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Linear representation theory of symmetric group:S3

No change in size, 11:44, 18 July 2011
Interpretation as general linear group of degree two
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| One-dimensional, factor through the determinant map || a homomorphism <math>\alpha: \mathbb{F}_q^\ast \to \mathbb{C}^\ast</math> || <math>x \mapsto \alpha(\det x)</math> || 1 || 1 || <math>q - 1</math> || 1 || trivial representation
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| Unclear || a homomorphism <math>\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast</math>, up to the equivalence <math>\! \varphi \simeq \varphi^q</math>, excluding the cases where <math>\varphi = \varphi^q</math> || unclear || <math>q - 1</math> || 1 || <math>q(q - 1)/2</math> || 1 || [[#Sign representation|sign representation]]
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| Tensor product of one-dimensional representation and the nontrivial component of permutation representation of <math>GL_2</math> on the projective line over <math>\mathbb{F}_q</math> || a homomorphism <math>\alpha: \mathbb{F}_q^\ast \to \mathbb{C}^\ast</math> || <math>x \mapsto \alpha(\det x)\nu(x)</math> where <math>\nu</math> is the nontrivial component of permutation representation of <math>GL_2</math> on the projective line over <math>\mathbb{F}_q</math> || <math>q</math> || 2 || <math>q - 1</math> || 1 || [[Standard representation of symmetric group:S3|standard representation]]
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| Induced from one-dimensional representation of Borel subgroup || <math>\alpha, \beta</math> homomorphisms <math>\mathbb{F}_q^\ast \to \mathbb{C}^\ast</math> with <math>\alpha \ne \beta</math>, where <math>\{ \alpha, \beta \}</math> is treated as unordered. || Induced from the following representation of the Borel subgroup: <math>\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\beta(d)</math> || <math>q + 1</math> || 3 || <math>(q - 1)(q - 2)/2</math> || 0 || --
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| Unclear || a homomorphism <math>\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast</math>, up to the equivalence <math>\! \varphi \simeq \varphi^q</math>, excluding the cases where <math>\varphi = \varphi^q</math> || unclear || <math>q - 1</math> || 1 || <math>q(q - 1)/2</math> || 1 || [[#Sign representation|sign representation]]
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| Total || NA || NA || NA || NA || <math>q^2 - 1</math> || 3 ||
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