Linear representation theory of general linear group:GL(2,3)

This article gives specific information, namely, linear representation theory, about a particular group, namely: general linear group:GL(2,3).
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This article describes the linear representation theory (in characteristic zero and other characteristics excluding 2,3) of general linear group:GL(2,3), which is the general linear group of degree two over field:F3.

Summary

Item	Value
degrees of irreducible representations over a splitting field	1,1,2,2,2,3,3,4 maximum: 4, lcm: 12, number: 8, sum of squares: 48

Irreducible representations

Interpretation as general linear group of degree two over field:F3

Compare with linear representation theory of general linear group of degree two.

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
One-dimensional, factor through the determinant map	a homomorphism $\alpha :{\mathbb{F}}_3^{\ast }\to {\mathbb{C}}^{\ast }$	$x\mapsto \alpha (detx)$	1	2	2
Tensor product of one-dimensional representation and the nontrivial component of permutation representation of $G{L}_2$ on the projective line over ${\mathbb{F}}_3$	a homomorphism $\alpha :{\mathbb{F}}_3^{\ast }\to {\mathbb{C}}^{\ast }$	$x\mapsto \alpha (detx)\nu (x)$ where $\nu$ is the nontrivial component of permutation representation of $G{L}_2$ on the projective line over ${\mathbb{F}}_3$	3	2	18
Induced from one-dimensional representation of Borel subgroup	Both distinct representations $\alpha ,\beta$ homomorphisms ${\mathbb{F}}_3^{\ast }\to {\mathbb{C}}^{\ast }$	Induced from the following representation of the Borel subgroup: $\left(\begin{array}{cc}a& b\\ 0& d\\ \end{array}\right)\mapsto \alpha (a)\beta (d)$	4	1	16
Unclear	a homomorphism $\phi :{\mathbb{F}}_9^{\ast }\to {\mathbb{C}}^{\ast }$	unclear	2	3	12
Total	NA	NA	NA	8	48