Applied Algebra Seminar/Abstracts F13: Difference between revisions
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== October 31 == | == October 31 == | ||
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|Functional Graphs of Affine-Linear Transformations over Finite Fields | |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 (\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. | |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 (\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. | ||
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Revision as of 15:47, 23 August 2013
October 31
| Title: | Functional Graphs of Affine-Linear Transformations over Finite Fields |
| Abstract: | 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 (\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. |