Difference between revisions of "Linear representation theory of projective general linear group of degree two over a finite field"

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(Irreducible representations)
(Irreducible representations)
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| Tensor product of sign representation and nontrivial component of permutation representation on projective line || -- || -- || <math>q</math> || 1 || <math>q^2</math>
 
| Tensor product of sign representation and nontrivial component of permutation representation on projective line || -- || -- || <math>q</math> || 1 || <math>q^2</math>
 
|-
 
|-
| Induced from one-dimensional representation of Borel subgroup || <math>\alpha</math> homomorphism <math>\mathbb{F}_q^\ast \to \mathbb{C}^\ast</math>, with <matH>\alpha</math> taking values other than <math>\pm 1</math>, up to inverses. || Induced from the following representation of the image of the Borel subgroup: <math>\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\alpha(d)^{-1}</math> || <math>q + 1</math> || <math>(q - 3)/2</math> || <math>(q + 1)^2(q - 3)/2 = (q^3 - q^2 -5q - 3)/2</math>
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| Induced from one-dimensional representation of Borel subgroup || <math>\alpha</math> homomorphism <math>\mathbb{F}_q^\ast \to \mathbb{C}^\ast</math>, with <math>\alpha</math> taking values other than <math>\pm 1</math>, up to inverses. || Induced from the following representation of the image of the Borel subgroup: <math>\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\alpha(d)^{-1}</math> || <math>q + 1</math> || <math>(q - 3)/2</math> || <math>(q + 1)^2(q - 3)/2 = (q^3 - q^2 -5q - 3)/2</math>
 
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| Unclear || a nontrivial homomorphism <math>\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast</math>, with the ... || unclear || <math>q - 1</math> || <math>(q - 1)/2</math> || <math>(q - 1)^3/2 = (q^3 - 3q^2 + 3q - 1)/2</math>
+
| Unclear || a nontrivial homomorphism <math>\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast</math>, with the property that <math>\varphi(x)^{q + 1} = 1</math> for all <math>x</math>, and <math>\varphi</math> takes values other than <math>\pm 1</math>. Identify <math>\varphi</math> and <math>\varphi^q</math>. || unclear || <math>q - 1</math> || <math>(q - 1)/2</math> || <math>(q - 1)^3/2 = (q^3 - 3q^2 + 3q - 1)/2</math>
 
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| Total || NA || NA || NA || <math>q + 2</math> || <math>q^3 - q</math>
 
| Total || NA || NA || NA || <math>q + 2</math> || <math>q^3 - q</math>
 
|}
 
|}

Revision as of 23:34, 29 October 2010

This article describes the linear representation theory of the general linear group of degree two over a finite field. The order (size) of the field is q, and the characteristic prime is p. q is a power of p.

See also linear representation theory of special linear group of degree two, linear representation theory of projective general linear group of degree two, and linear representation theory of general linear group of degree two.

Particular cases

Group p q Order of the group Number of irreducible representations Linear representation theory page
symmetric group:S3 2 2 6 3 linear representation theory of symmetric group:S3
symmetric group:S4 3 3 24 5 linear representation theory of symmetric group:S4
alternating group:A5 2 4 60 5 linear representation theory of alternating group:A5
symmetric group:S5 5 5 120 7 linear representation theory of symmetric group:S5
projective general linear group:PGL(2,7) 7 7 336 9 linear representation theory of projective general linear group:PGL(2,7)
special linear group:SL(2,8) 2 8 504 9 linear representation theory of special linear group:SL(2,8)
projective general linear group:PGL(2,9) 3 9 720 11 linear representation theory of projective general linear group:PGL(2,9)

Irreducible representations

Case p \ne 2, q odd

Description of collection of representations Parameter for describing each representation How the representation is described Degree of each representation Number of representations Sum of squares of degrees
Trivial -- x \mapsto 1 1 1 1
Sign representation -- Kernel is projective special linear group of degree two, image is \{ \pm 1 \} 1 1 1
Nontrivial component of permutation representation of PGL_2 on the projective line over \mathbb{F}_q -- -- q 1 q^2
Tensor product of sign representation and nontrivial component of permutation representation on projective line -- -- q 1 q^2
Induced from one-dimensional representation of Borel subgroup \alpha homomorphism \mathbb{F}_q^\ast \to \mathbb{C}^\ast, with \alpha taking values other than \pm 1, up to inverses. Induced from the following representation of the image of the Borel subgroup: \begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\alpha(d)^{-1} q + 1 (q - 3)/2 (q + 1)^2(q - 3)/2 = (q^3 - q^2 -5q - 3)/2
Unclear a nontrivial homomorphism \varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast, with the property that \varphi(x)^{q + 1} = 1 for all x, and \varphi takes values other than \pm 1. Identify \varphi and \varphi^q. unclear q - 1 (q - 1)/2 (q - 1)^3/2 = (q^3 - 3q^2 + 3q - 1)/2
Total NA NA NA q + 2 q^3 - q