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Thought I’d write up some old notes at symmetric product of circles (linked from unitary group, explanation to come on symmetric product of circles). Not finished yet, but have to leave it for now.
(I was incensed to discover that to look at the source article for the material for this to check that I’m remembering it right - I last looked at it about 10 years ago - I have to pay 30 UKP. The article is 3 pages long. That’s 10UKP per page! So I’m going from vague memories and “working it out afresh”.)
Andrew, I’m confused by your definition 1. I expect what you mean is to take the joint coequalizer of the maps $X^n \to X^n$ induced by permuting coordinates, and this can be phrased equivalently as the coequalizer of the pair of maps
$\coprod_{\sigma \in \Sigma_n} X^n \cong S_n \times X^n \stackrel{\overset{\pi}{\to}}{\underset{\alpha}{\to}} X^n$where $\pi$ is projection onto the second coordinate, and $\alpha$ is the canonical $S_n$-action. Is that what you meant?
Todd, probably! I was trying to be “fancy” and put a categorical definition (this is the nlab, after all). I was a bit surprised that we didn’t have a page symmetric product already as it seems a fairly obvious notion.
Shouldn’t the term be symmetric power? We have an article on that (with a definition that respects multiplicities), albeit restricted to a different context. (Probably that article should be left as it is, at symmetric algebra, with a new page symmetric power for the concept in an arbitrary symmetric monoidal category.)
I think that in topology it’s more usual to talk of the symmetric product. Nonetheless, as we have the page symmetric power then I took out the general definition from this page and linked to that page.
Also completed the proof of the description of $SP^n(S^1)$ and added that of $SP^n(\mathbb{R})$. Still to do is the connection to unitary matrices.
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