Talk:Projective representation

In simple terms, why are spin representations "projective" ?

I'm familiar that rotations can be computed using rotors/quaternions/spin-matrices, x → R x R^-1 instead of rotation matrices x → M x; and with Spin(n) as the group that describes the one-sided combination of those rotors/quaternions/spin-matrices, R_BA = R_B R_A.

I also have a very rudimentary idea of the construction of projective spaces for the lowest dimension Euclidean spaces.

Given that starting point, is it possible to understand in simple terms how the spin representations are "projective representations" of the orthogonal or special orthogonal groups? What is the meaning of "projective" here, and how does it come about?

I've read the lead and the introduction of the article, but I don't understand how it applies. (And without much of an algebra background, I don't really have much feel for most of the terminology, or the diagram).

So is it possible to explain in a more concrete way why, say, a representation of Spin(3) can be described as a "projective" representation of SO(3)? Is there some obvious thing that I'm missing? Jheald (talk) 10:06, 3 July 2013 (UTC)[reply]

Hmm. Let me see if I'm beginning to connect. We can start with

SO(3)={\mathit {Spin}}(3)\,/\,Z_{2}

Z₂ being a quotient group of Spin(3) because there is a homomorphism h from Spin(3) to SO(3), h(R_BA) = h(R_B) h(R_A), the kernel of h being {-1, +1}.

This is a surjection ("onto") because for every element in SO(3) there are corresponding elements in Spin(3). So any representation of Spin(3) is also a representation of SO(3).

A homomorphism to GL(V, F) means I need to think about a representation of Spin(3) in the group of matrices (or something equivalent). Then, for the representation to live in GL(V,F)/F*, I need to think about the significance of factoring out the group of scalar transformations F*. The word "projective" seems to be saying that a representation of Spin(3) can meet this requirement, but ordinary rotation matrices can't. Ordinary rotation matrices are constrained to have determinant 1, which one might think was factoring out their scale freedom. But they are not "projective" representations, so evidently that is not what's needed here. So what does it mean, to restrict the representation to living in GL(V,F)/F*, and what does it signify? Jheald (talk) 12:44, 3 July 2013 (UTC)[reply]

Cross-posted to Wikipedia:Reference_desk/Mathematics, where there was further discussion. Jheald (talk) 19:21, 3 July 2013 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Projective representation/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Examples and more references needed. How about the Witt algebra? Geometry guy 21:51, 14 September 2008 (UTC)[reply]

Last edited at 21:51, 14 September 2008 (UTC). Substituted at 02:31, 5 May 2016 (UTC)

Diagrams

In many group theory articles, for instance at Schur_multiplier#Relation_to_projective_representations, there are diagrams of projective representations. These diagrams use what I think of, for lack of any better names, as "single-headed arrows", "double-headed arrows", and "umbrellas". Does Wikipedia provide any simple, accessible, explanation of what these three types of arrows show? Maproom (talk) 22:07, 8 May 2021 (UTC)[reply]