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I am looking for a 527-dimensional irrep of $K(E_{10})$. Does any exist?
There is a mentioning of a would-be $K(E_{10})$-rep of this dimension on. p. 37 in arXiv:hep-th/0606105 — but I don’t understand the commentary there yet, maybe it says that such a rep might naively be expected but does not actually exist.(?)
And then there is a coset space of Weyl algebras $W(D E_{10})/W(E_{10})$ claimed to have 527 elements claimed on p. 54 of arXiv:hep-th/0409037, where the number 527 arises by the same combinatorial formula (on p. 55) that makes me look for a rep of that dimension — but I don’t know if any of this has directly to do with a rep of $K(E_{10})$. (?)
[edit: Re-reading the latter reference, I think they are saying that the known irrep $\mathbf{32} \in Rep\big( K(E_{10}) \big)$ does have symmetric square $\mathbf{32} \otimes_{sym} \mathbf{32} \,\simeq\, \mathbf{1} \oplus \mathbf{527}$ (which is all I’d be asking for), just that the fermionic constraints $\mathscr{S}$ considered in the article do not actually transform in $\mathbf{32}$ (which I dont’ currently care about). So it looks like this answers my question. But it would still be nice to have a more explicit reference. ]
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