Basic Notions Seminar talk on October 11, 2002
Speaker: Arkady Berenstein (University of Oregon)
Title: Macdonald's Identities
Abstract:
In 1748 Leonard Euler found a beautiful power series
expansion for the following infinite product:
phi(x)=(1-x)(1-x^2)(1-x^3)...
Surprisingly enough, after multiplying out, almost all
powers of x got canceled!
Seventy years later Karl-Friederich Gauss raised the same
question about the cube of phi(x), and the power series
expansion turned out to be even more beautiful.
Gauss' result is especially remarkable because
there is nothing special about the square of phi(x).
Since then, many mathematicians were puzzled
by existence of certain exponents n for which phi(x)^n
has a nice power series expansion with lots of cancelations:
the case n=8 was studied by Klein, n=14, 26 by Atkin,
n=24 by Ramanujan, and so on.
Apparently, the most comprehensive list of such "good" exponents
was obtained by Freeman Dyson in late sixties.
Here is the beginning
of the list:
n=1, 3, 8, 10, 14, 15, 21, 24, 26, 28, 35, 36, ...
Dyson tried very hard to understand the origin of these numbers
but with no success.
The real breaktrough in computing phi(x)^n was due to Yan Macdonald (who
worked at Princeton at the same time as Dyson). Macdonals noticed that:
- The good exponents n are dimensions of simple Lie algebras;
- The identity for phi(x)^n is a very special case of the
Denominator Identity for Kac-Moody Lie algebras. The Denominator
Identity itself is a generalization of the Wandermond
determinant identity.
In my talk I will describe Macdonald identities and will
outline their relation to the Kac-Moody theory.