Linear representation theory of maximal unipotent subgroup of symplectic group of degree four 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 of degree four. 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 of degree four | View other specific information about group families for rings of the type finite field

Let q be a prime power with underlying prime p (we may have q = p). This page covers the linear representation theory of the maximal unipotent subgroup of the symplectic group of degree four Sp(4,q) (in our naming convention, this stands for the group of 4 \times 4 symplectic matrices) over the finite field of size q. This group can also be described as the p-Sylow subgroup of Sp(4,q).

Summary


Item Value
number of conjugacy classes (equals number of irreducible representations over a splitting field) Case q even (i.e., a power of 2): 5q^2 - 6q + 2
Case q odd: 2q^2 - 1
See number of irreducible representations equals number of conjugacy classes, element structure of maximal unipotent subgroup of symplectic group of degree four over a finite field
degrees of irreducible representations over a splitting field (such as \overline{\mathbb{Q}} or \mathbb{C}) Case q even (i.e., a power of 2): 1 (occurs q^2 times), q/2 (occurs 4(q - 1)^2 times), q (occurs 2(q - 1) times). NOTE: For q = 2, 1 = q/2, so we have to count both types of occurences, and we get 1 (occurs q^2 + 4(q-1)^2 = 8 times), 2 (occurs 2 times)
Case q odd: 1 (occurs q^2 times), q (occurs q^2 - 1 times)
sum of squares of degrees of irreducible representations q^4 (equals order of the group)
see sum of squares of degrees of irreducible representations equals order of group
lcm of degrees of irreducible representations q
condition for a field (characteristic not equal to p) to be a splitting field The polynomial x^p - 1 should split completely.
For a finite field of size s, this is equivalent to s \equiv 1 \pmod p.
field generated by character values, which in this case also coincides with the unique minimal splitting field (characteristic zero) Field \mathbb{Q}(\zeta) where \zeta is a primitive p^{th} root of unity. This is a degree p - 1 extension of the rationals.
unique minimal splitting field (characteristic c \ne 0,p) The field of size c^r where r is the order of c mod p.


Related information

Particular cases

Field size q (a prime power) Field characteristic p \log_pq Maximal unipotent subgroup of Sp(4,q) Order of the group (equals q^4) Second part of GAP ID Number of irreducible representations equls number of conjugacy classes (equals 5q^2 - 6q + 2 if p = 2, equals 2q^2 - 1 otherwise) Degrees of irreducible representations (check against summary) Information on linear representation theory
2 2 1 direct product of D8 and Z2 16 11 10 1 (occurs 8 times), 2 (occurs 2 times) linear representation theory of direct product of D8 and Z2
3 3 1 wreath product of Z3 and Z3 81 7 17 1 (occurs 9 times), 3 (occurs 8 times) linear representation theory of wreath product of Z3 and Z3
4 2 2 free product of class two of two Klein four-groups 256 8935 58 1 (occurs 16 times), 2 (occurs 36 times), 4 (occurs 6 times) linear representation theory of free product of class two of two Klein four-groups
5 5 1 maximal unipotent subgroup of symplectic group:Sp(4,5) 625 7 49 1 (occurs 25 times), 5 (occurs 24 times) linear representation theory of maximal unipotent subgroup of symplectic group:Sp(4,5)