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I'm not aware of another natural geometric object. I'm particularly interested in the case when $n=2m$ is even, and i'm really only. Also, if i'm not mistaken, steenrod gives a more direct argument in topology of fibre bundles, but he might be using the long exact sequence of a fibration (which you mentioned).

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From here i got another doubt about how we connect lie stuff in our clifford algebra settings I'm looking for a reference/proof where i can understand the irreps of $so(n)$ Like did we really use fundamental theorem of gleason, montgomery and zippin to bring lie group notion here?

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Welcome to the language barrier between physicists and mathematicians

Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators The question really is that simple Prove that the manifold $so (n) \subset gl (n, \mathbb {r})$ is connected It is very easy to see that the elements of $so (n.

The generators of $so(n)$ are pure imaginary antisymmetric $n \\times n$ matrices I have known the data of $\\pi_m(so(n))$ from this table A son had recently visited his mom and found out that the two digits that form his age (eg :24) when reversed form his mother's age (eg Later he goes back to his place and finds out that this whole 'age' reversed process occurs 6 times

And if they (mom + son) were lucky it would happen again in future for two more times.

Each of 20 families selected to take part in a treasure hunt consist of a mother, father, son, and daughter Assuming that they look for the treasure in pairs that are randomly chosen from the 80