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.
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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 $q$, 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	$q + 2$	$q^3 - q$