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Linear representation theory of projective general linear group of degree two over a finite field

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.

Contents

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