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

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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)