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

The notation $Vodot V $ means the symmetric tensor product similar to that $Vwedge V$ means the antisymmetric tensor product.

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