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

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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

m n = 2m maximal unipotent subgroup of symplectic group of degree n = 2m 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 subgroup
Size 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 subgroup
Cumulative 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 q and the underlying prime by p. Let r = \log_p q.

First we consider the case where p \ne 2, so q is odd.

m n = 2m Total number of irreps equals number of conjugacy classes in maximal unipotent subgroup of Sp(n,q) Number of irreps of degree 1 Number of irreps of degree q Number of irreps of degree q^2 Number of irreps of degree q^3
1 2 q q
2 4 2q^2 - 1 q^2 q^2 - 1
equals
(q - 1)(q + 1)
3 6 q^4 + 4q^3 - 2q^2 - 3q + 1 q^3 q^4 - q
equals
q(q - 1)(q^2 + q + 1)
2q^3 - 2q^2
equals
2q^2(q - 1)
q^3 - 2q + 1
equals
(q - 1)(q^2 + q - 1)

Partial sum values of squares of degrees

First we consider the case where p \ne 2, so q is odd.

m n = 2m Order of maximal unipotent subgroup of Sp(n,q) equals q^{m^2} = q^{n^2/4} Sum of squares of degrees for degree 1 irreps Sum of squares of degrees for irreps of degree dividing q Sum of squares of degrees for irreps of degree dividing q^2 Sum of squares of degrees for irreps of degree dividing q^3
1 2 q q
2 4 q^4 q^2 q^4
3 6 q^9 q^3 q^6 2q^7 - q^6 q^9