added these pointers now:

Discussion of central product spin groups as subgroups of semi-spin groups (motivated by analysis of the gauge groups and Green-Schwarz anomaly cancellation of heterotic string theory) is in

Brett McInnes, p. 9 of

*The Semispin Groups in String Theory*, J. Math. Phys. 40:4699-4712, 1999 (arXiv:hep-th/9906059)Brett McInnes,

*Gauge Spinors and String Duality*, Nucl. Phys. B577:439-460, 2000 (arXiv:hep-th/9910100)

As such these also appear as U-duality groups and their subgroups, e.g.

- Arjan Keurentjes, p. 10 of
*The topology of U-duality (sub-)groups*, Class.Quant.Grav. 21 (2004) 1695-1708 (arXiv:hep-th/0309106)

Should also add pointer to

arxiv.org/abs/hep-th/0309106

and

arxiv.org/abs/hep-th/9906059

arxiv.org/abs/hep-th/9910100

]]>there is a curious appearance of $Sp(1)\cdot Sp(1) \cdot Sp(1)$ in

- Peter Freund, p. 634 of
*World topology and gauged internal symmetries*, Proc. 19th Int. Conf. High Energy Physics, Tokyo 1978 (spire:137780, pdf)

Have added the pointer. Also I suspect the following two really mean $SU(1)\cdot SU(2) \cdot SU(2)$ instead of $SU(2) \times SU(2) \times SU(2)$, but not sure yet:

Peter Goddard (auth.), Peter Freund, K. T. Mahanthappa, p. 128 of

*Superstrings*, NATO ASI Series 175, Springer 1988Kazuo Hosomichi, Sangmin Lee, Sungjay Lee, Jaemo Park, slide 13 of

*New SuperconformalChern-Simons Theories*(pdf)

cross-linked with *central product of groups*

made the example $Spin(n) \cdot Spin(2) \simeq Spin^c(n)$ more explicit, giving it its own Examples-subsection here

]]>I am thinking:

Since the subgroup isomorphism on the left of

$\array{ Sp(2)Sp(1) &\hookrightarrow& Spin(8) \\ \mathllap{\simeq}\big\downarrow && \big\downarrow\mathrlap{=} \\ Sp(1)Sp(2) &\hookrightarrow& Spin(8) }$manifestly comes from exchanging factors, and since the subgroup isomorphism on the left of

$\array{ Sp(2)Sp(1) &\hookrightarrow& Spin(8) \\ \mathllap{\simeq}\big\downarrow && \big\downarrow\mathrlap{=} \\ Spin(5)Spin(3) &\hookrightarrow& Spin(8) }$is the dot-product of the isomorphisms $Sp(2) \overset{\simeq}{\to} Spin(5)$ with $Sp(1) \overset{\simeq}{\to} Spin(3)$ (by the proof of Lemma 2.4 here), the proof of my conjecture (that the first and second inner circles here commute) is reduced to checking that the diagram

$\array{ Sp(1)Sp(1) &\hookrightarrow& Sp(2) \\ \mathllap{\simeq}\big\downarrow && \big\downarrow\mathrlap{\simeq} \\ Spin(3)Spin(3) &\hookrightarrow& Spin(5) }$commutes, where the iso on the left is either the dot-product of the iso $Sp(1) \stackrel{\simeq}{\to} Spin(3)$ with itself, or that followed by switching dot-factors.

If there is any justice in the world, then it does. But I’d need to dig deeper into the details to prove this.

]]>In other words:

Is $Sp(1)Sp(1)Sp(1)$ maybe the homotopy-fixed locus of triality?

As now shown here.

]]>Does anyone know the following:

The group $Sp(1)\cdot Sp(1) \cdot Sp(1)$ has an evident action of the symetric group $\Sigma_3$ by automorphisms permuting the three dot factors.

This $\Sigma_3$-action is probably related to triality?! How?

I am guessing as follows:

Probably there are inclusions of $Sp(1)\cdot Sp(1) \cdot Sp(1)$ into each of

$Sp(1)\cdot Sp(2)$

$Sp(2)\cdot Sp(1)$

$Spin(3)\cdot Spin(5)$

(these now all understood under their canonical embedding as subgroups of $Spin(8)$ as here) such that as these three get permuted into each other under the action of triality, their common subgroup $Sp(1)\cdot Sp(1) \cdot Sp(1)$ is fixed up to isomorphism, and these fixing isomorphisms are the $\Sigma_3$ action from before!?

