Math 620, Representation Theory

All of mathematics is some kind of representation theory. I. Gelfand.

This course is an introduction to the representation theory of groups. Although the catalog title refers to finite groups, we will consider both finite and infinite groups. Representation theory studies the way in which a given group may act on vector spaces; in other words, it is concerned with representing groups as groups of matrices. We are mostly interested in irreducible representations, which are the building blocks for the construction of all representations of a given group. The main questions in representation theory are related to: (1) the construction of irreducible representations; (2) the calculation of certain algebraic invariants of these; (3) the decomposition of other representations into irreducibles.

Representation theory is a fundamental tool for studying group symmetry - geometric, analytic, or algebraic - by means of linear algebra. Its origins are mostly in the work of F. Frobenius, H. Weyl, I. Schur, and A. Young, about a century ago; Weyl's work, for instance, is a milestone in the representation theory of Lie groups, which play a central role in many areas of mathematics. Important advances have been made during last century, through the study of representations of more and more general groups, and through a better understanding of the subtle combinatorics involved, which led to some very explicit constructions and computations. Today, representation theory plays an important role in many recent developments of mathematics and theoretical physics. A considerable amount of recent work is devoted to the representation theory of quantum groups, which are certain deformations of Lie groups, with applications to physics.

Here are some pretty pictures that will come up in this course, created by John Stembridge; they illustrate the hyperplane arrangements corresponding to the root systems A3 and B3 .

Tentative syllabus: The main concepts of representation theory (characters, induced representations, irreducible representations). Symmetric functions (Young tableaux, Schur functions). Representations of the symmetric group (construction of the irreducible representations, Frobenius's formula). Representations of the general linear group (Weyl's construction, characters). Extension of Weyl's construction to other Lie groups and Lie algebras. The main concepts of Lie theory and the classification of complex semisimple Lie algebras. Representations of complex simple Lie groups. The Weyl character formula. A hint about what lies ahead.

Prerequisites for this course are Math 220 (Linear Algebra) and Math 327 (Elementary Abstract Algebra). On the other hand, Math 420 (Abstract Algebra) and Math 424 (Advanced Linear Algebra) are helpful, but not required.

Textbook: W. Fulton, J. Harris, Representation Theory. A First Course, Graduate Texts in Mathematics, 129, Springer-Verlag, 1991, ISBN 0-387-97495-4, 3-540-97495-4.

Time: MWF 1:25-2:20, ES 143.

Cristian Lenart, Department of Mathematics, ES 118, SUNY at Albany