KTH/SU Mathematics Colloquium
2005-01-19
Dennis Stanton, University of Minnesota
Roots of unity in enumeration
Let X be a finite set. Suppose that each x in X
corresponds to a monomial q^{c(x)}. The generating
function X(q) for X is the sum of all of these
monomials, and is a polynomial in q. Clearly X(1)=|X|,
but there are important examples in which X(-1) is
the size of a specified set related to X. In this talk,
I will give a generalization of this phenomena to roots
of unity w, where X(w) counts the number of fixed points
of a cyclic group which acts on X. An equivalent representation
theory reformulation is given. Examples of the phenomenon will
be given for integer partitions, Coxeter groups, and
finite fields. Conjectured Bruhat-like cell decompositions
in finite fields and new results on modular invariant theory
will be given.