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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.

  1. The first line is the trivial representation $\mathbb{C}$.
  2. 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 $\mathbb{Z}/2\mathbb{Z}$.
  3. 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.
  4. 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.

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    $\begingroup$ It might help to use the fact that ${\rm Sp}(4,2) \cong S_6$, and the representations of $S_n$ are well understood. The $5$-dimensional representations come from permutation representations. $\endgroup$
    – Derek Holt
    Nov 20, 2014 at 1:45
  • $\begingroup$ @DerekHolt : Is there an easy to way to see that they come from permutation representations? $\endgroup$
    – Melanie
    Nov 20, 2014 at 1:59
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    $\begingroup$ I assume "coming from permutation representations" means "permutation representation of $S_6$ modulo trivial subrepresentation". In this case, it is easy to see that $S_6$ has two such representations of dimension $5$. One of them is obtained by having $S_6$ act on $\left\{1,2,3,4,5,6\right\}$ in the usual way, and the other by composing this representation with the infamous outer automorphism of $S_6$. It might be a bit of work to check that these are nonequivalent. See §1.5 in math.stanford.edu/~vakil/files/sixjan2308.pdf . $\endgroup$ Nov 20, 2014 at 2:12
  • $\begingroup$ Ah, I see! This actually answers the question. Somehow I had forgotten the isomorphism that Derek referred to; I was reading quickly and thought he was talking about $Sp(2,2) \cong S_3$. $\endgroup$
    – Melanie
    Nov 20, 2014 at 2:16
  • $\begingroup$ @Melanie: You should look at the extensive information about this and related groups in the Atlas of Finite Groups (pages 4-5). There are of course various ways of constructing the actual representations, but for this it's always helpful to consider other incarnations of the group. You might also find this online material useful: brauer.maths.qmul.ac.uk/Atlas/v3/… $\endgroup$ Nov 20, 2014 at 14:13

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