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

## Contents

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.
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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 $p$.

## 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)