Excellent, thanks!

(For bystanders: It’s about this part of the entry.)

]]>Notbohm agrees with Aguadé, referring to a misprint in Dwyer-Wilkerson, so I’ve corrected and added this location.

]]>Thanks. Made a brief note in the entry, here. Please feel invited to edit.

]]>J. Aguadé, "The torsion index of a p-compact group", Proceedings of the AMS 138(11) , 2010, p. 4133.

- Tilman Bauer ]]>

Added a reference:

- Michael Aschbacher, Andrew Chermak,
*A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver*, (paper)

It was there but had ) instead of }, so nothing appeared. Fixed now.

This is the reference mentioned in #22, with the warning about $B$ and $C$ groups.

It would be fun to work out how the sporadic finite simple groups show up. Over at the Café, John posted on Theo Johnson-Freyd and David Treumann’s calculation of the fourth cohomology of $Co_0$. There’s plenty of ’physics’ talk, e.g., Theo:

]]>My motivation was the following. There is a holomorphic $N=1$ SCFT called $V^{f\natural}$ with automorphism group $Co_1$, first constructed by John Duncan. This $Co_1$ action is

anomalous, meaning that there is an obstruction to gauging it. The anomaly comes in two pieces. First, “the” Ramond sector of an SCFT is only well-defined up to isomorphism, and so symmetries of an SCFT act projectively on the Ramond sector, but to gauge a symmetry requires promoting that action to an honest action. This requires lifting from $Co_1$ to its double cover $Co_0$. Second, there is still an $H^4(Co_0; Z)$ valued anomaly, which can be canceled by anomaly-inflow from a 3D Dijkgraaf—Witten model. Equivalently, to trivialize the anomaly requires pulling back from $Co_0$ to the appropriate 3-group.I claim that of the 24 3-groups, the one that trivializes this anomaly is specifically the one corresponding to our generator of $H^4(Co_0; Z)$, namely the fractional Pontryagin class of the 24-dimensional representation.

Thanks. What’s your references Benson98 ?

]]>Added some rationale for the 3 dim proposal.

]]>Yeah, I don’t know if people have more than plain numerology 45 = 45 for the $3 \times 3$ picture. I was initially assuming they must have, for otherwise it would not seem worth mentioning even. But if so, I haven’t seen it either.

]]>Putting things figuratively, I guess when a concept is put under this much pressure, we should only expect at best traces of features that were there before. $G_3$ seems to be a vestige, hanging on by its finger tips in the 2-adic world. We might plausibly expect then a trace of the sedenions.

Do we have a rationale for the $3 \times 3$-aspect, beyond the numerology of 45? Benson in #22 speaks of a “tempting candidate”, but I haven’t seen an explanation of 3 dimensions, unless Wilson (#12) is pointing to this with his 3 octonionic dimensional construction of the Leech lattice.

Does $G_2$ have any such thing going on?

]]>Okay, if that’s the pattern, then the number 15 here should be identified with the dimension of the space of imaginary sedenions, inside the 16-dimensional space of all sedenions.

That goes against the grain of having $G_3$ be the automorphisms of the Albert algebra, but okay. Or might there be a natural way in which the sedenions sit inside the Albert algebra?

]]>I guess given the action of $SO(3)$ on imaginary quaternions (dim 3) and $G_2$ on imaginary octonions (dim 7), 15 might be expected.

]]>Added tentative claim of a 15-dim homotopy representation.

]]>Thanks! Unfortunatly, GoogleBooks decides to hide this from me. I’ll see if I get a copy elsewhere.

Just started to look around if anyone had made a proposal for $U(3,\mathbb{O})$. There is an MO-question here, but no real reply,

]]>If you can see the bottom of this page from Google Books, Benson is wondering about a 45-dimensional algebra of skew-hermitian matrices over a 2-adic version of the octonions, and points to a problem with a 21-dim subalgebra being $C_3$ instead of $B_3$. “But there may be some twisted version of this which works.”

]]>[ … ]

]]>Hm, now there is an obvious relation, isn’t there:

If it were not for non-associativity of the octonions, the skew-hermitian $3 \times 3$ octonion matrices would be the Lie algebra of the unitary $3 \times 3$ matrices, which in turn would act on the $3 \times 3$ hermitian matrices in the Albert algebra by automorphisms.

Only that non-associativity prevents these unitary matrices from actually forming a group… but so maybe they form the $\infty$-group $G_3$?!

]]>I have recorded this in the entry, here

]]>Ah, interesting, thanks. So for one I learn that Wilson’s “skew-symmetric” did not refer to “hermitian” (as in the Albert algebra), but to “skew-hermitian”. But it’s still curious how close this is now to the Albert algebra.

]]>I was wondering whether that idea of centrally extending $\mathbb{R}^{0|3}$ might be relevant.

Not sure how Wilson’s reasoning has gone. Elsewhere there is

]]>As the Dwyer–Wilkerson 2-compact group is, in some sense, a 45-dimensional object, it is a plausible conjecture that it might have some connection to the 45-dimensional algebra $S H(3, \mathbb{O})$ of $3 \times 3$ skew hermitian matrices over the octonions with bracket multiplication). (p. 175 of Conjectures on finite and p-local groups)

Now if what Wilson means by “skew-symmetric $3 \times 3$-matrices over the octonions” connects to the Albert algebra, that would be something, and a rough physics picture would immediately spring to mind.

So where does he get this from, that $G_3$ is “some kind of twist skew-symmetric $3 \times 3$-matrices over the octonions”. Does this come from $CO_3$ inside it?

]]>A little more on the $Co_3$ link.

]]>Should make it “Robert A. Wilson”, but you are editing now…

]]>Thanks for the pointer! Have added that to the entry, too (here)

]]>