tr;dr intro: I came up with this question when I couldn't remember how many terms are in an $E_7$-ish (representing $\bigotimes$ adjoint) clebsch. Tried it on $G_2$, $7 \bigotimes 14=7+...$ argh, is it 27+64 or 14+77 or 14+77'? (The first. I looked it up afterwards :-)
I didn't actually check it, but my hypothesis is that in any Lie algebra there are more "random" sum rules than in a "comparable" random set. Let's formalize the stuff a bit:
Let $S$ be a "random" set of different (positive) integers increasing as some $O(f(n))$.
Question 1: What is $f(n)$ for the dimensions of a Lie algebra $L$ (counting equal dimensions once, which biases only against them)? Graphing the first 45 terms of $G_2$ from OEIS was no help. Sqrt? Log?
Let $S_i=S_j+S_k$ be a statement $A$ in $S$, with all summands $<m$. (We could also consider more summands.)
Question 2: What is the probability $p=O(g(m))$ that $A$ is true for a "random" set $S$, given $f$? ($G_2$: There are 36 three-term sum rules among the first 45 dimensions. Also, couldn't try a "random" $S$ to compare as I don't know $f$...)
Question 3: What is $g$ given $L$? (Again, we bias against $L$ because of multiplicities, but they are hard to model. $G_2$: not enough data to guess $g$)
Question 4: Is $p$ really larger for some $L$ than for a "random" $S$?
Question 5: If yes, why, and can you explain the "accidental" sum rules in terms of Lie algebra theory? (For a start, many dimensions of $G_2$ are a multiple of 7, also 11 and 13 are prominent, this increases the probability of a "hit". But explain the taxi identity $1+1728=1729$ that way! In any case, it would be helpful to have at least 10000 or so dimensions of say $G_2$ for some numerical experiments...but I was too lazy :-)
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asked Oct 12, 2018 at 9:38
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3$\begingroup$ I find this question very hard to parse, but it seems that several of the questions should be answerable by simply writing out Weyl's dimension formula explicitly for type $G_2$. $\endgroup$– Tobias KildetoftCommented Oct 12, 2018 at 9:57
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2$\begingroup$ Seconded. What are dimensions of $L$? I assume it's the dimensions of representations. What are "sum rules"? Equalities among dimensions coming from decompositions into irreducibles? $\endgroup$– Vít TučekCommented Oct 12, 2018 at 9:59
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$\begingroup$ Yes, of course dimensions of the irreps. And yes, since the dimensions of $G_2$ can be written as a fairly simple function f(weight1,weight2), one could find sum rules...but there are much more than those Vit mentions (which are necessary) - I'm more after the "accidental" ones. E.g. for a start - is there a symmetry or map that directly shows that 77 and 77' have the same dimension? $\endgroup$– Hauke ReddmannCommented Oct 14, 2018 at 17:52
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$\begingroup$ @HaukeReddmann I repeat my question: What are sum rules? Since you made no efforts in two days to improve your question I voted to close since it is not clear what you are asking. $\endgroup$– Vít TučekCommented Oct 15, 2018 at 8:51
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