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

Contents

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 $q + 2$ $q^3 - q$