(Note that symmetric algebra and symmetric tensor do not coincide when the characteristic is not $0$.)
I'm reading this aricle:http://en.m.wikipedia.org/wiki/Symmetric_tensor
And here it defines $v_1\odot ...\odot v_k= \frac{1}{k!}\sum_{\sigma \in S_k} v_{\sigma(1)}\otimes...\otimes v_{\sigma(k)}$.
Is this the standard definition of $\odot$? I think it should be used for symmetric algebra. That is:
Let $R$ be a commutativ ring and $M$ be an $R$-module. Let $T(M)$ be the tensor algenra of $M$ and $I$ be the ideal of $T(M)$ generated by elements of the form $m\otimes m' - m'\otimes m$. Then, the symmetric algebra is $T(M)/I$.
Isn't it more natural to use $\odot$ as follows?: $v_1\odot...\odot v_k=v_1\otimes...\otimes v_k + I$?