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

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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 <math>q</math>, and the characteristic prime is <math>p</math>. <math>q</math> is a power of <math>p</math>.
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{{group family-specific information|
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group family = projective general linear group of degree two|
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information type = linear representation theory|
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connective = of}}
  
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]].
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This article describes the linear representation theory of the [[projective general linear group of degree two]] over a [[finite field]]. The order (size) of the field is <math>q</math>, and the characteristic prime is <math>p</math>. <math>q</math> is a power of <math>p</math>. The group is denoted <math>PGL(2,q)</math> or <math>PGL_2(q)</math>.
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See also the linear representation theory for: [[linear representation theory of special linear group of degree two over a finite field|special linear group]], [[linear representation theory of projective special linear group of degree two over a finite field|projective special linear group]], and [[linear representation theory of general linear group of degree two over a finite field|general linear group]].
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==Summary==
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<section begin="summary"/>
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{| class="sortable" border="1"
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! Item !! Value
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|-
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| [[degrees of irreducible representations]] over a [[splitting field]] || Case <math>q</math> odd: 1 (2 times), <math>q - 1</math> (<math>(q - 1)/2</math> times), <math>q</math> (2 times), <math>q + 1</math> (<math>(q - 3)/2</math> times) <br> Case <math>q</math> even: 1 (1 time), <math>q - 1</math> (<math>q/2</math> times), <math>q</math> (1 time), <math>q + 1</math> (<math>(q - 2)/2</math> times)
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|-
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| number of irreducible representations || Case <math>q</math> odd: <math>q + 2</math>, case <math>q</math> even: <math>q + 1</math><br>See [[number of irreducible representations equals number of conjugacy classes]], [[element structure of projective general linear group of degree two over a finite field#Conjugacy class structure]]
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|-
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| [[quasirandom degree]] (minimum degree of nontrivial ireducible representation) || 1
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|-
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| [[maximum degree of irreducible representation]] || <math>q + 1</math>
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|-
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| [[lcm of degrees of irreducible representations]] || Case <math>q</math> odd: <math>q(q + 1)(q - 1)/2 = (q^3 - q)/2</math>; Case <math>q</math> even: <math>q(q+1)(q-1) = q^3 - q</math>
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|-
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| sum of squares of degrees of irreducible representations ||  <math>q(q + 1)(q - 1) = q^3 - q</math>, equal to the group order; see [[sum of squares of degrees of irreducible representations equals group order]]
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|}
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<section end="summary"/>
  
 
==Particular cases==
 
==Particular cases==
  
 
{| class="sortable" border="1"
 
{| class="sortable" border="1"
! Group !! <math>p</math> !! <math>q</math> !! Order of the group !! Number of irreducible representations !! Linear representation theory page
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! <math>q</math> (field size) !! <math>p</math> (underlying prime, field characteristic) !! Case for <math>q</math> !! Group <math>PGL(2,q)</math> !! Order of the group (<math>= q^3 - q</math>) !! Degrees of irreducible representations (ascending order) !! Number of irreducible representations (<math>= q + 1</math> if <math>q</math> even, <math>q + 2</math> if <math>q</math> odd) !! Linear representation theory page
 
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| [[symmetric group:S3]] || 2 || 2 || 6 || 3 || [[linear representation theory of symmetric group:S3]]
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|  2 || 2 || even || [[symmetric group:S3]] ||6 || 1,1,2 || 3 || [[linear representation theory of symmetric group:S3]]
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|-
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|  3 || 3 || odd || [[symmetric group:S4]] ||24 || 1,1,2,3,3 || 5 || [[linear representation theory of symmetric group:S4]]
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|-
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|  4 || 2 || even || [[alternating group:A5]] ||60 || 1,3,3,4,5 || 5 || [[linear representation theory of alternating group:A5]]
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|-
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|  5 || 5 || odd || [[symmetric group:S5]] ||120 || 1,1,4,4,5,5,6 || 7 || [[linear representation theory of symmetric group:S5]]
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|-
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| 7 || 7 || odd || [[projective general linear group:PGL(2,7)]] ||336 || 1,1,6,6,6,7,7,8,8 ||9 || [[linear representation theory of projective general linear group:PGL(2,7)]]
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|-
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| 8 || 2 || even || [[projective special linear group:PSL(2,8)]] || 504 || 1,7,7,7,7,8,9,9,9 || 9 || [[linear representation theory of projective special linear group:PSL(2,8)]]
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|-
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| 9 || 3 || odd || [[projective general linear group:PGL(2,9)]] || 720 || 1,1,8,8,8,8,9,9,10,10,10 || 11 || [[linear representation theory of projective general linear group:PGL(2,9)]]
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|}
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==Irreducible representations==
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===Case <math>p \ne 2</math>, <matH>q</math> odd===
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{| class="sortable" border="1"
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! 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
 
