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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 . Does any exist?

   There is a mentioning of a would-be -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 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 . (?)

   [edit: Re-reading the latter reference, I think they are saying that the known irrep does have symmetric square (which is all I’d be asking for), just that the fermionic constraints considered in the article do not actually transform in (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 sigma model (in progress…)

   diff, v22, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 18th 2025
   • (edited Mar 18th 2025)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * [[Krzysztof A. Meissner]], [[Hermann Nicolai]]: *Standard Model Symmetries and $K(E_{10})$* \[<a href="https://arxiv.org/abs/2503.13155">arXiv:2503.13155</a>\] In the course of this I re-grouped all the previous related article to a new References-subsection *[E10 -- References -- Phenomenology](https://ncatlab.org/nlab/show/E10#StandardModelPhenomenology)* <a href="https://ncatlab.org/nlab/revision/diff/E%E2%82%81%E2%82%80/28">diff</a>, <a href="https://ncatlab.org/nlab/revision/E%E2%82%81%E2%82%80/28">v28</a>, <a href="https://ncatlab.org/nlab/show/E%E2%82%81%E2%82%80">current</a>

   added pointer to today’s

   • Krzysztof A. Meissner, Hermann Nicolai: Standard Model Symmetries and [arXiv:2503.13155]

   In the course of this I re-grouped all the previous related article to a new References-subsection E10 – References – Phenomenology

   diff, v28, current