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Results tagged with spin-geometry
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user 13972
For questions about spin manifolds, the groups $\operatorname{Spin}(n)$, as well as generalisations such as $\operatorname{Pin}^{\pm}(n)$ and $\operatorname{Spin}^c(n)$. This tag should also be used for any questions about the geometry of spin manifolds, including questions involving Dirac operators and the Lichnerowicz formula.
5
votes
Accepted
Decomposition into irreducible components of a representation of $Spin(9)$
This is easily computed via LiE: $Sym^2(\mathbb{R}^{16})$ breaks into three irreducible components:
The trivial representation, i.e., $\mathbb{R}$,
The standard representation of $\mathrm{SO}(9)$, …
- 108k
5
votes
Accepted
Frame-bundle reduction from spinor-bundle reduction
Well, in one sense, this is always true. If $Q\subset SP(M)$ is a principal right $H$-bundle, where $H\subset\mathrm{Spin}(n)$ is a subgroup, then $\delta(Q)\subset F(M)$ is a principal right $\pi(H) …
- 108k
15
votes
Triality of Spin(8)
In addition to the above answers involving spinors and/or octonions, you might be interested in Cartan's original construction of the triality automorphisms, which is very explicit and takes just a co …
- 108k
7
votes
Accepted
Quadratic forms on $\mathbb{R}^{16}$ coming from octonions
I'm revising my answer because whether the image of the map $j$ that the OP defines is equal to the $10$-dimensional subspace $H$ of $\mathrm{Sym}^2(\mathbb{R}^{16})$ that is invariant under $\mathrm{ …
- 108k
7
votes
Accepted
Explicit generators of the Lie algebra $spin(9)$
There are various places where you can see this written down, but let me suggest some notes that I wrote about spinors in the low dimensions that includes what you want, assuming that you know somethi …
- 108k
4
votes
Accepted
A representation of Spin(9,1)
In this case if your 16-dimensional $\mathrm{Spin}(9,1)$-representation is the one of highest weight $(0,0,0,0,1)$, then the $\mathrm{Spin}(9,1)$-irreducible decomposition of its symmetric square is j …
- 108k
3
votes
Accepted
Cayley Subspaces in a Calibrated 8-Space
All your questions (and much more) are answered in the 1982 paper Calibrated Geometries by Harvey and Lawson. See Acta Mathematica July 1982, Volume 148, Issue 1, pp. 47-157.
Just so you'll know: H …
- 108k
5
votes
Accepted
$Spin(7)$ as stabilizer of a $4$-form revisited
I think that you want to look at the relevant passages in Spin Geometry by Lawson and Michelsohn, particularly Chapter IV, Sections 9 and 10, where they explain in general how the square of a spinor c …
- 108k
12
votes
Accepted
Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$
Here's a standard explicit formula: Let $\mathbb{O}\simeq\mathbb{R}^8$ denote the algebra of octonions, and for $x\in\mathbb{O}$, let $L_x$ (respectively $R_x)$ denote the linear map from $\mathbb{O} …
- 108k
2
votes
Injective group homomorphism on $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/2}$ or $\frac{Spin(...
You really should have a look at F. Reese Harvey's book Spinors and Calibrations, where all of your questions are answered.
For example, your 'inclusion' (1) is not correct for sufficiently large $n$. …
- 108k
3
votes
Accepted
Isomorphisms of Positive and Negative Spinor Bundles
You're really asking an algebra question about how the various representations of $\mathrm{Spin}(8)$ interact. There are lots of places where you can read about this, but here is a set of notes that …
- 108k
11
votes
Accepted
The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$?
You can work out the answers to these questions using the material in Chapter 11 of the book Spinors and Calibrations by F. Reese Harvey. You will also need to recall that, for $N\not=4$, the group o …
- 108k