# Changes

Jump to: navigation, search

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

, 23:34, 29 October 2010
Irreducible representations
| 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 <matHmath>\alpha[/itex] 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 || $q + 2$ || $q^3 - q$
|}
Bureaucrats, emailconfirmed, Administrators
38,924
edits