INTRODUCTION TO REPRESENTATION THEORY

Course Code:

MATH 423

Course Type:

Area Elective

P:

3

Lab:

0

Laboratuvar Saati:

0

Credits:

3

ECTS:

7

Course Language:

English

Course Objectives:

To introduce basic facts about representation theory of groups and to find a representation of a group as a group of matrices in order to have a concrete description of this group.

Course Content:

Generalities and basic definitions. Sums, quotients, tensor products, characters and decompositions of representations. Group algebra. Generalities on algebras and modules, semi-simple modules. Invertible and nilpotent elements. Idempotents. The Jacobson radical. Semi-simple and local algebras. Projective modules. Primitive decompositions and points. Blocks of an algebra. Duality. Symmetric algebras.

Course Methodology:

1: Lecture, 2: Problem Solving

Course Evaluation Methods:

A: Written examination, B: Homework

Vertical Tabs

Course Learning Outcomes

Learning Outcomes	Teaching Methods	Assessment Methods
1) Visualizes groups as matrices	1,2	A
2) Uses group algebra to construct the regular representation of a group	1,2	A
3) Uses FG-modules to obtain information about representations of a group G over a field F	1,2	A
4) Computes the character table of a group	1,2	A
5) Applies tensor products to find all the irreducible characters of a direct product of groups	1,2	A
6) Uses blocks of an algebra to get information about its modules	1,2	A

Course Flow

Week	Topics	Study Materials
1	Generalities and basic definitions	Textbook
2	Sums, quotients, tensor products, characters	Textbook
3	Decompositions of representations	Textbook
4	Group algebra	Textbook
5	Generalities on algebras and modules, semi-simple modules	Textbook
6	Invertible and nilpotent elements	Textbook
7	Idempotents	Textbook
8	The Jacobson radical	Textbook
9	Semi-simple and local algebras	Textbook
10	Projective modules	Textbook
11	Primitive decompositions and points	Textbook
12	Blocks of an algebra	Textbook
13	Duality	Textbook
14	Symmetric algebras	Textbook

Recommended Sources

Textbook	Representations and characters of groups. Gordon James, Martin Liebeck.
Additional Resources	Representations of finite groups and associative algebras. C.W. Curtis, I. Reiner.

Material Sharing

Documents
Assignments
Exams

Assessment

IN-TERM STUDIES	NUMBER	PERCENTAGE
Mid-terms	2	100
Quizzes
Assignments
Total		100
CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE		40
CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE		60
Total		100

COURSE CATEGORY

Expertise/ Field Courses

Course’s Contribution to Program

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry)			x
2	Acquiring fundamental knowledge on fundamental research fields in mathematics					x
3	Ability form and interpret the relations between research topics in mathematics					x
4	Ability to define, formulate and solve mathematical problems					X
5	Consciousness of professional ethics and responsibilty					x
6	Ability to communicate actively		x
7	Ability of self-development in fields of interest					x
8	Ability to learn, choose and use necessary information technologies			x
9	Lifelong education					x

ECTS

Activities	Quantity	Duration (Hour)	Total Workload (Hour)
Course Duration (14x Total course hours)	14	3	42
Hours for off-the-classroom study (Pre-study, practice)	14	6	84
Mid-terms (Including self study)	2	15	30
Quizzes	-	-	-
Assignments	-	-	-
Final examination (Including self study)	1	20	20
Total Work Load			176
Total Work Load / 25 (h)			7.04
ECTS Credit of the Course			7

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