# Difference between revisions of "Linear representation theory of projective general linear group of degree two over a 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 $p$.

## Particular cases

Group $p$ $q$ Order of the group Number of irreducible representations Linear representation theory page
symmetric group:S3 2 2 6 3 linear representation theory of symmetric group:S3
symmetric group:S4 3 3 24 5 linear representation theory of symmetric group:S4
alternating group:A5 2 4 60 5 linear representation theory of alternating group:A5
symmetric group:S5 5 5 120 7 linear representation theory of symmetric group:S5
projective general linear group:PGL(2,7) 7 7 336 9 linear representation theory of projective general linear group:PGL(2,7)
special linear group:SL(2,8) 2 8 504 9 linear representation theory of special linear group:SL(2,8)
projective general linear group:PGL(2,9) 3 9 720 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
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: $\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\alpha(d)^{-1}$ $q + 1$ $(q - 3)/2$ $(q + 1)^2(q - 3)/2 = (q^3 - q^2 -5q - 3)/2$
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 $q - 1$ $(q - 1)/2$ $(q - 1)^3/2 = (q^3 - 3q^2 + 3q - 1)/2$
Total NA NA NA $q + 2$ $q^3 - q$