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2 (corrected some more typos)

Here is another proof of the same result (it is probably the same as the one given by darij grinberg but I do not understand all the words. It has perhaps more his place as a comment, but I do not know how to write comments). In the case $t=2$, it is Claim 1 in Lemma 1.3.2 of http://arxiv.org/pdf/1002.4595v2

Denote by $P : M_d(\mathbb C)^{\otimes t} \to M_d(\mathbb C)^{\otimes t}$ the linear map such that $P(A_1 \otimes \dots \otimes A_t) = 1/n!\sum 1/t!\sum A_{\sigma(1)} \otimes \dots \otimes A_{\sigma(t)}$, where the sum os over all permutations of ${1,\dots ,t}$. $P$ is the natural projection on the symmetric tensors.

Let $(H_k)$ be the assertion "$\mathbb V$ contains the space $F_k$ spanned by $P(h_1 \otimes h_2 \otimes \dots \otimes h_t)$ for any hermitian matrices $h_1,...,h_t$ with $h_{k+1} = h_{k+2} = \dots h_t=1$" ($1$ is the identity matrix).

By polarity, $(H_k)$ is equivalent to the assertion "$\mathbb V$ contains $P(h_1 \otimes h_2 \otimes \dots \otimes h_t)$ for any hermitian matrices $h_1,...,h_t$ with $h_1 = h_2 = \dots =h_k$ and $h_{k+1} = h_{k+2} = \dots h_t=1$".

Prove $(H_k)$ by induction on $k$. Take indeed a symmetric matrix $h$, and consider $f(x) = \exp(ixh)^{\otimes t}$ and develop $f$ in power series: $f(x) = \sum_n x^n a_n$. Then $a_n$ belongs to $\mathbb V$ for all $n$. For example $a_1 = i n t P(h\otimes 1 \ \dots \otimes 1)$ which proves $(H_1)$.

One moment of thinking gives you that you can write $a_k=i^k C(n,kC(t,k) P(h\otimes \dots h \otimes 1 \dots \otimes 1) +$something in $F_{k-1}$, where $C(n,k)$ C(t,k)$is the binomial coefficient and there are$kh$'s and$(t-k)1$'s in$h\otimes \dots h \otimes 1 \dots \otimes 1$. This allows to proceed by induction. Thus for$k=n$, k=t$, and using that $M_d(\mathbb C)$ is the linear span of the hermitian matrices, you get that $\mathbb V$ is the whole space of symmetric tensors.

1

Here is another proof of the same result (it is probably the same as the one given by darij grinberg but I do not understand all the words. It has perhaps more his place as a comment, but I do not know how to write comments). In the case $t=2$, it is Claim 1 in Lemma 1.3.2 of http://arxiv.org/pdf/1002.4595v2

Denote by $P : M_d(\mathbb C)^{\otimes t} \to M_d(\mathbb C)^{\otimes t}$ the linear map such that $P(A_1 \otimes \dots \otimes A_t) = 1/n!\sum A_{\sigma(1)} \otimes \dots \otimes A_{\sigma(t)}$, where the sum os over all permutations of ${1,\dots ,t}$. $P$ is the natural projection on the symmetric tensors.

Let $(H_k)$ be the assertion "$\mathbb V$ contains the space $F_k$ spanned by $P(h_1 \otimes h_2 \otimes \dots \otimes h_t)$ for any hermitian matrices $h_1,...,h_t$ with $h_{k+1} = h_{k+2} = \dots h_t=1$" ($1$ is the identity matrix).

By polarity, $(H_k)$ is equivalent to the assertion "$\mathbb V$ contains $P(h_1 \otimes h_2 \otimes \dots \otimes h_t)$ for any hermitian matrices $h_1,...,h_t$ with $h_1 = h_2 = \dots =h_k$ and $h_{k+1} = h_{k+2} = \dots h_t=1$".

Prove $(H_k)$ by induction on $k$. Take indeed a symmetric matrix $h$, and consider $f(x) = \exp(ixh)^{\otimes t}$ and develop $f$ in power series: $f(x) = \sum_n x^n a_n$. Then $a_n$ belongs to $\mathbb V$ for all $n$. For example $a_1 = i n P(h\otimes 1 \ \dots \otimes 1)$ which proves $(H_1)$.

One moment of thinking gives you that you can write $a_k=i^k C(n,k) P(h\otimes \dots h \otimes 1 \dots \otimes 1) +$something in $F_{k-1}$, where $C(n,k)$ is the binomial coefficient. This allows to proceed by induction.

Thus for $k=n$, and using that $M_d(\mathbb C)$ is the linear span of the hermitian matrices, you get that $\mathbb V$ is the whole space of symmetric tensors.