# Linear representation theory of special linear group of degree two over a finite field

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: special linear group of degree two. This article restricts attention to the case where the underlying ring is a finite field.

View linear representation theory of group families | View other specific information about special linear group of degree two | View other specific information about group families for rings of the type finite field

This article describes the linear representation theory of the special linear group of degree two over a finite field. The order (size) of the field is , and the characteristic prime is . is a power of . We denote the group as or .

See also the linear representation theories of: general linear group of degree two, projective general linear group of degree two, and projective special linear group of degree two.

For linear representation theory in characteristics that divide the order of the group, refer:

- Modular representation theory of special linear group of degree two over a finite field in its defining characteristic: In characteristic , same as the field characteristic.
- (Modular representation theory in other characteristics that divide or -- link to be added)

## Summary

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

degrees of irreducible representations over a splitting field (such as or ) | Case odd: 1 (1 time), (2 times), (2 times), ( times), (1 time), ( times) Case even: 1 (1 time), ( times), (1 time), ( times) |

number of irreducible representations | Case odd: Case even: See number of irreducible representations equals number of conjugacy classes, element structure of special linear group of degree two over a finite field#Conjugacy class structure |

quasirandom degree (minimum degree of nontrivial irreducible representation) | Case odd: Case even: |

maximum degree of irreducible representation over a splitting field | if if |

lcm of degrees of irreducible representations over a splitting field | Case : We get 6 Case odd, : Case even: |

sum of squares of degrees of irreducible representations over a splitting field | , equal to the group order. See sum of squares of degrees of irreducible representations equals group order |

## Particular cases

## Irreducible representations

### Case odd

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