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

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(October 31)
(October 31)
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|Andrew Bridy<br>UW-Madison
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|Andrew Bridy<br>UW-Madison [[Image:Aasf13 andrewbridy.jpg|right|200px]]
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| '''Title:'''
 
 
|Functional Graphs of Affine-Linear Transformations over Finite Fields
 
|Functional Graphs of Affine-Linear Transformations over Finite Fields
 
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| '''Abstract:'''
 
 
|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 10:09, 23 August 2013

October 31

Andrew Bridy
UW-Madison
Aasf13 andrewbridy.jpg
Functional Graphs of Affine-Linear Transformations over Finite Fields
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.