# Changes

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

, 11:43, 18 July 2011
Case p \ne 2, q odd
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| Sign representation || -- || Kernel is [[projective special linear group of degree two]], image is $\{ \pm 1 \}$ || 1 || 1 || 1
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| 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 || $q - 1$ || $(q - 1)/2$ || $(q - 1)^3/2 = (q^3 - 3q^2 + 3q - 1)/2$
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| Nontrivial component of permutation representation of $PGL_2$ on the projective line over $\mathbb{F}_q$ || -- || -- || $q$ || 1 || $q^2$
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| 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: $\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\alpha(d)^{-1}$ || $q + 1$ || $(q - 3)/2$ || $(q + 1)^2(q - 3)/2 = (q^3 - q^2 -5q - 3)/2$
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| 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 || $q - 1$ || $(q - 1)/2$ || $(q - 1)^3/2 = (q^3 - 3q^2 + 3q - 1)/2$
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| Total || NA || NA || NA || $q + 2$ || $q^3 - q$
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