MA 792J Classical Representation Theory and Vertex Operators
Spring 2006
Instructor: N. Jing, Office: 234 HA, 3-3584,
email:
Tentative Schedule, TBA

Course Description:
In this course we will present the following topics:
A) Representation theory of S_n and GL_n
B) Symmetric functions
C) Vertex operator approach to symmetric functions

Representation theory of symmetric groups and general linear groups
 are widely used in many mathematical researches. The importance can be partly explained
by the fact that the subject has been associated with famous names such as Euler, Gauss, Jacobi, Cauchy, Schur,
Weyl, etc. This classical subject has become a central part of algebraic combinatorics during the last several
decades. Moreover the study of symmetric polynomials has turned out to be a multi-disciplinary field that has
connections with algebraic geometry, invariant theory, representation theory, Lie groups and algebras,
PDE (KdV and KP hierarchy equations), statistical mechanics, random matrix models, to name a few.

Detailed materials are as follows.
1) Constructions of irreducible representations of S_n and GL_n;
2) Schur algebras;
3) Rudiments of block theory of S_n;
4) Basic material about Schur functions;
5) Vertex operator construction of classical symmetric functions;
6) q-deformed Fock spaces and the quantum affine algebra U_q(\hat{gl}_n);
7) Polynomial functors.

We intend to survey basic techniques of vertex operators to help students understand this part of
vertex operator calculus. Students with maturity in linear algebra will have sufficient background for the
course. Group theory and related algebraic notions will be reviewed when needed. I will try to build the
materials from scratch and present the course in a down-to-earth fashion.

References:
There will be no formal textbooks and the materials are drawn
from the following:
1. R. Stanley, Enumerative combinatorics, II
2. Fulton and Harris, Representation theory, a first course, Springer-Verlag, 1991
3. I. G. Macdonald, Symmetric functions and Hall polynomials, 1995
4. I. Frenkel, J. Lepowsky, A. Meurman, Vertex operator algebras and the Monstor, 1988
and research papers of myself and others.
5. S. Martin, Schur algebras and representation theory, Cambridge University Press, 1993.

Grading Policy:
The student is required to give a short presentation at the end of the semester.