Special Colloquium

Abstract: Given a nice piece of combinatorics for the symmetric group S_n, there is often a corresponding nice piece of combinatorics for the general linear group GL_n(F_q) over a finite field F_q, called a q-analogue.  In this talk, we'll describe an example of this phenomenon coming from the enumeration of factorizations.  In S_n, the number of ways to write an n-cycle as a product of n - 1 transpositions is Cayley's number n^(n - 2).  In GL_n(F_q), the corresponding problem is to write a Singer cycle as a product of n reflections.  We show that the number of such factorizations is (q^n - 1)^(n - 1), and give some extensions.  Mysteriously, the second answer is closely related to the first as q approaches 1.  Our proofs do not provide an explanation for this relationship; instead, they proceed by exploiting the (complex) representation theory of GL_n(F_q).