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1. • CommentRowNumber1.
   • CommentAuthorSamuel Adrian Antz
   • CommentTimeJul 17th 2024
   • (edited Jul 17th 2024)
   • PermaLink
   Author: Samuel Adrian Antz
   Format: MarkdownItexUsed unicode subscripts for indices of [[exceptional Lie groups]] including title and links. When not linked, usual formulas are used. See discussion [here](https://nforum.ncatlab.org/discussion/18215/title-convention). Links will be re-checked after all titles have been changed. (Removed two redirects for "E10" from the top and added one for "E10" at the bottom of the page.) <a href="https://ncatlab.org/nlab/revision/diff/E%E2%82%81%E2%82%80/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/E%E2%82%81%E2%82%80/11">v11</a>, <a href="https://ncatlab.org/nlab/show/E%E2%82%81%E2%82%80">current</a>

   Used unicode subscripts for indices of exceptional Lie groups including title and links. When not linked, usual formulas are used. See discussion here. Links will be re-checked after all titles have been changed. (Removed two redirects for “E10” from the top and added one for “E10” at the bottom of the page.)

   diff, v11, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 16th 2024
   • (edited Sep 16th 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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](https://arxiv.org/pdf/hep-th/0606105#page=37) in [arXiv:hep-th/0606105](https://arxiv.org/abs/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](https://arxiv.org/pdf/hep-th/0409037v2#page=54) of [arXiv:hep-th/0409037](https://arxiv.org/abs/hep-th/0409037), where the number 527 arises by the same combinatorial formula (on [p. 55](https://arxiv.org/pdf/hep-th/0409037v2#page=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. ]

   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. ]

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 16th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have checked with the authors, and it's indeed true. This is remarkable. Have made a brief note of the matter [here](https://ncatlab.org/nlab/show/E10#MaximalCompactSubalgebra), will expand tomorrow. <a href="https://ncatlab.org/nlab/revision/diff/E%E2%82%81%E2%82%80/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/E%E2%82%81%E2%82%80/14">v14</a>, <a href="https://ncatlab.org/nlab/show/E%E2%82%81%E2%82%80">current</a>

   I have checked with the authors, and it’s indeed true. This is remarkable.

   Have made a brief note of the matter here, will expand tomorrow.

   diff, v14, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 2nd 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexam adding a bunch more references on the $E_{10}/K(E_{10})$ sigma model (in progress...) <a href="https://ncatlab.org/nlab/revision/diff/E%E2%82%81%E2%82%80/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/E%E2%82%81%E2%82%80/22">v22</a>, <a href="https://ncatlab.org/nlab/show/E%E2%82%81%E2%82%80">current</a>

   am adding a bunch more references on the $E_{10}/K(E_{10})$ sigma model (in progress…)

   diff, v22, current