Understanding symmetric tensor products Let $H$ be a Hilbert space, and let
$$H_n = \otimes_n H = \Big\{\sum_{i_1,\ldots, i_n} \alpha_{i_1, \ldots, i_n} \big(e_{i_1} \otimes \cdots \otimes e_{i_n}\big) : \sum_{i_1,\ldots, i_n} |\alpha_{i_1, \ldots, i_n}|^2<\infty \Big\}$$
denote the $n$-fold tensor product of $H$. Here $\{e_i\}$ denotes a basis element for $H_i$ and $\alpha_{i_1, \ldots, i_n} \in \mathbb{C}$. Please correct me if this definition is incorrect.
I am trying to understand a particular subset of $H_n$, namely the symmetric tensor product. This is defined as the space
$$H_n^s = \Big\{\sum_{i_1,\ldots, i_n} \alpha_{i_1, \ldots, i_n} \big(e_{i_1} \otimes \cdots \otimes e_{i_n}\big) \in H_n: ~\alpha_{i_1, \ldots, i_n} = \alpha_{i_{\sigma(1)}, \ldots, i_{\sigma(n)}}~ \forall \sigma \in S_n,\Big\}$$
where $S_n$ is of course the permutation group of $n$ objects.
To me, this says that for every vector in $H_n^s$ if we swap the order of the coefficient's components then the result is the same coefficient we started with. However this is clearly wrong, as this would imply that the coefficients must all be the same. If anyone can provide a detailed breakdown of this definition that would be much appreciated.
 A: The symmetric tensors are all symmetrized tensor products of vectors:
$$
  H^s_n = \left\{\sum_{\sigma \in S_n}v_{\sigma(1)}\otimes\cdots\otimes v_{\sigma(n)} \;:\; v_1,\dotsc,v_n \in H\right\}.
$$
It follows that each basis element of $H^s_n$ corresponds to a selection of integers $1 \leq i_1 \leq i_2 \leq \cdots \leq i_n$, and that basis element is
$$
  \sum_{\sigma\in S_n}e_{i_{\sigma(1)}}\otimes\cdots\otimes e_{i_{\sigma(n)}}.
$$
(Of course, you could normalize however you please.) This means we can write
$$
  H^s_n = \left\{\sum_{1\leq i_1\leq\cdots\leq i_n}a_{i_1,\cdots, i_n}\sum_{\sigma\in S_n}e_{i_{\sigma(1)}}\otimes\cdots\otimes e_{i_{\sigma(n)}} \;:\; a_{i_1,\cdots,i_n} \in \mathbb C\right\}
$$
In terms of coordinates, this means that every $e_{i_1}\otimes\cdots\otimes e_{i_n}$ for any selection of $i_1,\dots, i_n$ must have the same coefficient as $e_{i_{\sigma(1)}}\otimes\cdots\otimes e_{i_{\sigma(n)}}$ for any $\sigma \in S_n$. Hence we could also write
$$
  H^s_n = \left\{\sum_{i_1,\cdots,i_n} a_{i_1,\cdots,i_n}e_{i_1}\otimes\cdots\otimes e_{i_n} \;:\; \forall\sigma\in S_n.\: a_{i_{\sigma(1)},\cdots,i_{\sigma(n)}} = a_{i_1,\cdots,i_n}\right\}.
$$
This says that if any two coefficients $a_{i_1,\cdots,i_n}$ and $a_{i'_1,\cdots,i'_n}$ draw their indices from the same multiset, then $a_{i_1,\cdots,i_n} = a_{i'_1,\cdots,i'_n}$; it does not say anything about how e.g. $a_{1,2,3}$ and $a_{1,1,3}$ are related.
A: A (small) example to get the feel of these objects:
$$
2 \, e_3 \otimes e_{17} - 5 \, e_4 \otimes e_4 
$$
is an arbitrary element of the two-fold tensor product space $H_2$. Here the multi-index $(i_1, i_2) = (3, 17)$ for the first term and $(i_1, i_2) = (4, 4)$ for the second. The coefficients are $\alpha_{3, 17} = 2$, $\alpha_{4, 4} = -5$, and $\alpha_{i_1, i_2} = 0$ for all others. But this tensor is not symmetric. If you swap the indices (the only non-trivial permutation of $2$ indices), you get
$$
2 \, e_{17} \otimes e_3 - 5 \, e_4 \otimes e_4 
$$
which is a different tensor.
An example of a symmetric tensor in $H_2$ would be
$$
2 \, e_3 \otimes e_{17} - 5 \, e_4 \otimes e_4 + 2 \, e_{17} \otimes e_3.
$$
You can easily see that either permutation in $S_2$ swaps the first and third terms, but their coefficients are the same, so the tensor is fixed.
Here's how to define the tensor spaces:
$$
H_n = H^{\otimes n} 
= \biggl\{\, \sum_{i_1, \ldots, i_n} 
\alpha_{i_1, \ldots, i_n} 
\bigl(e_{i_1} \otimes \cdots \otimes e_{i_n} \bigr): \;
\sum_{i_1, \ldots, i_n} \lvert \alpha_{i_1, \ldots, i_n}\rvert^2 
< \infty \bigg\}
$$
and
$$
\operatorname{Sym}^n H 
= \biggl\{\, \sum_{i_1, \ldots, i_n} 
\alpha_{i_1, \ldots, i_n} 
\bigl(e_{i_1} \otimes \cdots \otimes e_{i_n} \bigr) \in H_n: \;
\alpha_{i_1, \ldots, i_n} 
= \alpha_{i_{\sigma(1)}, \ldots, i_{\sigma(n)}} \, 
\forall \sigma \in S_n \bigg\}.
$$
Almost by definition, these tensors are invariant under any permutation $\sigma \in S_n$. All that happens when you act on the sub-indices is that the order of the sum of term is permuted.
Does this make sense?
