View source for Linear representation theory of general linear group:GL(2,3)

{{group-specific information|
group = general linear group:GL(2,3)|
information type = linear representation theory|
connective = of}}

This article describes the linear representation theory (in characteristic zero and other characteristics excluding 2,3) of [[general linear group:GL(2,3)]], which is the [[general linear group of degree two]] over [[field:F3]].

==Summary==

{| class="sortable" border="1"
! Item !! Value
|-
| [[degrees of irreducible representations]] over a [[splitting field]] || 1,1,2,2,2,3,3,4<br>[[maximum degree of irreducible representation|maximum]]: 4, [[lcm of degrees of irreducible representations|lcm]]: 12, [[number of irreducible representations equals number of conjugacy classes|number]]: 8, [[sum of squares of degrees of irreducible representations equals group order|sum of squares]]: 48
|-
| [[ring generated by character values]] (characteristic zero) || <math>\mathbb{Z}[\sqrt{-2}]</math>, same as <math>\mathbb{Z}[t]/(t^2 + 2)</math>
|-
| [[field generated by character values]] (characteristic zero) || <math>\mathbb{Q}(\sqrt{-2})</math>, same as <math>\mathbb{Q}[t]/(t^2 + 2)</math>
|}

Note that [[general linear group:GL(2,3)]] and the [[binary octahedral group]] are [[isoclinic groups]] of the same order. We know that [[isoclinic groups have same proportions of degrees of irreducible representations]], therefore, in this case, the [[degrees of irreducible representations]] are the same for both groups. However, the character tables themselves are not identical. In fact, the fields generated by character values also differ from one another.

==Irreducible representations==

===Interpretation as general linear group of degree two===

The group is a general linear group of degree two over [[field:F3]]. Compare with [[linear representation theory of general linear group of degree two over a finite field]].

{| class="sortable" border="1"
! Description of collection of representations !! Parameter for describing each representation !! How the representation is described !! Degree of each representation (generic <math>q</math>) !! Degree of each representation (<math>q = 3</math>) !! Number of representations (generic <math>q</math>) !! Number of representations (<math>q = 3</math>) !! Sum of squares of degrees (generic <math>q</math>) !! Sum of squares of degrees (<math>q = 3</math>)
|-
| One-dimensional, factor through the determinant map || a homomorphism <math>\alpha: \mathbb{F}_3^\ast \to \mathbb{C}^\ast</math> || <math>x \mapsto \alpha(\det x)</math> || 1 || 1 || <math>q - 1</math> || 2 || <math>q - 1</math> || 2
|-
| Unclear || a homomorphism <math>\varphi:\mathbb{F}_{9}^\ast \to \mathbb{C}^\ast</math> || unclear || <math>q - 1</math> || 2 || <math>q(q - 1)/2</math> || 3 || <math>q(q - 1)^3/2</math> ||12
|-
| Tensor product of one-dimensional representation and the nontrivial component of permutation representation of <math>GL_2</math> on the projective line over <math>\mathbb{F}_3</math> || a homomorphism <math>\alpha: \mathbb{F}_3^\ast \to \mathbb{C}^\ast</math> || <math>x \mapsto \alpha(\det x)\nu(x)</math> where <math>\nu</math> is the nontrivial component of permutation representation of <math>GL_2</math> on the projective line over <math>\mathbb{F}_3</math> || <math>q</math> || 3 || <math>q - 1</math> || 2 || <math>q^2(q - 1)</math> || 18
|-
| Induced from one-dimensional representation of Borel subgroup || Both distinct representations <math>\alpha, \beta</math> homomorphisms <math>\mathbb{F}_3^\ast \to \mathbb{C}^\ast</math> || Induced from the following representation of the Borel subgroup: <math>\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\beta(d)</math> || <math>q + 1</math> || 4 || <math>(q - 1)(q - 2)/2</math> || 1 ||<math>(q+1)^2(q-1)(q-2)/2</math> || 16
|-
! Total || NA || NA || NA || NA || <math>q^2 - 1</math> || 8 || <math>q^4 - q^3 - q^2 + q</math> ||48
|}

==Character table==

{{character table facts to check against}}

In the table below, we denote by <math>\sqrt{-2}</math> a fixed square root of -2.

{| class="sortable" border="1"
! Representation/conjugacy class representative and size !! <math>\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}</math> (size 1) !! <math>\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix}</math> (size 1) !! <math>\begin{pmatrix} 0 & 1 \\ -1 & 0 \\\end{pmatrix}</math> (size 6) !! <math>\begin{pmatrix} 0 & 1 \\ 1 & -1 \\\end{pmatrix}</math> (size 6) !! <math>\begin{pmatrix} 0 & 1 \\ 1 & 1 \\\end{pmatrix}</math> (size 6) !! <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> (size 8) !! <math>\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}</math> (size 8) !! <math>\begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}</math> (size 12)
|-
| trivial || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|-
| nontrivial one-dimensional || 1 || 1 || 1 || -1 || -1 || 1 || 1 || -1
|-
| two-dimensional (unclear) || 2 || 2 || 2 || 0 || 0 || -1 || -1 || 0
|-
| two-dimensional (unclear) || 2 || -2 || 0 || <math>\sqrt{-2}</math> || <math>-\sqrt{-2}</math> || -1 || 1 || 0
|-
| two-dimensional (unclear) || 2 || -2 || 0 || <math>-\sqrt{-2}</math> || <math>\sqrt{-2}</math> || -1 || 1 || 0
|-
| three-dimensional, factors through [[standard representation of symmetric group:S4]] || 3 || 3 || -1 || -1 || -1 || 0 || 0 || 1
|-
| three-dimensional, factors through tensor product of standard and sign representations of <math>S_4</math> || 3 || 3 || -1 || 1 || 1 || 0 || 0 || -1
|-
| four-dimensional, induced from one-dimensional representation of Borel subgroup || 4 || -4 || 0 || 0 || 0 || 1 || -1 || 0
|}

