# Combinatorics Seminar- Counting real conjugacy classes with generating functions

Speaker:C. Ryan Vinroot, William & Mary

Date and time: Thursday, November 21, 2:15–3:15pm

Place: Phillips 110

Title: Counting real conjugacy classes with generating functions

Abstract: It was noticed by Lehrer (1975) and Macdonald (1981) that the number of conjugacy classes in the finite projective linear group PGL(n,q) is equal to the number of conjugacy classes of GL(n,q) which are contained in SL(n,q), and the number of classes in the finite projective unitary group PGU(n,q) is equal to the number of classes in the finite unitary group U(n,q) which are contained in SU(n,q). Meanwhile, Gow (1981) showed that the number of real classes of GL(n,q) (classes which are inverse-invariant) is equal to the number of real classes of U(n,q). More recently, Gill and Singh (2011) showed that when n is odd or q is even, the number of real classes of PGL(n,q) is equal to the number of real classes of GL(n,q) which are contained in SL(n,q). In this talk, we give a result which generalizes and specializes all of these results. Using generating functions, along with a nice application of Euler's trick, we show that for any n and any q, we have

# real classes of PGL(n,q)

= # real classes of GL(n,q) contained in SL(n,q)

= # real classes of PGU(n,q)

= # real classes of U(n,q) contained in SU(n,q).

This is joint work with my former honors thesis student Elena Amparo, now a PhD student at UC-Santa Barbara in physics.