Why do the adjoint representations of three exceptional groups have the same first eight moments? For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The sequence of moments determines the distribution of the trace.
For the adjoint representations of $G_2,F_4$, and $E_7$, the $0$th through $7$th moments are all 1,0,1,1,5,16,80,436.
Is there a conceptual reason for this, or is it just a numerical coincidence? 
Nick Katz asked this question in his graduate class.
One possible way to make this more conceptual is to note that there is more structure on the vector spaces of invariants than just their dimensions. There is an action of $S_n$, and maps front the $n$th invariants tensor the $m$th invariants to the $n+m$th invariants. Is this structure the same for these three groups? Is there a single structure with a conceptual definition that contains all three?
Some potentially related sequences, where the first $6$ elements agree with this one, are the moment sequences of the adjoint representations of $SP_6$ and $E_8$, and the sequence with exponential generating function  $e^{-\int_{0}^x \log(1-y)dy}$
Is there some kind of stabilization phenomenon occurring where high-dimensional exceptional groups, if they existed, would also agree with this moment sequence? If so it doesn't seem to agree with the first few stable moments of the adjoint representation of any sequence of classical groups. For $A_n$ the moment sequence is $1,0,1,2,9, 44, 265, 1854$ (counting derangments) and $C_n$ and possibly the others have moment sequence $1,0,1,1,6,22,130, 822$ (counting graphs where each vertex has two edges, with multiple edges but without loops).
 A: Yes, look for "Deligne's exceptional series". There are no theorems, but several beautiful conjectures.
The basic idea is that there should be a symmetric pivotal category generated by a trivalent vertex, with just a few local relations, depending on a parameter. At special values of the parameter, the category becomes degenerate, and the quotient by the negligible ideal recovers the representation category of one of the exceptional Lie algebras. (More or less; in some cases you get an equivariantization or subcategory.)
Working over rational functions in the parameter instead, it is expected that the category is semisimple, and its moments should agree with the sequence you describe. The exceptional algebra $F_4$ is the `least degenerate' point, in that its moments fall short the least.
Here are some pointers to the literature.
Pierre Deligne, La série exceptionnelle de groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 4, 321--326.
Pierre Deligne and Ronald de Man, La série exceptionnelle de groupes de Lie. II, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 6, 577--582.
Arjeh M. Cohen and Ronald de Man, Computational evidence for Deligne’s conjecture regarding exceptional Lie groups, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 5, 427--432.
Pierre Deligne and Benedict H. Gross, On the exceptional series, and its descendants, C. R. Math. Acad. Sci. Paris 335 (2002), no. 11, 877--881.
J. M. Landsberg and L. Manivel, Series of Lie groups, Michigan Math. J. 52 (2004), no. 2, 453--479.
J. M. Landsberg and L. Manivel, Triality, exceptional Lie algebras and Deligne dimension formulas, Adv. Math. 171 (2002), no. 1, 59--85.
