Linear representation theory of general affine group of degree one over a finite field

This article gives specific information, namely, linear representation theory, about a family of groups, namely: general affine group of degree one.
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This article discusses the linear representation theory of the general affine group of degree one over a finite field of size $q$ and characteristic $p$, where $q = p^r$.

Summary

Item	Value
degrees of irreducible representations over a splitting field	1 ($q - 1$ times), $q - 1$ (1 time)
number of irreducible representations	$q$ See number of irreducible representations equals number of conjugacy classes, element structure of general affine group of degree one over a finite field
maximum degree of irreducible representation over a splitting field	$q - 1$
lcm of degrees of irreducible representations over a splitting field	$q - 1$
sum of squares of degrees of irreducible representations	$q(q - 1)$ sum of squares of degrees of irreducible representations equals group order
condition for a field to be a splitting field	The field should be a splitting field for the multiplicative group and its characteristic should not be $p$. Equivalently, its order should be relatively prime to $q$ and $q - 1$ and it should contain all the primitive $(q - 1)^{th}$ roots of unity, i.e., the polynomial $\Phi_{q - 1}(x)$ must split over the field.
smallest size splitting field	The finite field whose size is the smallest prime power other than $q$ that is congruent to 1 modulo $q - 1$.

Particular cases

$q$ (field size)	$p$ (underlying prime, field characteristic)	Group $GA(1,q)$	Order of the group	Second part of GAP ID	Degrees of irreducible representations	Number of irreducible representations	Linear representation theory page
2	2	cyclic group:Z2	2	1	1,1	2	linear representation theory of cyclic group:Z2
3	3	symmetric group:S3	6	1	1,1,2	3	linear representation theory of symmetric group:S3
4	2	alternating group:A4	12	3	1,1,1,3	4	linear representation theory of alternating group:A4
5	5	general affine group:GA(1,5)	20	3	1,1,1,1,4	5	linear representation theory of general affine group:GA(1,5)
7	7	general affine group:GA(1,7)	42	1	1,1,1,1,1,1,6	7	linear representation theory of general affine group:GA(1,7)
8	2	general affine group:GA(1,8)	56	11	1,1,1,1,1,1,1,7	8	linear representation theory of general affine group:GA(1,8)
9	3	general affine group:GA(1,9)	72	39	1,1,1,1,1,1,1,1,8	9	linear representation theory of general affine group:GA(1,9)

Irreducible representations

We denote by the pair $(a,\mu)$ the element $x \mapsto a + \mu x$. The multiplicative part is $\mu \in \mathbb{F}_q^\ast$ and the additive part is $a \in \mathbb{F}_q$.

Description of collection of representations	Parameter for describing the representation	How the representation is described	Degree of each representation	Number of representations	Sum of squares of degrees
One-dimensional, depends only on multiplicative part, kernel contains additive subgroup	a homomorphism $\nu: \mathbb{F}_q^\ast \to \mathbb{C}^\ast$	$(a,\mu) \mapsto \nu(\mu)$	1	$q - 1$	$q - 1$
Nontrivial component of permutation representation on $q$ elements.	None	Consider the permutation representation of $GA(1,q)$ induced by its action on $\mathbb{F}_q$. This has a one-dimensional trivial subrepresentation. The complementary subrepresentation has degree $q - 1$ and is irreducible.	$q - 1$	1	$(q - 1)^2$
Total	--	--	--	$q$	$q(q - 1)$