# Linear representation theory of special linear group:SL(2,3)

This article gives specific information, namely, linear representation theory, about a particular group, namely: special linear group:SL(2,3).

View linear representation theory of particular groups | View other specific information about special linear group:SL(2,3)

## Summary

Item | Value |
---|---|

degrees of irreducible representations over a splitting field (such as or ) | 1,1,1,2,2,2,3 grouped: 1 (3 times), 2 (3 times), 3 (1 time) maximum: 3, lcm: 6, number: 7, sum of squares: 24, quasirandom degree: 1 |

Schur index values of irreducible representations over a splitting field | 1,1,1,2,1,1,1 |

smallest ring of realization for all irreducible representations (characteristic zero) | , same as Same as ring generated by character values |

minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero) | Same as field generated by character values |

condition for a field to be a splitting field | characteristic not equal to 2 or 3 and must contain a primitive cube root of unity. For a finite field of size , this is equivalent to 6 dividing . |

minimal splitting field in characteristic | Case : prime field Case : , a quadratic extension of . |

smallest size splitting field | field:F7, i.e., the field of 7 elements. |

degrees of irreducible representations over a non-splitting field (such as the field of rational numbers) | 1,2,3,4,4 number: 5 |

orbits of irreducible representations over a splitting field under action of automorphism group | 1 orbit of size 1 of degree 1 representations, 1 orbit of size 2 of degree 1 representations, 1 orbit of size 1 of degree 2 representations, 1 orbit of size 2 of degree 2 representations, 1 orbit of size 1 of degree 3 representations number: 5 |

orbits of irreducible representations over a splitting field under action of Galois group | Characteristic : all orbits of size 1. Characteristic 0 or : 1 orbit of size 1 of degree 1 representations, 1 orbit of size 2 of degree 1 representations, 1 orbit of size 1 of degree 2 representations, 1 orbit of size 2 of degree 2 representations, 1 orbit of size 1 of degree 3 representations. number: 5 |

orbits of irreducible representations over a splitting field under action of one-dimensional representations by multiplication, i.e., up to projective equivalence | Orbit sizes: 3 (degree 1 representations), 3 (degree 2 representations), 1 (degree 3 representations) number: 3 |

## Family contexts

Family name | Parameter values | General discussion of linear representation theory of family |
---|---|---|

double cover of alternating group | degree , i.e., the group | linear representation theory of double cover of alternating group |

special linear group of degree two over a finite field of size | , i.e., field:F3, i.e., the group is | linear representation theory of special linear group of degree two over a finite field |

COMPARE AND CONTRAST: View linear representation theory of groups of order 24 to compare and contrast the linear representation theory with other groups of order 24.

## Irreducible representations

### Summary information

Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2 or 3, except in the last two columns, where we consider what happens in characteristic 2 and characteristic 3.

Name of representation type | Number of representations of this type | Degree | Schur index | Criterion for field | Kernel | Quotient by kernel (on which it descends to a faithful representation) | Characteristic 2 | Characteristic 3 |
---|---|---|---|---|---|---|---|---|

trivial | 1 | 1 | 1 | any | whole group | trivial group | works | works |

one-dimensional nontrivial | 2 | 1 | 1 | must contain a primitive cube root of unity, or equivalently, the polynomial must split. | Q8 in SL(2,3) (the 2-Sylow subgroup) | cyclic group:Z3 | works | works, same as trivial |

irreducible two-dimensional (quaternionic, rational character) | 1 | 2 | 2 | ? | trivial subgroup, i.e., the representation is faithful | special linear group:SL(2,3) | ? | ? |

two-dimensional with non-rational character | 2 | 2 | 1 | must contain a primitive cube root of unity, or equivalently, the polynomial must split. | trivial subgroup | special linear group:SL(2,3) | ? | ? |

three-dimensional | 1 | 3 | 1 | any | center of special linear group:SL(2,3) | alternating group:A4 | indecomposable but not irreducible | ? |

## Character table

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):Orthogonality relations: Character orthogonality theorem | Column orthogonality theoremSeparation results(basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zeroNumerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integersCharacter value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

In the table below, we denote by a primitive cube root of unity.

Representation/conjugacy class representative and size | (size 1) | (size 1) | (size 6) | (size 4) | (size 4) | (size 4) | (size 4) |
---|---|---|---|---|---|---|---|

trivial | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

nontrivial one-dimensional | 1 | 1 | 1 | ||||

nontrivial one-dimensional | 1 | 1 | 1 | ||||

irreducible two-dimensional (quaternionic, rational character) | 2 | -2 | 0 | -1 | -1 | 1 | 1 |

irreducible two-dimensional | 2 | -2 | 0 | ||||

irreducible two-dimensional | 2 | -2 | 0 | ||||

three-dimensional | 3 | 3 | -1 | 0 | 0 | 0 | 0 |

## Orthogonality relations and numerical checks

General statement | Verification in this case |
---|---|

number of irreducible representations equals number of conjugacy classes | Both numbers are equal to 7 (the degrees of irreducible representations are 1,1,1,2,2,2,3, and the sizes of conjugacy classes are 1,1,4,4,4,4,6). See also element structure of special linear group:SL(2,3). As special linear group , odd: Both numbers are equal to . |

