I'm trying to construct the irreducible representations (over $\mathbb{C}$) of the finite group $Sp(4,\mathbb{F}_2)$.
Using GAP, the character table is as follows:
$$ \left(\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 \\ 5 & -3 & 1 & 1 & 2 & 0 & -1 & -1 & -1 & 0 & 1 \\ 5 & 3 & 1 & -1 & 2 & 0 & -1 & 1 & -1 & 0 & -1 \\ 5 & -1 & 1 & 3 & -1 & -1 & 2 & 1 & -1 & 0 & 0 \\ 5 & 1 & 1 & -3 & -1 & 1 & 2 & -1 & -1 & 0 & 0 \\ 9 & -3 & 1 & -3 & 0 & 0 & 0 & 1 & 1 & -1 & 0 \\ 9 & 3 & 1 & 3 & 0 & 0 & 0 & -1 & 1 & -1 & 0 \\ 10 & -2 & -2 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & -1 \\ 10 & 2 & -2 & -2 & 1 & -1 & 1 & 0 & 0 & 0 & 1 \\ 16 & 0 & 0 & 0 & -2 & 0 & -2 & 0 & 0 & 1 & 0 \\ \end{matrix}\right)$$
Here are the representations I understand.
- The first line is the trivial representation $\mathbb{C}$.
- The second line is a sort of "alternating representation" $\mathbb{C}_a$ coming from the fact that the abelianization of $Sp(4,\mathbb{F}_2)$ is $\mathbb{Z}/2\mathbb{Z}$.
- One of the $9$-dimensional representations is "almost" a permutation representation. Here is what I mean. Let $\omega$ be the symplectic form on $\mathbb{F}_2^4$. Define $\mathcal{P}$ to be the set of $2$-dimensional subspaces of $\mathbb{F}_2^4$ on which $\omega$ restricts to a nonzero form. The group $Sp(4,\mathbb{F}_2)$ acts transitively on $\mathcal{P}$ and the stabilizer of an element of $\mathcal{P}$ is isomorphic to $SL(2,\mathbb{F}_2) \times SL(2,\mathbb{F}_2)$. We thus have $$|\mathcal{P}| = |Sp(4,\mathbb{F}_2)| / |SL(2,\mathbb{F}_2) \times SL(2,\mathbb{F}_2)| = 720/6^2=20.$$ There is a fixed-point free involution $\sigma$ of $\mathcal{P}$ that takes $V$ to the $\omega$-orthogonal complement of $V$. Define $\widehat{\mathcal{P}}$ to be the set of unordered pairs $\{V,\sigma(V)\}$ with $V \in \mathcal{P}$, so $|\widehat{\mathcal{P}}| = 10$. The group $Sp(4,\mathbb{F}_2)$ acts transitively on $\widehat{\mathcal{P}}$, so the trivial subrepresentation of the permutation representation $\mathbb{C}[\widehat{\mathcal{P}}]$ is $1$-dimensional. The quotient $W$ of $\mathbb{C}[\widehat{\mathcal{P}}]$ by this trivial subrepresentation is thus a $9$-dimensional representation with no trivial subrepresentations. Looking at the above character table, we see that the only possibility is that $W$ is irreducible.
- The last line is the Steinberg representation.
The other thing I know is that other than the Steinberg representation, the representations come in pairs $X$ and $X \otimes \mathbb{C}_a$ (the Steinberg representation is not changed when you tensor it with $\mathbb{C}_a$).
Can anyone help me construct the missing representations? Modulo tensoring with $\mathbb{C}_a$, there are two $5$-dimensional representations and one $10$-dimensional one.