Linear representation theory of dihedral group:D10

This article gives specific information, namely, linear representation theory, about a particular group, namely: dihedral group:D10.
View linear representation theory of particular groups | View other specific information about dihedral group:D10

We shall use the dihedral group with the following presentation (here, $e$ is used to denote the identity element):

$\langle a,x\mid a^{5}=x^{2}=e,xax^{-1}=a^{-1}\rangle$.

Family contexts

Family name	Parameter values	General discussion of linear representation theory of family
dihedral group	degree $n=5$, order $2n=10$	linear representation theory of dihedral groups

COMPARE AND CONTRAST: View linear representation theory of groups of order 10 to compare and contrast the linear representation theory with other groups of order 10.

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)	$\{x,ax,a^{2}x,a^{3}x,a^{4}x\}$ (size 5)	$\{a,a^{-1}\}$ (size 2)	$\{a^{2},a^{-2}\}$ (size 2)
Trivial representation	$1$	$1$	$1$	$1$
Non-trivial one-dimensional	$1$	$-1$	$1$	$1$
Faithful irreducible representation	$2$	$0$	${\frac {-1-{\sqrt {5}}}{2}}$	${\frac {-1+{\sqrt {5}}}{2}}$
Faithful irreducible representation	$2$	$0$	${\frac {-1+{\sqrt {5}}}{2}}$	${\frac {-1-{\sqrt {5}}}{2}}$

See also

Linear representation theory of dihedral groups