Spherical functions and root systems

Minh-Tri Vo and Bernard Mares

We have been studying Lie groups/algebras, their root systems, and associated polynomials such as characters and spherical functions. These functions have applications to many different areas of mathematics, physics and statistics.

There are extensive character tables in the literature, but not so for spherical functions. We have been using a recently discovered formula to devise an algorithm for computing spherical functions and more general polynomials.

Using a recently discovered formula, we are writing programs to compute spherical functions for various groups. The results of this program will eventually appear on a webpage.

Our current activities include:

• Understanding the foundations of spherical functions and their precise relationship with root systems and symmetric space geometry
• Finding an optimal algorithmic implementation of the formula that can run efficiently on a computer
• Comparing our algorithm and results with existing algorithms and results
• Implementing extensions of the formula that will allow us to compute more general quantities

Ben (Bernard) got sick with strep throat and did his presentation through the computer. You can download the audio from his presentation.

• A quick review of root systems
• A rapid introduction to spherical functions

References

Sahi, Siddhartha  A New Formula for Weight Multiplicities and Characters,  Duke Mathematical Journal, Vol. 101, No. 1 (2000), 77-84

Bremner, M.R.; Moody, R.V.; Patera, J.  Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras,  Pure and Applied Mathematics: A series of monographs and textbooks 90, Marcel Dekker Inc. 1985

Helgason, Sigurdur  Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions,  Pure and Apllied Mathematics: A series of monographs and textbooks 113, Academic Press Inc. 1984

Helgason, Sigurdur  Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics: A series of monographs and textbooks 80, Academic Press Inc. 1978

Humphreys, James E.  Reflection Groups and Coxeter Groups,  Cambridge Studies in Advanced Mathematics 29, Cambridge University Press 1990

Humphreys, James E.  Introduction to Lie Algebras and Representation Theory,  Graduate Texts in Mathematics 9, Springer-Verlag 1972

James, A.T.  Calculation of zonal polynomial coefficients by use of the Laplace-Beltrami operator,  Annals of Mathematical Statistics, 39 (1968) 1711-1718

Macdonald, I.G.  Affine Hecke Algebras and Orthogonal Polynomials,  Asterisque 237, (1996), 189-207, Seminaire Bourbaki 1994/95 exp. no. 797

Macdonald, I.G.  Symmetric Functions and Hall Polynomials 2nd ed.,  Oxford Mathematical Monographs, Oxford University Press, 1995

Stembridge, John   Coxeter graph paper   http://www.math.lsa.umich.edu/~jrs/archive.html#cox