We will mostly follow parts of the textbook
- Fulton, W. & Harris, J.: Representation theory: a first course. 2004.
It has a strong emphasis on developing the theory by using examples.
Another book which could be used as the textbook for this course is
- Simon, B.: Representations of finite and compact groups. 1996.
This one has an analysis flavor.
Books on related topics, basic and advanced.
Hall, Brian C., Lie Groups, Lie Algebras, and Representations, An Elementary Introduction. 2015
- Armstrong, Mark, Groups and Symmetry, 1988
Steinberg, B., Representation Theory of Finite Groups - An Introductory Approach. 2012
Berndt, Rolf, Representations of Linear Groups - An Introduction Based on Examples from Physics and Number Theory. 2007
Georgi, H., Lie Algebras In Particle Physics: from Isospin To Unified Theories. 1999 (a classic).
Fuchs, J. and Schweigert, C., Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists. 2003 (The first chapter is attached below which is an overview of different appearances of symmetry in physics).
- Very quick overview of basic terminology and standard examples in group theory
- Very interesting short article on the history and role of Lie groups
Sophus Lie and the Role of Lie Groups in Mathematics, by Sigurdur Helgason
- Representations of finite groups and finite dimensional algebras are treated in a different style in:
Etingof, P. & Golberg, O. & Hensel, S. & Liu, T. & Schwendner, A. & Vaintrob, D. & Yudovina, E.: Introduction to representation theory. Lecture notes of courses given in the Clay Mathematics Institute and Massachusetts Institute of Technology, 2009.
- Representations of Lie algebras are treated also in:
Shlomo Sternberg: Lie algebras. Online book.
- Lie groups and Lie algebras are treated also in:
- Diaconis, Persi, Group Representations in Probability and Statistics. Book
- The web page Representation theory online resources lists many more links to freely available books and lecture notes.