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 $\alpha :{\mathbb{F}}_q^{\ast }\to {\mathbb{C}}^{\ast }$	$x\mapsto \alpha (detx)$	1	$q-1$	$q-1$
Unclear	a homomorphism $\phi :{\mathbb{F}}_{q^2}^{\ast }\to {\mathbb{C}}^{\ast }$, up to the equivalence $\phi \simeq {\phi }^q$, excluding the cases where $\phi ={\phi }^q$	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 $G{L}_2$ on the projective line over ${\mathbb{F}}_q$	a homomorphism $\alpha :{\mathbb{F}}_q^{\ast }\to {\mathbb{C}}^{\ast }$	$x\mapsto \alpha (detx)\nu (x)$ where $\nu$ is the nontrivial component of permutation representation of $G{L}_2$ 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: $\left(\begin{array}{cc}a& b\\ 0& d\\ \end{array}\right)\mapsto \alpha (a)\beta (d)$	$q+1$	$(q-1)(q-2)/2$	$(q+1{)}^2(q-1)(q-2)/2=(q^4-q^3-3q^2+q+2)/2$
Total	NA	NA	NA	$q^2-1$	$q^4-q^3-q^2+q$