Changes

Jump to: navigation, search
Case p \ne 2, q odd
|-
| Sign representation || -- || Kernel is [[projective special linear group of degree two]], image is <math>\{ \pm 1 \}</math> || 1 || 1 || 1
|-
| Unclear || a nontrivial homomorphism <math>\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast</math>, with the property that <math>\varphi(x)^{q + 1} = 1</math> for all <math>x</math>, and <math>\varphi</math> takes values other than <math>\pm 1</math>. Identify <math>\varphi</math> and <math>\varphi^q</math>. || unclear || <math>q - 1</math> || <math>(q - 1)/2</math> || <math>(q - 1)^3/2 = (q^3 - 3q^2 + 3q - 1)/2</math>
|-
| Nontrivial component of permutation representation of <math>PGL_2</math> on the projective line over <math>\mathbb{F}_q</math> || -- || -- || <math>q</math> || 1 || <math>q^2</math>
|-
| Induced from one-dimensional representation of Borel subgroup || <math>\alpha</math> homomorphism <math>\mathbb{F}_q^\ast \to \mathbb{C}^\ast</math>, with <math>\alpha</math> taking values other than <math>\pm 1</math>, up to inverses. || Induced from the following representation of the image of the Borel subgroup: <math>\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\alpha(d)^{-1}</math> || <math>q + 1</math> || <math>(q - 3)/2</math> || <math>(q + 1)^2(q - 3)/2 = (q^3 - q^2 -5q - 3)/2</math>
|-
| Unclear || a nontrivial homomorphism <math>\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast</math>, with the property that <math>\varphi(x)^{q + 1} = 1</math> for all <math>x</math>, and <math>\varphi</math> takes values other than <math>\pm 1</math>. Identify <math>\varphi</math> and <math>\varphi^q</math>. || unclear || <math>q - 1</math> || <math>(q - 1)/2</math> || <math>(q - 1)^3/2 = (q^3 - 3q^2 + 3q - 1)/2</math>
|-
| Total || NA || NA || NA || <math>q + 2</math> || <math>q^3 - q</math>
|}
Bureaucrats, emailconfirmed, Administrators
38,756
edits

Navigation menu