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

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): PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Case q odd: q^4 + 4q^3 - 2q^2 - 3q + 1
See number of irreducible representations equals number of conjugacy classes, element structure of maximal unipotent subgroup of symplectic group of degree six 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): PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Case q odd: 1 (occurs q^3 times), q (occurs q^4 - q times), q^2 (occurs 2q^3 - 2q^2 times), q^3 (occurs q^3 - 2q + 1 times)
sum of squares of degrees of irreducible representations q^9 (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^3
condition for a field (characteristic not equal to p) to be a splitting field PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
field generated by character values, which in this case also coincides with the unique minimal splitting field (characteristic zero) PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
unique minimal splitting field (characteristic c \ne 0,p) PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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