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Linear representation theory of symmetric group:S3

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Standard representation
===Standard representation===
{{quotationfurther|See the discussion of this representation in general for symmetric groups at [[standard Standard representationof symmetric group:S3]]}}
This is a faithful two-dimensional {{#lst:standard representation. One way of obtaining this representation is as follows: consider a three-dimensional vector space with basis <math>e_1, e_2, e_3</math>. Let the symmetric group permute the basis vectors, and consider the induced action of the symmetric group on the vector space. This is a three-dimensional :S3|representation. Consider the two-dimensional subspace of all vectors of the form <math>x_1e_1 + x_2e_2 + x_3e_3</math> where <math>x_1 + x_2 + x_3 = 0</math>. When the characteristic of the field is not two or three, this is a faithful, irreducible, two-dimensional representation. Note that <math>e_1 - e_2</math> and <math>e_2 - e_3</math> can be taken as a basis for this, with <math>e_3 - e_1</math> being the negative of the sum of these.table}}
ThusSince the representation is realized over <math>\mathbb{Z}</math>, for fields whose it makes sense over all characteristics. The only characteristic where it is not two or threeirreducible is characteristic 3. In characteristic 3, every the representation of the symmetric group that is realized over the field, can be realized over its prime subfieldindecomposable but not irreducible.
Over the real numbers, this representation is conjugate to the representation as ''orthogonal'' matrices, where we view the symmetric group of degree three as the dihedral group acting on three elements. Here, the <math>3</math>-cycles act as rotations by multiples of <math>2\pi/3</math>, and the transpositions act as reflections about suitable axes. {| class="sortable" border="1"! Element !! Matrix for standard representation with basis <math>e_1 - e_2</math>, <math>e_2 - e_3</math> !! Matrix for standard representation viewed as quotient with basis <math>\overline{e_1}, \overline{e_2}</math> !! Matrix for real representation as dihedral group !! Matrix for complex representation as dihedral group !! Characteristic polynomial !! Minimal polynomial !! Trace, character value|-| Identity element || <math>\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}</math> || <math>\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}</math> || <math>\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}</math> || <math>\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}</math> || <math>(x - 1)^2</math> || <math>x - 1</math> || 2|-| <math>(1,2,3)</math> || <math>\begin{pmatrix} 0 & -1 \\ 1 & -1 \\\end{pmatrix}</math> || <math>\begin{pmatrix} 0 & -1 \\ 1 & -1 \\\end{pmatrix}</math> || <math>\begin{pmatrix} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & -1/2\end{pmatrix}</math> || <math>\begin{pmatrix} e^{2\pi i/3} & 0 \\ 0 & e^{-2\pi i/3}\end{pmatrix}</math> || <math>x^2 + x + 1</math>|| <math>x^2 + x + 1</math> || -1|-| <math>(1,3,2)</math> || <math>\begin{pmatrix} -1 & 1 \\ -1 & 0 \\\end{pmatrix}</math> || <math>\begin{pmatrix} -1 & 1 \\ -1 & 0 \\\end{pmatrix}</math> || <math>\begin{pmatrix} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2 & -1/2 \end{pmatrix}</math> || <math>\begin{pmatrix} e^{-2\pi i/3} & 0 \\ 0 & e^{2\pi i/3}\end{pmatrix}</math> || <math>x^2 + x + 1</math>|| <math>x^2 + x + 1</math> || -1|-| <math>(1,2)</math> || <math>\begin{pmatrix} -1 & 1 \\ 0 & 1\\\end{pmatrix}</math> || <math>\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}</math> ||<math>\begin{pmatrix}-1/2 & \sqrt{3}/2 \\ \sqrt{3}/2 & 1/2 \end{pmatrix}</math> || <math>\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}</math> || <math>x^2 - 1</math> || <math>x^2 - 1</math>|| 0|-| <math>(2,3)</math> || <math>\begin{pmatrix} 1 & 0 \\ 1 & - 1 \\\end{pmatrix}</math> || <math> \begin{pmatrix} 1 & -1 \\ 0 & -1 \end{pmatrix}</math> || <math> \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}</math> || <math>\begin{pmatrix} 0 & e^{-2\pi i/3} \\ e^{2\pi i/3} & 0\end{pmatrix}</math> || <math>x^2 - 1</math> || <math>x^2 - 1</math>|| 0|-| <math>(1,3)</math> || <math>\begin{pmatrix} 0 & -1 \\ -1 & 0\\\end{pmatrix}</math> || <math>\begin{pmatrix} -1 & 0 \\ -1 & 1 \\\end{pmatrix}</math> || <math>\begin{pmatrix} -1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2 & 1/2 \end{pmatrix}</math> || <math>\begin{pmatrix} 0 & e^{2\pi i/3} \\ e^{-2\pi i/3} & 0 \end{pmatrix}</math> || <math>x^2 - 1</math> || <math>x^2 - 1</math>|| 0|} We consider three cases: {| class="sortable" border="1"! Case for the field !! What happens in this case|-| Characteristic not 2 or 3 || Irreducible|-| Characteristic equal to 2 || Irreducible. In fact, the representation maps bijectively to the [[general linear group of degree two]] over [[field:F2]], so [[symmetric group:S3]] is isomorphic to <math>GL(2,2)</math>. One way of viewing this is to think of the vector space of dimension two over the field of two elements as an abelian group: in this case, it is the [[Klein four-group]]. The automorphisms of this Klein four-group are completely described by the way they permute the three non-identity elements, hence, this automorphism group is isomorphic to the symmetric group on three elements.|-| Characteristic equal to 3 || Indecomposable, but not irreducible. Here is an alternative perspective on this representationin characteristic 3.<br>In characteristic three, the The symmetric group is identified with the [[general affine group of degree ]] one over the [[field:F3|field of three elements]]. In other words, it is the semidirect product of the additive group of this field (a [[cyclic group:Z3|cyclic group of order three]]) and the [[cyclic group:Z2|multiplicative group of this field]], where the multiplicative group acts on the additive group by multiplication. Via the general fact that embeds a general affine group in a general linear group of one size higher, we get a faithful representation of the symmetric group on three elements in the [[general linear group of degree two]] over [[field:F3]], i.e., in <math>GL(2,3)</math>.|}
==Degrees of irreducible representations==
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