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

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

Contents

Particular cases

Related information

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