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.
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 in
Have added the pointer. Also I suspect the following two really mean instead of , but not sure yet:
Peter Goddard (auth.), Peter Freund, K. T. Mahanthappa, p. 128 of Superstrings, NATO ASI Series 175, Springer 1988
Kazuo 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 more explicit, giving it its own Examples-subsection here
]]>I am thinking:
Since the subgroup isomorphism on the left of
manifestly comes from exchanging factors, and since the subgroup isomorphism on the left of
is the dot-product of the isomorphisms with (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
commutes, where the iso on the left is either the dot-product of the iso 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 maybe the homotopy-fixed locus of triality?
As now shown here.
]]>Does anyone know the following:
The group has an evident action of the symetric group by automorphisms permuting the three dot factors.
This -action is probably related to triality?! How?
I am guessing as follows:
Probably there are inclusions of into each of
(these now all understood under their canonical embedding as subgroups of as here) such that as these three get permuted into each other under the action of triality, their common subgroup is fixed up to isomorphism, and these fixing isomorphisms are the action from before!?
Is this discussed anywhere?
]]>added pointer to
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
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 with central subgroup , such that is known, and calculate ?
]]>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 , Annali di Matematica 1977 (doi:10.1007/BF02413792)
Paolo Piccinni, Giuliano Romani, A generalization of symplectic Pontrjagin classes to vector bundles with structure group , Annali di Matematica pura ed applicata (1983) 133: 1 (doi:10.1007/BF01766008)
James, might you know a source that gives ? We are struggling with a factor of 1/2 in there…
]]>added one more case to the section of “Spin Grassmannians”:
Similarly,
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:
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,
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.
]]>