Linear representation theory of special linear group:SL(2,9)

This article gives specific information, namely, linear representation theory, about a particular group, namely: special linear group:SL(2,9).
View linear representation theory of particular groups | View other specific information about special linear group:SL(2,9)

Summary

Item	Value
degrees of irreducible representations over a splitting field (such as ${\overline {\mathbb {Q} }}$ or $\mathbb {C}$)	1,4,4,5,5,8,8,8,8,9,10,10,10 number: 13, maximum: 10, lcm: 360, sum of squares: 720

Family contexts

Family	Parameter values	General discussion of linear representation theory of family
special linear group of degree two over a finite field, $SL(2,q)$ where $q$ is the field size	$q=9$, i.e., field:F9, so the group is $SL(2,9)$	linear representation theory of special linear group of degree two over a finite field
double cover of alternating group $2\cdot A_{n}$	$n=6$, so the group is $2\cdot A_{6}$	linear representation theory of double cover of alternating group

GAP implementation

The degrees of irreducible representations can be computed using CharacterDegrees as follows:

gap> CharacterDegrees(SL(2,9));
[ [ 1, 1 ], [ 4, 2 ], [ 5, 2 ], [ 8, 4 ], [ 9, 1 ], [ 10, 3 ] ]

This says that there is 1 irreducible representation of degree 1, 2 of degree 4, 2 of degree 5, 4 of degree 8, 1 of degree 9, and 3 of degree 10.

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

gap> Irr(CharacterTable(SL(2,9)));
[ Character( CharacterTable( SL(2,9) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
      1, 1 ] ), Character( CharacterTable( SL(2,9) ),
    [ 4, 2, -1, -4, -2, 1, -1, -1, 1, 1, 0, 0, 0 ] ),
  Character( CharacterTable( SL(2,9) ), [ 4, -1, 2, -4, 1, -2, -1, -1, 1, 1,
      0, 0, 0 ] ), Character( CharacterTable( SL(2,9) ),
    [ 5, 2, -1, 5, 2, -1, 0, 0, 0, 0, -1, 1, -1 ] ),
  Character( CharacterTable( SL(2,9) ), [ 5, -1, 2, 5, -1, 2, 0, 0, 0, 0,
      -1, 1, -1 ] ), Character( CharacterTable( SL(2,9) ),
    [ 8, 1, 1, -8, -1, -1, -E(5)-E(5)^4, -E(5)^2-E(5)^3, E(5)+E(5)^4,
      E(5)^2+E(5)^3, 0, 0, 0 ] ), Character( CharacterTable( SL(2,9) ),
    [ 8, 1, 1, -8, -1, -1, -E(5)^2-E(5)^3, -E(5)-E(5)^4, E(5)^2+E(5)^3,
      E(5)+E(5)^4, 0, 0, 0 ] ), Character( CharacterTable( SL(2,9) ),
    [ 8, -1, -1, 8, -1, -1, -E(5)^2-E(5)^3, -E(5)-E(5)^4, -E(5)^2-E(5)^3,
      -E(5)-E(5)^4, 0, 0, 0 ] ), Character( CharacterTable( SL(2,9) ),
    [ 8, -1, -1, 8, -1, -1, -E(5)-E(5)^4, -E(5)^2-E(5)^3, -E(5)-E(5)^4,
      -E(5)^2-E(5)^3, 0, 0, 0 ] ), Character( CharacterTable( SL(2,9) ),
    [ 9, 0, 0, 9, 0, 0, -1, -1, -1, -1, 1, 1, 1 ] ),
  Character( CharacterTable( SL(2,9) ), [ 10, 1, 1, 10, 1, 1, 0, 0, 0, 0, 0,
      -2, 0 ] ), Character( CharacterTable( SL(2,9) ),
    [ 10, -1, -1, -10, 1, 1, 0, 0, 0, 0, -E(8)+E(8)^3, 0, E(8)-E(8)^3 ] ),
  Character( CharacterTable( SL(2,9) ), [ 10, -1, -1, -10, 1, 1, 0, 0, 0, 0,
      E(8)-E(8)^3, 0, -E(8)+E(8)^3 ] ) ]

The irreducible representations can be computed using the IrreducibleRepresentations function as follows:

gap> IrreducibleRepresentations(SL(2,9));