Applied Algebra Seminar/Abstracts F13

From DEV UW-Math Wiki
Revision as of 15:39, 23 August 2013 by Dynerman (talk | contribs) (New page: == October 31 == Title: Functional Graphs of Affine-Linear Transformations over Finite Fields Abstract: A linear transformation A : (\F_q)^n \to (\F_q)^n gives rise to a directed graph by...)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

October 31

Title: Functional Graphs of Affine-Linear Transformations over Finite Fields

Abstract: A linear transformation A : (\F_q)^n \to (\F_q)^n gives rise to a directed graph by regarding the elements of (\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 (\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 GL_n(q). This is joint work with Eric Bach.

Retrieved from "https://wiki-dev.math.wisc.edu/index.php?title=Applied_Algebra_Seminar/Abstracts_F13&oldid=5554"