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Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively.

Then there is an induced principal $\Spin_{n+n'}$ bundle over the product $M\times M'$. Is there a standard notation or symbol used to denote this bundle?

Also,

Is there a notation for the associated $\mathbb R^{n+n'}$ bundle over $M \times M'$, in terms of the associated $\mathbb R^n$ bundle on $M$ and the associated $\mathbb R^{n'}$ bundle on $M'$?

This is not particular to $\Spin$ at all; one could ask this question of any group $H$ containing a direct product $K \times K'$ inside it, and constructing an $H$ bundle over $M \times M'$ given a $K$ bundle over $M$ and a $K'$ bundle over $M'$.

In case there's any ambiguity: The bundle structure is induced by the homomorphism $$ \Spin_n \times \Spin_{n'} \to \Spin_{n+n'} $$ which lives over the homomorphism $$ O(n) \times O(n') \to O(n+n') $$ given by sending matrices $A_n$ and $A_{n'}$ to a block diagonal matrix with entries $A_n$ and $A_{n'}$. Then the induced transition maps are defined by $$ g_{U \times U', V \times V'}: (x,x') \mapsto (\phi_{UV}(x),\phi'_{U'V'}(x')) \in \Spin_n \times \Spin_{n'} \subset \Spin_{n+n'}. $$ where $\phi$ and $\phi'$ are the transition maps of the bundles $P$ and $P'$, respectively.

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    $\begingroup$ Maybe "external product" is the term you're looking for, denoted by a $\boxtimes$ for vector bundles. $\endgroup$ – WimC Apr 29 '13 at 8:56
  • $\begingroup$ Geometrically I'd just write $\pi^* P \oplus (\pi')^*P'$ where $(\pi,\pi')$ is the projection maps from $M\times M'$ to the components. @WimC's comment would be more pertinent if you are interested in describing the structure; and honestly I am not sure from your phrasing of the question which you are more interested about. $\endgroup$ – Willie Wong Apr 29 '13 at 9:04
  • $\begingroup$ @WimC : Wouldn't external product, for instance, produce a line bundle when $P$ and $P'$ are line bundles? It seems like the external product uses a tensor product where I would want a direct sum. $\endgroup$ – user54535 Apr 30 '13 at 3:12
  • $\begingroup$ @WillieWong, your notation describes the vector bundle correctly, but leaves out the structure of $spin(n+n')$'s action--if there's no standard notation, I guess I'll just have to live with it. $\endgroup$ – user54535 Apr 30 '13 at 3:13
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    $\begingroup$ @User24601 The notation $P$ itself also doesn't capture the spin structure; you have to say "$P$ is a Spin${}_n$ bundle...". I don't see why you cannot just define the appropriate vector bundle and state explicitly that it carries an induced Spin${}_{n+n'}$ structure. I'm sure if you say it once the people reading your paper will remember for the rest of the paper... $\endgroup$ – Willie Wong Apr 30 '13 at 9:12

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