Linear representation theory of 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: general linear group of degree two. This article restricts attention to the case where the underlying ring is 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 $q$, and the characteristic prime is $p$. $q$ is a power of $p$.

See also the linear representation theories of: special linear group of degree two, projective general linear group of degree two, and projective special linear group of degree two.

Summary

Item	Value
degrees of irreducible representations over a splitting field	1 ($q - 1$ times), $q$ ($q - 1$ times), $q + 1$ ($(q - 1)/(q - 2)/2$ times), $q - 1$ ($q(q - 1)/2$ times)
number of irreducible representations	$q^2 - 1$, equal to the number of conjugacy classes. See number of irreducible representations equals number of conjugacy classes, element structure of general linear group of degree two over a finite field#Conjugacy class structure
maximum degree of irreducible representation over a splitting field	$q + 1$
lcm of degrees of irreducible representations over a splitting field	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 over a splitting field	$q(q+1)(q-1)^2$, equal to the order of the group. See sum of squares of degrees of irreducible representations equals group order.

Particular cases

Group	$p$	$q$	Order of the group $= q^4 - q^3 - q^2 + q$	Number of irreducible representations $= q^2 - 1$	Degrees of irreducible representations	Linear representation theory page
symmetric group:S3	2	2	6	3	1,1,2	linear representation theory of symmetric group:S3
general linear group:GL(2,3)	3	3	48	8	1,1,2,2,2,3,3,4	linear representation theory of general linear group:GL(2,3)
direct product of A5 and Z3	2	4	180	15	1,1,1,3,3,3,3,3,3,4,4,4,5,5,5
general linear group:GL(2,5)	5	5	480	24	1 (4 times), 4 (10 times), 5 (4 times), 6 (6 times)	linear representation theory of general linear group:GL(2,5)
general linear group:GL(2,7)	7	7	2016	48	1 (6 times), 6 (21 times), 7 (6 times), 8 (15 times)	linear representation theory of general linear group:GL(2,7)

Irreducible representations

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
One-dimensional, factor through the determinant map	a homomorphism $\alpha: \mathbb{F}_q^\ast \to \mathbb{C}^\ast$	$x \mapsto \alpha(\det x)$	1	$q - 1$	$q - 1$
Unclear	a homomorphism $\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast$, up to the equivalence $\! \varphi \simeq \varphi^q$, excluding the cases where $\varphi = \varphi^q$	unclear	$q - 1$	$q(q - 1)/2$	$q(q - 1)^3/2 = (q^4 - 3q^3 + 3q^2 - q)/2$
Tensor product of one-dimensional representation and the nontrivial component of permutation representation of $GL_2$ on the projective line over $\mathbb{F}_q$	a homomorphism $\alpha: \mathbb{F}_q^\ast \to \mathbb{C}^\ast$	$x \mapsto \alpha(\det x)\nu(x)$ where $\nu$ is the nontrivial component of permutation representation of $GL_2$ on the projective line over $\mathbb{F}_q$	$q$	$q - 1$	$q^2(q - 1) = q^3 - q^2$
Induced from one-dimensional representation of Borel subgroup	$\alpha, \beta$ homomorphisms $\mathbb{F}_q^\ast \to \mathbb{C}^\ast$ with $\alpha \ne \beta$, where $\{ \alpha, \beta \}$ is treated as unordered.	Induced from the following representation of the Borel subgroup: $\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\beta(d)$	$q + 1$	$(q - 1)(q - 2)/2$	$(q + 1)^2(q - 1)(q - 2)/2 = (q^4 - q^3 - 3q^2 + q + 2)/2$
Total	NA	NA	NA	$q^2 - 1$	$q^4 - q^3 - q^2 + q$