# Changes

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

, 23:17, 29 October 2010
no edit summary
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| [[projective general linear group:PGL(2,9)]] || 3 || 9 || 720 || 11 || [[linear representation theory of projective general linear group:PGL(2,9)]]
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==Irreducible representations==

===Case $p \ne 2$, <matH>q[/itex] odd===

{| class="sortable" border="1"
! 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 || $q^2$
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| Induced from one-dimensional representation of Borel subgroup || $\alpha$ homomorphism $\mathbb{F}_q^\ast \to \mathbb{C}^\ast$, with <matH>\alpha[/itex] taking values other than $\pm 1$, up to inverses. || Induced from the following representation of the image of the Borel subgroup: $\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\alpha(d)^{-1}$ || $q + 1$ || $(q - 3)/2$ || $(q + 1)^2(q - 3)/2 = (q^3 - q^2 -5q - 3)/2$
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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 $x$ || unclear || $q - 1$ || ? || ?
|-
| Total || NA || NA || NA || $q^2 - 1$ || $q^4 - q^3 - q^2 + q$
|}