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have noted down the basic properties of the irreducible representations of the Lorentzian spin group, at spin representation – Properties.
have added to spin representation a bunch of classification results on spinor bilinears. It starts with a table in Real irreducible spin representations in Lorentz signature and then the full discussion is in Spinor bilinear forms.
Well, or a 0th-order approximation to a full discussion. It’s a bit tedious to write this out. But I threw in plenty of citations everywhere, so in the worst case the reader can follow them.
have expanded still a bit more under Spinor bilinear forms in that I have added more discussion of how the abstract classification translates to the standard matrix notation common in the physics literature. Also added, towards the end, a paragraph on how to count supersymmetries.
in the section Counting of numbers of supersymmetries I have added a paragraph on counting of symplectic Majorana reps in Lorentzin dim 5,6 and 7.
added to spin representation a section Expression of real representations via real normed division algebras summarizing the statements summarized in Baez-Huerta 09, 10.
for the fun of it, I added an Example spelling out in more detail the real reps in 2+1 dimensions, here.
Of course this is the case that comes out most simplistic, but it’s interesting exactly in how simple it is.
I have expanded that example just a tad more.
I have polished and considerably expanded the discussion of spin representations via real normed division algebras, here.
If you feel like making edits of this material at this point, please alert me, as I am working this into the lecture notes geometry of physics – supersymmetry.
There’s an ambiguity here, because some people use spin representation to mean a lift from $\mathbb ZG \to O(n)$ to $G \to Spin(n)$.
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