Linear representation theory of special linear group:SL(2,5)

This article gives specific information, namely, linear representation theory, about a particular group, namely: special linear group:SL(2,5).
View linear representation theory of particular groups | View other specific information about special linear group:SL(2,5)

This article gives information on the linear representation theory in characteristics other than 2,3,5 of special linear group:SL(2,5), which is the special linear group of degree two over field:F5. The group is also the binary icosahedral group and is one of the finite binary von Dyck groups.

Summary

Item	Value
Degrees of irreducible representations over a splitting field (such as $C$ or $\bar{Q}$ )	1,2,2,3,3,4,4,5,6 maximum: 6, lcm: 60, number: 9, sum of squares: 120, quasirandom degree: 2

Family contexts

Family	Parameter values	General discussion of linear representation theory of family
special linear group of degree two over a finite field (denoted $S L (2, q)$ for field size $q$ )	$q = 5$ , i.e., field:F5, so the group is $S L (2, 5)$	linear representation theory of special linear group of degree two over a finite field
double cover of alternating group $2 \cdot A_{n}$	$n = 5$ , so the group is $2 \cdot A_{5}$	linear representation theory of double cover of alternating group

GAP implementation

Degrees of irreducible representations

The degrees of irreducible representations can be determined using the CharacterDegrees function:

gap> CharacterDegrees(SL(2,5));
[ [ 1, 1 ], [ 2, 2 ], [ 3, 2 ], [ 4, 2 ], [ 5, 1 ], [ 6, 1 ] ]

This says that there is 1 irreducible representation of degree 1, 2 of degree 2, 2 of degree 3, 2 of degree 4, 1 of degree 5, 1 of degree 6.

Character table

The character table can be computed using the Irr and CharacterTable functions:

gap> Irr(CharacterTable(SL(2,5)));
[ Character( CharacterTable( SL(2,5) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( SL(2,5) ), [ 2, -E(5)-E(5)^4, -E(5)^2-E(5)^3,
      -2, E(5)+E(5)^4, E(5)^2+E(5)^3, -1, 1, 0 ] ),
  Character( CharacterTable( SL(2,5) ), [ 2, -E(5)^2-E(5)^3, -E(5)-E(5)^4,
      -2, E(5)^2+E(5)^3, E(5)+E(5)^4, -1, 1, 0 ] ),
  Character( CharacterTable( SL(2,5) ), [ 3, -E(5)^2-E(5)^3, -E(5)-E(5)^4,
      3, -E(5)^2-E(5)^3, -E(5)-E(5)^4, 0, 0, -1 ] ),
  Character( CharacterTable( SL(2,5) ), [ 3, -E(5)-E(5)^4, -E(5)^2-E(5)^3,
      3, -E(5)-E(5)^4, -E(5)^2-E(5)^3, 0, 0, -1 ] ),
  Character( CharacterTable( SL(2,5) ), [ 4, -1, -1, 4, -1, -1, 1, 1, 0 ] )
    , Character( CharacterTable( SL(2,5) ),
    [ 4, 1, 1, -4, -1, -1, 1, -1, 0 ] ),
  Character( CharacterTable( SL(2,5) ), [ 5, 0, 0, 5, 0, 0, -1, -1, 1 ] ),
  Character( CharacterTable( SL(2,5) ), [ 6, -1, -1, -6, 1, 1, 0, 0, 0 ] )
 ]

The character table can be displayed more nicely as follows:

gap> Display(CharacterTable(SL(2,5)));
CT17

     2  3   1   1  3   1   1  1  1  2
     3  1   .   .  1   .   .  1  1  .
     5  1   1   1  1   1   1  .  .  .

       1a 10a 10b 2a  5a  5b 3a 6a 4a

X.1     1   1   1  1   1   1  1  1  1
X.2     2   A  *A -2  -A -*A -1  1  .
X.3     2  *A   A -2 -*A  -A -1  1  .
X.4     3  *A   A  3  *A   A  .  . -1
X.5     3   A  *A  3   A  *A  .  . -1
X.6     4  -1  -1  4  -1  -1  1  1  .
X.7     4   1   1 -4  -1  -1  1 -1  .
X.8     5   .   .  5   .   . -1 -1  1
X.9     6  -1  -1 -6   1   1  .  .  .