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| [[symmetric group:S4]] || 3 || 3 || 24 || 5 || [[linear representation theory of symmetric group:S4]]
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| Trivial || -- || <math>x \mapsto 1</math>|| 1 || 1 || 1
 
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| [[alternating group:A5]] || 2 || 4 || 60 || 5 || [[linear representation theory of alternating group:A5]]
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| Sign representation || -- || Kernel is [[projective special linear group of degree two]], image is <math>\{ \pm 1 \}</math> || 1 || 1 || 1
 
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| [[symmetric group:S5]] || 5 || 5 || 120 || 7 || [[linear representation theory of symmetric group:S5]]
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| 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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|-
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| Nontrivial component of permutation representation of <math>PGL_2</math> on the projective line over <math>\mathbb{F}_q</math> || -- || -- || <math>q</math> || 1 || <math>q^2</math>
 
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| [[projective general linear group:PGL(2,7)]] || 7 || 7 || 336 || 9 || [[linear representation theory of projective general linear group:PGL(2,7)]]
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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>
 
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| [[special linear group:SL(2,8)]] || 2 || 8 || 504 || 9 || [[linear representation theory of special linear group:SL(2,8)]]
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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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| [[projective general linear group:PGL(2,9)]] || 3 || 9 || 720 || [[linear representation theory of projective general linear group:PGL(2,9)]]
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| Total || NA || NA || NA || <math>q + 2</math> || <math>q^3 - q</math>
 
|}
 
|}

Latest revision as of 17:52, 8 January 2012

This article gives specific information, namely, linear representation theory, about a family of groups, namely: projective general linear group of degree two.
View linear representation theory of group families | View other specific information about projective general linear group of degree two

This article describes the linear representation theory of the projective 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. The group is denoted PGL(2,q) or PGL_2(q).

See also the linear representation theory for: special linear group, projective special linear group, and general linear group.

Summary

Item Value
degrees of irreducible representations over a splitting field Case q odd: 1 (2 times), q - 1 ((q - 1)/2 times), q (2 times), q + 1 ((q - 3)/2 times)
Case q even: 1 (1 time), q - 1 (q/2 times), q (1 time), q + 1 ((q - 2)/2 times)
number of irreducible representations Case q odd: q + 2, case q even: q + 1
See number of irreducible representations equals number of conjugacy classes, element structure of projective general linear group of degree two over a finite field#Conjugacy class structure
quasirandom degree (minimum degree of nontrivial ireducible representation) 1
maximum degree of irreducible representation q + 1
lcm of degrees of irreducible representations Case q odd: q(q + 1)(q - 1)/2 = (q^3 - q)/2; Case q even: q(q+1)(q-1) = q^3 - q
sum of squares of degrees of irreducible representations q(q + 1)(q - 1) = q^3 - q, equal to the group order; see sum of squares of degrees of irreducible representations equals group order


Particular cases

q (field size) p (underlying prime, field characteristic) Case for q Group PGL(2,q) Order of the group (= q^3 - q) Degrees of irreducible representations (ascending order) Number of irreducible representations (= q + 1 if q even, q + 2 if q odd) Linear representation theory page
2 2 even symmetric group:S3 6 1,1,2 3 linear representation theory of symmetric group:S3
3 3 odd symmetric group:S4 24 1,1,2,3,3 5 linear representation theory of symmetric group:S4
4 2 even alternating group:A5 60 1,3,3,4,5 5 linear representation theory of alternating group:A5
5 5 odd symmetric group:S5 120 1,1,4,4,5,5,6 7 linear representation theory of symmetric group:S5
7 7 odd projective general linear group:PGL(2,7) 336 1,1,6,6,6,7,7,8,8 9 linear representation theory of projective general linear group:PGL(2,7)
8 2 even projective special linear group:PSL(2,8) 504 1,7,7,7,7,8,9,9,9 9 linear representation theory of projective special linear group:PSL(2,8)
9 3 odd projective general linear group:PGL(2,9) 720 1,1,8,8,8,8,9,9,10,10,10 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
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
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
Total NA NA NA q + 2 q^3 - q