# Changes

## Linear representation theory of projective general linear group of degree two over a finite field

, 16:23, 20 May 2011
no edit summary
{{group family-specific information|
group family = projective general linear group of degree two|
information type = linear representation theory|
connective = of}}

This article describes the linear representation theory of the [[general linear group of degree two]] over a [[finite field]]. The order (size) of the field is [itex]q[/itex], and the characteristic prime is [itex]p[/itex]. [itex]q[/itex] is a power of [itex]p[/itex].
See also the linear representation theory for: [[linear representation theory of special linear group of degree twoover a finite field|special linear group]], [[linear representation theory of projective special linear group of degree twoover a finite field|projective special linear group]], and [[linear representation theory of general linear group of degree twoover a finite field|general linear group]]. ==Summary==
{| class="sortable" border="1"
! Item !! Value
|-
| [[degrees of irreducible representations]] over a [[splitting field]] || Case [itex]q[/itex] odd: 1 (2 times), [itex]q[/itex] (2 times), [itex]q + 1[/itex] ([itex](q - 3)/2[/itex] times), [itex]q - 1[/itex] ([itex](q - 1)/2[/itex] times)<br> Case [itex]q[/itex] even:
|-
| number of irreducible representations || Case [itex]q[/itex] odd: [itex]q + 2[/itex], case [itex]q[/itex] even: [itex]q + 1[/itex]<br>See [[number of irreducible representations equals number of conjugacy classes]], [[element structure of projective general linear group of degree two over a finite field#Conjugacy class structure]]
|-
| [[maximum degree of irreducible representation]] || [itex]q + 1[/itex]
|-
| [[lcm of degrees of irreducible representations]] || [itex]q(q + 1)(q - 1) = q^3 - q[/itex]
|-
| sum of squares of degrees of irreducible representations || [itex]q(q + 1)(q - 1) = q^3 - q[/itex], equal to the group order; see [[sum of squares of degrees of irreducible representations equals group order]]
|}
==Particular cases==