Difference between revisions of "Applied Algebra Seminar/Abstracts F13"

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| [[Image:Aasf13 andrewbridy.jpg|200px]] [http://www.math.wisc.edu/~bridy/ Andrew Bridy], UW-Madison (Math)
 
| [[Image:Aasf13 andrewbridy.jpg|200px]] [http://www.math.wisc.edu/~bridy/ Andrew Bridy], UW-Madison (Math)
 
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|Functional Graphs of Affine-Linear Transformations over Finite Fields
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|''Functional Graphs of Affine-Linear Transformations over Finite Fields''
 
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|A linear transformation <math>A: (\mathbb{F}_q)^n \to (\mathbb{F}_q)^n</math> gives rise to a directed graph by regarding the elements of <math>(\mathbb{F}_q)^n</math> as vertices and drawing an edge from <math>v</math> to <math>w</math> if <math>Av = w</math>. In 1959, Elspas determined the "functional graphs" on <math>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>(\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>(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>\operatorname{GL}_n(q)</math>. This is joint work with Eric Bach.
 
|A linear transformation <math>A: (\mathbb{F}_q)^n \to (\mathbb{F}_q)^n</math> gives rise to a directed graph by regarding the elements of <math>(\mathbb{F}_q)^n</math> as vertices and drawing an edge from <math>v</math> to <math>w</math> if <math>Av = w</math>. In 1959, Elspas determined the "functional graphs" on <math>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>(\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>(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>\operatorname{GL}_n(q)</math>. This is joint work with Eric Bach.
 
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Revision as of 11:13, 23 August 2013

October 31

Aasf13 andrewbridy.jpg Andrew Bridy, UW-Madison (Math)
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.