Why do these two irreps of $E_6$ have the same dimension?

Question

$E_6$'s Dynkin diagram is a line of 5 vertices, which we will number 1...5, and a sixth one attached to #3, which we will ignore.

$\dim V_{\omega_2} = 351 = \dim V_{2\ \omega_1}$, where $\omega_i$ denotes the corresponding fundamental weight and $V$ the irrep with that high weight. (So $\dim V_{\omega_1} = 27$, the number of lines on a cubic surface.)

"Why" do these two irreps have the same dimension? I'm not sure what I'm asking, obviously, but here's an attempt: do they become isomorphic when restricted to some large isomorphic subgroups of $E_6$?

Incidentally, $V_{\omega_2} = \wedge^2 V_{\omega_1}$, and $Sym^2 V_{\omega_1} = V_{2\omega_1} \oplus (V_{\omega_1})^*$. So maybe the best answer is "the $Sym^2$, which is always larger than the $\wedge^2$, is in this case only larger by something of the same dimension as the original space ($V_{\omega_1}$)".

Answer 1

$\newcommand\Sym{\mathrm{Sym}}$

An extended comment which more or less suggests that your suggested answer might be as good as one can do.

If $G$ has a representation on $V$ which preserves a symmetric trilinear form on $V$, then $\Sym^3(V)$ has a $G$-invariant vector and thus $\Sym^2(V) \otimes V$ has a $G$-invariant vector and thus $V^{\vee}$ is a constituent of $\Sym^2(V)$. So for any such $G$ the virtual representations

$$\wedge^2(V), \quad [\Sym^2(V)] - [V^{\vee}]$$

are both actual representations and both have the same dimension. The Dickson invariant of $E_6$ acting on the $27$-dimensional representation $V$ is the corresponding form in this case.

To give a related example, if you take $V$ and restrict to $F_4$ then it decomposes as $U \oplus \mathbf{C}$. The action of $F_4$ on $U$ also admits an invariant cubic form. Hence you obtain a pair of corresponding representations of $F_4$ of dimensions $325$ which are not isomorphic. Unlike the case of $E_6$, however, neither of these are irreducible. This is clear in one case, because the action of $F_4$ on $U$ preserves a quadratic form. So now we have $[U^{\vee}] = [U]$ and decompositions

$$[\Sym^2(U)] - [U] = 1 + 324,$$ $$[\wedge^2(U)] = 273 + 52,$$

where the numbers refer to irreducible representations of the corresponding dimension.

You also see from this that the restrictions of your $351$ dimensional representations to $F_4$ are still different. But they are still different even when you restrict to the principal $\mathrm{SL}_2$. The $27$-dimensional representation $V$ restricts to the principal $\mathrm{SL}_2$ as a sum of representations $U_1 \oplus U_7 \oplus U_{19}$ where a representation of $\mathrm{SL}_2$ is determined by its dimension. But now:

$$\wedge^2(V) = \wedge^2(U_1 + U_7 + U_{19}) = \wedge^2(U_7) + \wedge^2(U_{19}) + U_7 + U_{19}+ U_7 \otimes U_{19},$$

from which we see that the $351$-dimensional representation $\wedge^2(V)$ has no $\mathrm{SL}_2$-invariants because odd dimensional irreducible representations don't admit symplectic forms. On the other hand,

$$\Sym^2(V) - [V^{\vee}] = \Sym^2(U_1 + U_7 + U_{19}) - (U_1 + U_7 + U_{19}) = \Sym^2(U_1) + \Sym^2(U_7) + \Sym^2(U_{19}) - U_1 + \ldots $$

has at least a $2$-dimensional space of invariants because all representations are self-dual and thus the the odd-dimensional representations are orthogonal (of course one can see they are orthogonal more directly by constrution).

So I think the conclusion is that the existence of an invariant symmetric cubic form guarantees the existence of two representations of the same dimension $\binom{n}{2}$ which have no reason to be related, and in the case of $E_6$ they just both happen to be irreducible.

Answer 2

A geometric picture of the answer given by user484566 is that (certain real form of) $E_6$ is octonionic version of the special linear group $SL_3$. The symmetric three-tensor is the polarization of the octonionic determinant of the 3x3 octonionic-hermitian matrices that constitute the 27 dimensional representation. The other 27 dimensional representation is the dual of this one and the duality pairing gives the incidence relation of the octonionic projective plane. Maybe it's a good exercise to prove irreducibility from the axioms of projective geometry and the fact that (the specific real form of) $E_6$ is the group of projective transformation?