Normal day Camperdown/Darlington, Sydney 6

### Enrolment rules

 Prohibitions ? MATH3966 None None Familiarity with abstract algebra, specifically vector space theory and basic group theory, e.g., MATH2922 or MATH2961 or equivalent. Yes

### Teaching staff and contact details

Coordinator Alexander Molev, alexander.molev@sydney.edu.au Alexander Molev

## Assessment

Type Description Weight Due Length
Assignment Final take home exam
Take home written exam
60% Formal exam period 16 pages
Outcomes assessed:
Assignment Assignment 1
Report
20% Week 07 3 to 5 questions
Outcomes assessed:
Assignment Assignment 2
Report
20% Week 12 3 to 5 questions
Outcomes assessed:
• Assignments: The assignments will require you to demonstrate ability to apply the theory developed in lectures and tutorials to produce your own arguments or perform calculations with particular classes of objects.
• Final exam: The exam will cover all material in the unit from both lectures and tutorials. The exam will consist of extended answer questions.

Detailed information for each assessment can be found on Canvas

### Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

At HD level, a student demonstrates a flair for the subject as well as a detailed and comprehensive understanding of the unit material. A ‘High Distinction’ reflects exceptional achievement and is awarded to a student who demonstrates the ability to apply their subject knowledge and understanding to produce original solutions for novel or highly complex problems and/or comprehensive critical discussions of theoretical concepts.

Distinction

75 - 84

At DI level, a student demonstrates an aptitude for the subject and a well-developed understanding of the unit material. A ‘Distinction’ reflects excellent achievement and is awarded to a student who demonstrates an ability to apply their subject knowledge and understanding of the subject to produce good solutions for challenging problems and/or a reasonably well-developed critical analysis of theoretical concepts.

Credit

65 - 74

At CR level, a student demonstrates a good command and knowledge of the unit material. A ‘Credit’ reflects solid achievement and is awarded to a student who has a broad general understanding of the unit material and can solve routine problems and/or identify and superficially discuss theoretical concepts.

Pass

50 - 64

At PS level, a student demonstrates proficiency in the unit material. A ‘Pass’ reflects satisfactory achievement and is awarded to a student who has threshold knowledge.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

### Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

The Current Student website provides information on academic honesty, academic dishonesty, and the resources available to all students.

The University expects students and staff to act ethically and honestly and will treat all allegations of academic dishonesty or plagiarism seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic dishonesty. If such matches indicate evidence of plagiarism or other forms of dishonesty, your teacher is required to report your work for further investigation.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Motivation and basics of representation theory Lecture and tutorial (4 hr)
Week 02 Internal structure Lecture and tutorial (4 hr)
Week 03 Uniqueness of decomposition Lecture and tutorial (4 hr)
Week 04 Artin-Wedderburn theory Lecture and tutorial (4 hr)
Week 05 Basics of representation of finite groups Lecture and tutorial (4 hr)
Week 06 Basics of multilinear algebra Lecture and tutorial (4 hr)
Week 07 Character tables Lecture and tutorial (4 hr)
Week 08 Frobenius-Schur indicators Lecture and tutorial (4 hr)
Week 09 Induced representations Lecture and tutorial (4 hr)
Week 10 Frobenius reciprocity Lecture and tutorial (4 hr)
Week 11 Representations of the symmetric group Lecture and tutorial (4 hr)
Week 12 Symmetric functions Lecture and tutorial (4 hr)
Week 13 Revision Lecture and tutorial (4 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University’s graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. Demonstrate a coherent and advanced understanding of the key concepts of representations of associative algebras, groups and other algebraic structures, and how they provide a unified approach to the study of symmetry.
• LO2. Apply the fundamental principles and results of representation theory to solve given problems.
• LO3. Distinguish and compare the properties of different types of representations, analysing them into constituent parts.
• LO4. Rephrase algebraic problems in representation-theoretic terms and determine the appropriate framework to solve them.
• LO5. Communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work.
• LO6. Devise computational solutions to complex problems in representation theory.
• LO7. Compose correct proofs of unfamiliar general results in representation theory.

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.