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

Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: general affine group of degree one.
View linear representation theory of group families | View other specific information about general affine group of degree one

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 -- -- -- $q$ $q(q - 1)$