What is the meaning, in tensor notation, of the dot-within-a-circle $\odot$? Usually shown as an exponent?

Question

This $\odot$ notation is e.g. used in this Phys.SE posts:

• General relativity: Why don't these two differentials commute?

• Why isn't there a second baryon octet?

Answer 1

1. The notation $V\odot V$ means the symmetric tensor product$^1$ similar to that $V\wedge V$ means the antisymmetric tensor product.

2. This can be generalized to higher tensor powers. e.g. ${\rm Sym}^3V~\equiv~ V\odot V\odot V~\equiv~V^{\odot 3},$ and $ \bigwedge{}^3V~\equiv~ V\wedge V\wedge V,$ and so forth.

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$^1$ The tensor product $V\otimes V$ is neither symmetric nor antisymmetric.