Linear representation theory of 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.

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 ] ] ], )   ,  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 ] ]     ], ), 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 ] ]     ], ), 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 ] ] ], ), 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 ] ] ], ), 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 ] ]     ], ), 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 ] ] ], ), 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 ] ]     ], ),  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) ] ] ], ) ]