## Confused about symmetrization of tensors and polynomials

Consider the symmetrization of tensor product $t_i\otimes t_j\otimes t_k$, i.e. $Sym^3(S)$, where $S=t_i$, can we say $t_1^2t_2\oplus t_2t_1^2$ is symmetrized (when n>=4)? or should we write it as "$t_1^2t_2\oplus t_2^2t_1$"? I'd prefer the later one, but someone told me the previous one, $t_1^2t_2\oplus t_2t_1^2$, is also rightly symmetrized.

This confused me.

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Are you looking for symmetry in two variables or more? It's important! If you're only in the land of two-variable polynomials then the symmetrization is the sum t_1^2t_2 + t_2^2t_1 (it must be symmetric in the roles of t_1 and t_2). But if you are working in the setting of three-variable polynomials then that sum of two terms in not symmetric in t_1, t_2, and t_3, so you'd need to add even more terms to create symmetry: t_1^2t_2 + t_2^2t_1 + t_1^2t_3 + t_3^2t_1 + t_2^2t_3 + t_3^2t_2. – KConrad Jun 10 2010 at 1:39
You don't state your problem right: how many variables $t_i$ do you have? The symmetrization in the case of $n$ variables $t_1,\dots,t_n$ will be $\sum_{\sigma\in S_n}t_{\sigma(1)}^2t_{\sigma(2)}$ where $\sigma$ runs over all permutations of $\lbrace 1,2,\dots,n\rbrace$. In particular, when $n=2$, your sum of 2 momomials in quotation marks is the right answer. – Wadim Zudilin Jun 10 2010 at 1:43
Please check with en.wikipedia.org/wiki/Symmetrization (especially, the "$n$ variables" section). – Wadim Zudilin Jun 10 2010 at 3:13
Another possibility for the reason of your confusion: the monomial $t_1^2t_2$ can be regarded as an element of the third symmetric power of the vector space spanned by $t_1,t_2$; as such, it is the symmetrization of the tensor $t_1\otimes t_1\otimes t_2$. So, to summarize, there are two concepts of "symmetric" that may overlap in your mind, symmetric tensors and symmetric polynomials. Hope that helps. – Vladimir Dotsenko Jun 10 2010 at 7:22
I'm afraid I'm voting to close, because your question doesn't seem to make any sense. You need to explain: 1) whether $S$ is a vector space or a vector, 2) the role of the integer $n$, 3) the number of variables $t_i$, 4) your reason for using $\oplus$ instead of $+$. While I don't know exactly what you want, my wild guess is that $t_1^2t_2 \oplus t_2^2t_1$ is the correct answer. – S. Carnahan Jun 11 2010 at 2:08
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