# 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?

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).

• These numbers are dimensions of spaces of morphisms between tensor powers of representations, so you could try to give direct descriptions of the categories of representations - I believe they have universal descriptions along the lines of "the free symmetric monoidal linear category with duals on an object of some dimension + a non-degenerate 3-form" but I'm not sure of the exact details. (So a reason for the 9th moments to differ is different behavior of 3 copies of the 3-form.) – Qiaochu Yuan Feb 20 '16 at 16:28
• @QiaochuYuan Sure. One could write a category in terms of generators and relations, remove some of the relations and note that the moment sequence simplifies, and then argue that the relations only affect moments starting at the $N$th one. This is precisely how I calculated the stable moments of $A_n$ - take the free symmetric monoidal linear category with duals on an object of dimension $n+1$, then remove the dimension $n+1$ condition and calculate the right $\operatorname{Hom}$. But the details in this case are beyond me. – Will Sawin Feb 20 '16 at 17:27

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.

• (By the way, all those references were formatted using MathOverflow's neat new tool for adding references! Just click the little "link" button in the editing toolbar, and search for titles and/or authors.) – Scott Morrison Feb 23 '16 at 22:52
• If there isn't an announcement about this new reference tool, perhaps there should be one on meta? (I can't find one). It looks to me like we're getting information straight from MathSciNet, which unfortunately means no arXiv links :( – Peter McNamara Feb 24 '16 at 1:16
• @PeterMcNamara, There's a not-so-great announcement at meta.mathoverflow.net/questions/2583/…, following on from the discussion and announcement of a userscript version at meta.mathoverflow.net/questions/1485/…. You're right that I should make a more prominent announcement somewhere. Adding arXiv links is definitely feasible, however reliably matching records from mathscinet and the arXiv is non-trivial. If you can think of a way to do this, or avoid needing to, let me know. – Scott Morrison Feb 24 '16 at 2:23