Linear representation theory of general affine group:GA(1,5)

This article gives specific information, namely, linear representation theory, about a particular group, namely: general affine group:GA(1,5).
View linear representation theory of particular groups | View other specific information about general affine group:GA(1,5)

Summary

This page regards the linear representation theory of the group General affine group:GA(1,5). We shall use the following presentation:

$\langle a,b\mid a^{5}=b^{4}=e,bab^{-1}=a^{2}\rangle$.

Item	Value
degrees of irreducible representations over a splitting field (such as $\mathbb {C}$ or ${\overline {\mathbb {Q} }}$)	1,1,1,1,4 maximum: 4, lcm: 4, number: 5, sum of squares: 20

Character table

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):
Orthogonality relations: Character orthogonality theorem | Column orthogonality theorem
Separation results (basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zero
Numerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integers
Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

This character table works over characteristic zero:

Representation/Conj class	$\{e\}$ (size 1)	(size 5)	(size 4)	(size 5)	(size 5)
Trivial representation	1	1	1	1	1
first non-trivial irrep of dimension 1	1	1	1	-1	-1
second non-trivial irrep of dimension 1	1	-1	1	i	-i
third non-trivial irrep of dimension 1	1	-1	1	-i	i
irreducible representation of dimension 4	4	0	-1	0	0

References

1. Loeffler, D. Calculating Group Characters Algorithmically Spring 2007. (Link to PDF)