Representation Theory and Young Tableaux

Course Advertisement, Summer, 1998

Math 690, Topics in Mathematics, credits and meeting times to be arranged

Course description:

We shall study the book `Young Tableaux: With Applications to Representation Theory and Geometry' by William Fulton (Springer-Verlag, 1997). We shall develop the combinatorics of Young tableaux and see them in action in the algebra of symmetric functions and representations of the symmetric and general linear groups. We shall try to make the various main approaches to the calculus of tableaux, and their applications, accessible to nonexperts.

A `Young diagram' is a collection of boxes or cells, arranged in left-justified rows, with a (weakly) decreasing number of boxes in each row. Listing the number of boxes in each row gives a partition of the integer n which is the total number of boxes. The purpose of writing a Young diagram is to put something into the boxes. We shall fill them with positive integers. A `Young tableau' is a filling that is (1) weakly increasing across each row and (2) strictly increasing down each column. A `standard tableau' is a tableau in which the entries are the numbers 1 to n, each occuring once. In the first part of the course we shall discuss the combinatorial calculus of tableaux. In the second part we shall discuss representation theory. In paricular, we shall discuss some uses of tableaux in the study of the representations of the symmetric group S_n and the general linear group Gl(m,C).

Representation theory is easy to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquititous in 20th century mathematics.

We shall see that to each partition \lambda of n we can construct an irreducible representation S^{\lambda} of the symmetric group called the `Specht module' and an irreducible representation E^{\lambda} of the general linear group Gl(m,C) called the `Schur module'. We shall see that we get all the irreducible representations of S_n as Specht modules and that they have bases corresponding to standard tableaux with n boxes. We shall also see that we get all the irreducible representatons of Gl(m,C) as Schur modules; these modules are parametrized by Young diagrams with at most m rows and have bases corresponding to Young tableaux on \lambda with entries from 1 to m.

For more information about the course contact Dr. Kenneth R. Driessel (office: Carver 410, e-mail: driessel@iastate.edu, internet: http://www.public.iastate.edu/~driessel.)