==GAP implementation==

===Degrees of irreducible representations===

The degrees of irreducible representations can be computed using GAP's [[GAP:CharacterDegrees|CharacterDegrees]] function, as follows:

<pre>gap> CharacterDegrees(GL(2,3));
[ [ 1, 2 ], [ 2, 3 ], [ 3, 2 ], [ 4, 1 ] ]</pre>

===Character table===

The character table can be computed using GAP's [[GAP:CharacterTable|CharacterTable]] function, as follows:

<pre>gap> Irr(CharacterTable(GL(2,3)));
[ Character( CharacterTable( GL(2,3) ), [ 1, 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( GL(2,3) ), [ 1, 1, 1, 1, 1, -1, -1, -1 ] ),
  Character( CharacterTable( GL(2,3) ), [ 2, -1, 2, -1, 2, 0, 0, 0 ] ),
  Character( CharacterTable( GL(2,3) ), [ 2, 1, -2, -1, 0, -E(8)-E(8)^3,
      E(8)+E(8)^3, 0 ] ), Character( CharacterTable( GL(2,3) ),
    [ 2, 1, -2, -1, 0, E(8)+E(8)^3, -E(8)-E(8)^3, 0 ] ),
  Character( CharacterTable( GL(2,3) ), [ 3, 0, 3, 0, -1, 1, 1, -1 ] ),
  Character( CharacterTable( GL(2,3) ), [ 3, 0, 3, 0, -1, -1, -1, 1 ] ),
  Character( CharacterTable( GL(2,3) ), [ 4, -1, -4, 1, 0, 0, 0, 0 ] ) ]</pre>

A visual display of the character table can be achieved as follows:

<pre>gap> Display(CharacterTable(GL(2,3)));
CT1

     2  4  1  4  1  3  3  3  2
     3  1  1  1  1  .  .  .  .

       1a 6a 2a 3a 4a 8a 8b 2b

X.1     1  1  1  1  1  1  1  1
X.2     1  1  1  1  1 -1 -1 -1
X.3     2 -1  2 -1  2  .  .  .
X.4     2  1 -2 -1  .  A -A  .
X.5     2  1 -2 -1  . -A  A  .
X.6     3  .  3  . -1  1  1 -1
X.7     3  .  3  . -1 -1 -1  1
X.8     4 -1 -4  1  .  .  .  .

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2</pre>

===Irreducible representations===

The irreducible representations of <math>GL(2,3)</math> can be computed using GAP's [[GAP:IrreducibleRepresentations]] function, as follows:

<pre>gap> IrreducibleRepresentations(GL(2,3));
[ CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8)
     ] -> [ [ [ -1 ] ], [ [ 1 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 0, E(3) ], [ E(3)^2, 0 ] ],
      [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] ->
    [ [ [ -1/2*E(24)^11-1/2*E(24)^17, -1/2*E(24)-E(24)^11-E(24)^17-1/2*E(24)^19 ], [ -1/2*E(8)-1/2*E(8)^3, 1/2*E(24)^11+1/2*E(24)^17 ] ],
      [ [ E(3), E(3) ], [ 0, E(3)^2 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] ->
    [ [ [ E(24)+E(24)^19, -1 ], [ -E(3)+E(3)^2, -E(24)-E(24)^19 ] ], [ [ E(3)+2*E(3)^2, E(8)+E(8)^3 ], [ -E(24)-E(24)^19, -E(3)^2 ] ] ], <action isomorphism> ),
  CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 0, 0, 1 ], [ 0, -1, 0 ], [ 1, 0, 0 ] ], [ [ 1, 0, 0 ], [ 0, 0, 1 ], [ 1, -1, -1 ] ] ],
     <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ],
      [ [ 1, 0, 0 ], [ -1, -1, -1 ], [ 0, 1, 0 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] ->
    [ [ [ 0, 0, -E(3), 0 ], [ 0, 0, -E(3), -E(3) ], [ -E(3)^2, 0, 0, 0 ], [ E(3)^2, -E(3)^2, 0, 0 ] ],
      [ [ -E(3)^2, E(3)^2, 0, 0 ], [ -E(3)^2, 0, 0, 0 ], [ 0, 0, E(3)^2, 0 ], [ 0, 0, 1, 1 ] ] ], <action isomorphism> ) ]</pre>