A = -E(5)-E(5)^4
  = (1-ER(5))/2 = -b5

Irreducible representations

The irreducible linear representations can be computed explicitly using the IrreducibleRepresentations function:

gap> IrreducibleRepresentations(SL(2,5));
[ CompositionMapping(
    [ (2,5,4,3)(6,11,16,21)(7,15,19,23)(8,12,20,24)(9,13,17,25)(10,14,18,
        22), (2,16,9)(3,21,15)(4,6,17)(5,11,23)(7,22,10)(8,12,13)(14,18,
        19)(20,24,25) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ], <action isomorphism> )
    ,
  CompositionMapping( [ (2,5,4,3)(6,11,16,21)(7,15,19,23)(8,12,20,24)(9,13,
        17,25)(10,14,18,22), (2,16,9)(3,21,15)(4,6,17)(5,11,23)(7,22,10)(8,
        12,13)(14,18,19)(20,24,25) ] ->
    [ [ [ -E(5)^2-E(5)^4, E(5)-E(5)^2 ], [ -1, E(5)^2+E(5)^4 ] ],
      [ [ E(5)^3, E(5)^3 ], [ E(5)+E(5)^4, E(5)+E(5)^2+E(5)^4 ] ]
     ], <action isomorphism> ),
  CompositionMapping( [ (2,5,4,3)(6,11,16,21)(7,15,19,23)(8,12,20,24)(9,13,
        17,25)(10,14,18,22), (2,16,9)(3,21,15)(4,6,17)(5,11,23)(7,22,10)(8,
        12,13)(14,18,19)(20,24,25) ] ->
    [ [ [ -E(5)^4, -1 ], [ -E(5)-E(5)^2-E(5)^4, E(5)^4 ] ],
      [ [ E(5)+E(5)^2+E(5)^4, -E(5)^4 ], [ E(5)+E(5)^2+E(5)^4, E(5)^3 ] ]
     ], <action isomorphism> ),
  CompositionMapping( [ (2,5,4,3)(6,11,16,21)(7,15,19,23)(8,12,20,24)(9,13,
        17,25)(10,14,18,22), (2,16,9)(3,21,15)(4,6,17)(5,11,23)(7,22,10)(8,
        12,13)(14,18,19)(20,24,25) ] ->
    [ [ [ -E(5)^2-E(5)^3, -E(5)^2-E(5)^3, 1 ], [ 0, -1, 0 ],
          [ E(5)^2+E(5)^3, -1, E(5)^2+E(5)^3 ] ],
      [ [ 0, 0, 1 ], [ -E(5)^2-E(5)^3, -E(5)^2-E(5)^3, 1 ],
          [ E(5)^2+E(5)^3, -1, E(5)^2+E(5)^3 ] ] ], <action isomorphism> ),
  CompositionMapping(
    [ (2,5,4,3)(6,11,16,21)(7,15,19,23)(8,12,20,24)(9,13,17,25)(10,14,18,
        22), (2,16,9)(3,21,15)(4,6,17)(5,11,23)(7,22,10)(8,12,13)(14,18,
        19)(20,24,25) ] ->
    [ [ [ 1, -E(5)^2-E(5)^3, E(5)^2+E(5)^3 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ],
      [ [ -E(5)^2-E(5)^3, -E(5)^2-E(5)^3, -1 ], [ 0, 0, -1 ],
          [ 1, -E(5)^2-E(5)^3, E(5)^2+E(5)^3 ] ] ], <action isomorphism> ),
  CompositionMapping(
    [ (2,5,4,3)(6,11,16,21)(7,15,19,23)(8,12,20,24)(9,13,17,25)(10,14,18,
        22), (2,16,9)(3,21,15)(4,6,17)(5,11,23)(7,22,10)(8,12,13)(14,18,
        19)(20,24,25) ] ->
    [ [ [ -E(5)^2-E(5)^4, 2*E(5)+E(5)^2+2*E(5)^3+2*E(5)^4, -E(5)-2*E(5)^3,
              -E(5)^3 ], [ -1, E(5)^4, -E(5)-E(5)^3-E(5)^4, -E(5)^3 ],
          [ E(5)^3, E(5)+E(5)^4, -E(5)+E(5)^2, -E(5)-E(5)^3 ],
          [ -E(5)-E(5)^3, -E(5)-E(5)^2-E(5)^4, E(5)-E(5)^2, E(5) ] ],
      [ [ 0, 0, E(5)^3, 0 ], [ E(5), 0, 0, 0 ], [ 0, E(5), 0, 0 ],
          [ E(5)+E(5)^2+E(5)^4, -E(5)+E(5)^2, -E(5)-E(5)^2-E(5)^4, 1 ] ]
