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

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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.
View linear representation theory of group families | View other specific information about general linear group of degree two | View other specific information about group families for rings of the type finite field

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