Applied Algebra Seminar/Abstracts F13

 Andrew Bridy, UW-Madison (Math) Functional Graphs of Affine-Linear Transformations over Finite Fields A linear transformation $A: (\mathbb{F}_q)^n \to (\mathbb{F}_q)^n$ gives rise to a directed graph by regarding the elements of $(\mathbb{F}_q)^n$ as vertices and drawing an edge from $v$ to $w$ if $Av = w$. In 1959, Elspas determined the "functional graphs" on $q^n$ vertices that are realized in this way. In doing so he showed that there are many non-similar linear transformations which have isomorphic functional graphs (and so are conjugate by a non-linear permutation of $(\mathbb{F}_q)^n$). I review some of this work and prove an new upper bound on the number of equivalence classes of affine-linear transformations of $(F_q)^n$ under the equivalence relation of isomorphism of functional graphs. This bound is significantly smaller than the number of conjugacy classes of $\operatorname{GL}_n(q)$. This is joint work with Eric Bach.