# Linear representation theory of maximal unipotent subgroup of symplectic group over a finite field

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: maximal unipotent subgroup of symplectic group. 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 maximal unipotent subgroup of symplectic group | View other specific information about group families for rings of the type finite field

## Particular cases

maximal unipotent subgroup of symplectic group of degree | Linear representation theory page for such a group over a finite field | ||
---|---|---|---|

1 | 2 | additive group of the field, which, for a finite field, is an elementary abelian group | -- |

2 | 4 | maximal unipotent subgroup of symplectic group of degree four | linear representation theory of maximal unipotent subgroup of symplectic group of degree four over a finite field |

3 | 6 | maximal unipotent subgroup of symplectic group of degree six | linear representation theory of maximal unipotent subgroup of symplectic group of degree six over a finite field |

## Related information

### More information about this family

### Similar families

## Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroupSize bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroupCumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

### Listing of degrees

We denote the field size by and the underlying prime by . Let .

First we consider the case where , so is odd.

Total number of irreps equals number of conjugacy classes in maximal unipotent subgroup of | Number of irreps of degree 1 | Number of irreps of degree | Number of irreps of degree | Number of irreps of degree | ||
---|---|---|---|---|---|---|

1 | 2 | |||||

2 | 4 | equals | ||||

3 | 6 | equals |
equals |
equals |

### Partial sum values of squares of degrees

First we consider the case where , so is odd.

Order of maximal unipotent subgroup of equals | Sum of squares of degrees for degree 1 irreps | Sum of squares of degrees for irreps of degree dividing | Sum of squares of degrees for irreps of degree dividing | Sum of squares of degrees for irreps of degree dividing | ||
---|---|---|---|---|---|---|

1 | 2 | |||||

2 | 4 | |||||

3 | 6 |