number of orbits of irreducible representations equals number of orbits under automorphism group | Both numbers are 5. The orbit sizes for irreducible representations are: 1 (degree 1 representation), 2 (degree 1 representations), 1 (degree 2 representation), 2 (degree 2 representations), 1 (degree 3 representation). The orbit sizes for elements are 1,1,6,8 (2 conjugacy classes of size 4 each), 8 (2 conjugacy classes of size 4 each). See also element structure of special linear group:SL(2,3). |

number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy | Both numbers are equal to 5. The irreducible representations over rationals have degrees 1, 2 (the sum of the two one-dimensional nontrivial representations), 3, 4 (double of the quaternionic representation), 4 (sum of the two other two-dimensional representations). The equivalence classes under rational conjugacy have sizes 1,1,6,8,8. See also element structure of special linear group:SL(2,3). |

number of one-dimensional representations equals order of abelianization | The derived subgroup is Q8 in SL(2,3), a 2-Sylow subgroup of order eight and index three. The abelianization is the corresponding quotient group, isomorphic to cyclic group:Z3, and has order three. The number of one-dimensional representations is also equal to three. |

sum of squares of degrees of irreducible representations equals order of group | The degrees of the irreducible representations are . We have . |

degree of irreducible representation divides index of abelian normal subgroup | The center (see center of special linear group:SL(2,3) is the unique nontrivial abelian normal subgroup, and it is a subgroup of order 2 and index 12. All the degrees of irreducible representations divide 12. |

order of inner automorphism group bounds square of degree of irreducible representation | The center of special linear group:SL(2,3) has order 2 and index 12, and the quotient group, the inner automorphism group, has order 12 (the quotient is and is isomorphic to alternating group:A4). The squares of the degrees of irreducible representations are all less than or equal to this number (the largest square being ). |

degree of irreducible representation is bounded by index of abelian subgroup | There is an abelian 2-subnormal subgroup of order four and index six, isomorphic to cyclic group:Z4 (in fact, there is a conjugacy class of three such subgroups). All the degrees of irreducible representations are less than or equal to . |

## Isoclinism and projective representations

Template:Irrep isoclinism information facts to check against

Please compare this with projective representation theory of alternating group:A4.

### Grouping by restriction to center

Restriction to center as a representation of center of special linear group:SL(2,3) which is isomorphic to cyclic group:Z2 (this determines, essentially, the cohomology class of the projective representation for the inner automorphism group) | List of irreducible projective representations of the inner automorphism group (which is alternating group:A4) | List of corresponding linear representations of special linear group:SL(2,3) | List of degrees | Sum of squares of degrees (should equal order of inner automorphism group, which is 12) |
---|---|---|---|---|

trivial representation | trivial representation | all the three one-dimensional representations, and the three-dimensional irreducible representation (which descends to the standard representation for alternating group:A4) | 1,1,1,3 | 12 |

sign representation | nontrivial two-dimensional projective representation | all the two-dimensional representations | 2,2,2 | 12 |

### Grouping by projective representation

Irreducible projective representation of inner automorphism group (which is alternating group:A4) | Degree | Size of stabilizer under action of one-dimensional representations of special linear group:SL(2,3) (or equivalently, of its abelianization, cyclic group:Z3) | Size of orbit (equals order of abelianization (3) divided by size of stabilizer) | List of irreducible representations of special linear group:SL(2,3) |
---|---|---|---|---|

trivial representation | 1 | 1 | 3 | trivial representation and the two other one-dimensional representations |

nonlinear projective two-dimensional representation | 2 | 1 | 3 | all the three two-dimensional representations |

linear three-dimensional representation (standard representation of alternating group:A4) | 3 | 3 | 1 | the unique three-dimensional representation |

## GAP implementation

### Degrees of irreducible representations

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

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

This says that there are three irreducible representations of degree one, three irreducible representations of degree two, and one irreducible representation of degree three.

### Character table

The characters of irreducible representations can be computed using Irr and CharacterTable as follows:

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

A nicer display of the character table can be achieved with the Display function:

gap> Display(CharacterTable(SL(2,3))); CT1 2 3 1 1 3 1 1 2 3 1 1 1 1 1 1 . 1a 6a 6b 2a 3a 3b 4a X.1 1 1 1 1 1 1 1 X.2 1 A /A 1 /A A 1 X.3 1 /A A 1 A /A 1 X.4 2 1 1 -2 -1 -1 . X.5 2 /A A -2 -A -/A . X.6 2 A /A -2 -/A -A . X.7 3 . . 3 . . -1 A = E(3)^2 = (-1-ER(-3))/2 = -1-b3

### Irreducible representations

The irreducible representations can be computed using IrreducibleRepresentations as follows:

gap> IrreducibleRepresentations(SL(2,3)); [ CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] -> [ [ [ E(3)^2 ] ], [ [ 1 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] -> [ [ [ E(3) ] ], [ [ 1 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] -> [ [ [ E(3), -E(3) ], [ 0, E(3)^2 ] ], [ [ E(3), 1 ], [ E(3), -E(3) ] ] ], <action isomorphism> ), CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] -> [ [ [ E(3)^2, E(3) ], [ 0, 1 ] ], [ [ -E(3)^2, -E(3) ], [ -E(3), E(3)^2 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] -> [ [ [ E(3), E(3)^2 ], [ 0, 1 ] ], [ [ 0, 1 ], [ -1, 0 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] -> [ [ [ 0, 0, 1 ], [ 0, 1, 0 ], [ -1, -1, -1 ] ], [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ -1, -1, -1 ] ] ], <action isomorphism> ) ]