     ], <action isomorphism> ),
  CompositionMapping( [ (2,5,4,3)(6,11,16,21)(7,15,19,23)(8,12,20,24)(9,13,
        17,25)(10,14,18,22), (2,16,9)(3,21,15)(4,6,17)(5,11,23)(7,22,10)(8,
        12,13)(14,18,19)(20,24,25) ] ->
    [ [ [ -E(5)^3-E(5)^4, -E(5)^2-E(5)^3-E(5)^4,
              2/5*E(5)-1/5*E(5)^2+1/5*E(5)^3+3/5*E(5)^4,
              6/5*E(5)+7/5*E(5)^2+3/5*E(5)^3-1/5*E(5)^4 ],
          [ -E(5)-2*E(5)^2-E(5)^3-E(5)^4, -E(5)-E(5)^2, -1,
              -E(5)^2-E(5)^3-E(5)^4 ],
          [ -E(5)^4, -E(5)^2-E(5)^3-E(5)^4, 3/5*E(5)+1/5*E(5)^2-1/5*E(5)^3
                 +2/5*E(5)^4, 4/5*E(5)+3/5*E(5)^2+2/5*E(5)^3+1/5*E(5)^4 ],
          [ E(5)+E(5)^2, E(5), -6/5*E(5)-2/5*E(5)^2-3/5*E(5)^3-4/5*E(5)^4,
              2/5*E(5)+4/5*E(5)^2+6/5*E(5)^3+3/5*E(5)^4 ] ],
      [ [ 0, 0, 0, -E(5) ],
          [ E(5)+E(5)^2, -E(5)^3-E(5)^4, -2/5*E(5)-4/5*E(5)^2-1/5*E(5)^3
                 -3/5*E(5)^4, 4/5*E(5)+8/5*E(5)^2+7/5*E(5)^3+1/5*E(5)^4 ],
          [ -E(5)^3, -E(5)-E(5)^2-E(5)^3,
              -2/5*E(5)-4/5*E(5)^2-1/5*E(5)^3-3/5*E(5)^4,
              -1/5*E(5)-2/5*E(5)^2-3/5*E(5)^3-4/5*E(5)^4 ],
          [ E(5)^2+E(5)^3, E(5)^2,
              4/5*E(5)-2/5*E(5)^2+2/5*E(5)^3+1/5*E(5)^4,
              -3/5*E(5)-1/5*E(5)^2+1/5*E(5)^3+3/5*E(5)^4 ] ]
     ], <action isomorphism> ),
  CompositionMapping( [ (2,5,4,3)(6,11,16,21)(7,15,19,23)(8,12,20,24)(9,13,
        17,25)(10,14,18,22), (2,16,9)(3,21,15)(4,6,17)(5,11,23)(7,22,10)(8,
        12,13)(14,18,19)(20,24,25) ] ->
    [ [ [ 1, 0, 0, 0, 0 ], [ -1, -1, -1, -1, -1 ], [ 0, 0, 0, 1, 0 ],
          [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1 ] ],
      [ [ 0, 0, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ],
          [ 0, 1, 0, 0, 0 ], [ -1, -1, -1, -1, -1 ] ]
     ], <action isomorphism> ),
  CompositionMapping( [ (2,5,4,3)(6,11,16,21)(7,15,19,23)(8,12,20,24)(9,13,
        17,25)(10,14,18,22), (2,16,9)(3,21,15)(4,6,17)(5,11,23)(7,22,10)(8,
        12,13)(14,18,19)(20,24,25) ] ->
    [ [ [ -E(5)^2, -E(5), E(5)^3-E(5)^4, E(5), 0, -E(5)^3 ],
          [ 0, 0, 1, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0 ],
          [ E(5)+E(5)^2+E(5)^4, 0, E(5)+E(5)^4, E(5)^2+E(5)^4, E(5)^3,
              -E(5)^4 ],
          [ E(5)+E(5)^3, E(5)^2, -E(5)-E(5)^2-E(5)^4, E(5)+E(5)^3+E(5)^4,
              -E(5)^4, 0 ],
          [ E(5)^2-E(5)^3, E(5), E(5)^4, E(5)^2, E(5), 0 ] ],
      [ [ 0, 0, E(5)^3, 0, 0, 0 ], [ -E(5)^3, 0, 0, 0, 0, 0 ],
          [ 0, -E(5)^4, 0, 0, 0, 0 ],
          [ E(5)^2, E(5), -E(5)^3+E(5)^4, -E(5), 0, E(5)^3 ],
          [ 0, 0, 0, 0, 0, -E(5)^4 ],
          [ -E(5)-E(5)^3-E(5)^4, 0, -E(5)-E(5)^3, -E(5)-E(5)^4, -1, E(5) ]
         ] ], <action isomorphism> ) ]