Is this discussed anywhere?

]]>added pointer to

- Andreas Kollross, Prop. 3.3 of
*A Classification of Hyperpolar and Cohomogeneity One Actions*, Transactions of the American Mathematical Society Vol. 354, No. 2 (Feb., 2002), pp. 571-612 (jstor:2693761)

and used that for a new section *Examples - Triality*. (Same material now also at Spin(8) in the section “Subgroup lattice”)

added pointer to Pro. 17.3.1 of

- Joachim Hilgert, Karl-Hermann Neeb,
*Structure and Geometry of Lie Groups*, Springer Monographs in Mathematics, Springer-Verlag New York, 2012 (doi:10.1007/978-0-387-84794-8)

for an example of usage of the dot-notation applied to general Spin-groups

]]>@Urs did you manage to resolve it?

]]>[ also removed, sorry for the noise]

]]>[ removed ]

]]>Surely! ;-)

But never mind, the factor came to me under the shower.

]]>@Urs #15

surely there is a general method to take a simply-connected connected compact Lie group $G$ with central subgroup $\mathbb{Z}/2$, such that $H^4(B G,\mathbb{Z})$ is known, and calculate $H^4(B[G/(\mathbb{Z}/2)],\mathbb{Z})$?

]]>added also pointer to these here:

Paolo Piccinni,

*Vector fields and characteristic numbers on hyperkàhler and quaternion Kâhler manifolds*, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1992) Volume: 3, Issue: 4, page 295-298 (dml:244204)Dmitri Alekseevskii S. Marchiafava,

*Quaternionic structures on a manifold and subordinated structures*, Annali di Matematica pura ed applicata (1996) 171: 205 (doi:10.1007/BF01759388)

added these pointers:

Stefano Marchiafava, Giuliano Romani,

*Alcune osservazioni sui sottogruppi abeliani del gruppo $Sp(n)\cdot Sp(1)$*, Annali di Matematica 1977 (doi:10.1007/BF02413792)Paolo Piccinni, Giuliano Romani,

*A generalization of symplectic Pontrjagin classes to vector bundles with structure group $Sp(n)\cdot Sp(1)$*, Annali di Matematica pura ed applicata (1983) 133: 1 (doi:10.1007/BF01766008)

James, might you know a source that gives $H^4\Big( B \big(Sp(1)Sp(1)Sp(1)\big), \mathbb{Z}\Big)$? We are struggling with a factor of 1/2 in there…

]]>added one more case to the section of “Spin Grassmannians”:

Similarly,

$Spin(6)/ \big( Spin(3)\cdot Spin(3) \big) \;\simeq\; SU(6)/ SO(4)$is the Grassmannian of those Cayley 4-planes that are also special Lagrangian submanifolds (BBMOOY 96, p. 8).

]]>started an Examples-subsection “Spin-Grassmannians”. Currently I have this, to be polished and expanded:

We have the following coset spaces of spin groups by dot-products of Spin groups as above:

$Spin(7)/ \big( Spin(4)\cdot Spin(3) \big) \;\simeq\; SO(7) / \big( SO(4) \times SO(3) \big) \;\simeq\; Gr(4, 7)$is the space of Cayley 4-planes (Cayley 4-form-calibrated submanifolds in 8d Euclidean space), which in turn is homeomorphic to just the plain Grassmannian of 4-planes in 7d (recalled e.g. in Ornea-Piccini 00, p. 1)

Moreover,

$Spin(8)/ \big( Spin(5)\cdot Spin(3) \big) \;\simeq\; Gr(3, 8)$is the Grassmannian of 3-planes in 8d. (Cadek-Vanzura 97, Lemma 2.6).

]]>I have added hyperlinks for authors: *Dmitry Vladimirovich Alekseevsky* and *Alfred Gray*

Exccellent, thanks! I have added pointers to your new references from the first lines of the main text.

]]>Added references and some discussion of notation in relation to quaternionic geometries.

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