# Applied Algebra Seminar/Abstracts F13: Difference between revisions

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== October 31 == | == October 31 == | ||

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|Andrew Bridy<br>UW-Madison [[Image:Aasf13 andrewbridy.jpg|right|200px]] | |Andrew Bridy<br>UW-Madison [[Image:Aasf13 andrewbridy.jpg|right|200px]] | ||

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## Revision as of 16:10, 23 August 2013

## October 31

Andrew Bridy UW-Madison |

Functional Graphs of Affine-Linear Transformations over Finite Fields |

A linear transformation [math]\displaystyle{ A: (\mathbb{F}_q)^n \to (\mathbb{F}_q)^n }[/math] gives rise to a directed graph by regarding the elements of [math]\displaystyle{ (\mathbb{F}_q)^n }[/math] as vertices and drawing an edge from [math]\displaystyle{ v }[/math] to [math]\displaystyle{ w }[/math] if [math]\displaystyle{ Av = w }[/math]. In 1959, Elspas determined the "functional graphs" on [math]\displaystyle{ q^n }[/math] 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 [math]\displaystyle{ (\mathbb{F}_q)^n) }[/math]. I review some of this work and prove an new upper bound on the number of equivalence classes of affine-linear transformations of [math]\displaystyle{ (F_q)^n }[/math] under the equivalence relation of isomorphism of functional graphs. This bound is significantly smaller than the number of conjugacy classes of [math]\displaystyle{ \operatorname{GL}_n(q) }[/math]. This is joint work with Eric Bach. |