Let $V_\bullet = (V_k)_{k \in \mathbb{Z}}$ and $W_\bullet = (W_k)_{k \in \mathbb{Z}}$ be two graded vector spaces on 0 caracteristic field. We define the tensor product of $V_\bullet$ by $W_\bullet$ to be, for any integer $k$, $$ (V_\bullet \otimes W_\bullet)_k = \bigoplus_{i + j = k} V_i \otimes W_j $$ where $V_i \otimes W_j$ is the usual tensor product on vector spaces. We define the Koszul braiding $\tau_{V_\bullet,W_\bullet}$to be, $$ v \otimes w \mapsto (-1)^{|v| |w|} w \otimes v, $$ where $v$ and $w$ are homogenous. This endows the category of graded vector spaces with the structure of a symmetric monoidal category.
Let $V_\bullet = (V_k)_{k \in \mathbb{Z}}$ a graded vector space. The $n$-th tensor product $(V_\bullet)^{\otimes n}$ has a natural left action of the symetric group $\mathfrak{S}_n$ by, $$ \tau_{i,i+1} \mapsto id_V\otimes \cdots \otimes \tau_{V_\bullet,V_\bullet}\otimes \cdots \otimes id_V $$ where $\tau_{V_\bullet,V_\bullet}$ is in the i-th position.
Let $v_1,\dots, v_n$ be homogenous (in degree) vectors of $V_\bullet$ and $\sigma \in \mathfrak{S}_n$. My first question is:
is there any way to get a formula for the sign of: $$ \sigma \cdot (v_1 \otimes \cdots \otimes v_n) = \pm v_{\sigma^{-1}(1)}\otimes \cdots \otimes v_{\sigma^{-1}(n)} ? $$
This is related to my second question which is the one that I wanted to awnser when I came a cross the first.
Let $T(V_\bullet) = \bigoplus_{n} (V_\bullet)^n$ be the tensor algebra of $V_\bullet$. Let $p_n:(V_\bullet)^{\otimes} \to (V_\bullet)^{\otimes}$ defined by $\frac{1}{n!}\sum_{\sigma \in \mathfrak{S}_n} \sigma$ and $q_n$ defined by $\frac{1}{n!}\sum_{\sigma \in \mathfrak{S}_n} |\sigma|\cdot \sigma$ where $|\sigma|$ is the signature of $\sigma$. We define $S(V)$ to be the cokernel of $\oplus_n p_n$ and $\Lambda(V)$ to be the cokernel of $\oplus_n q_n$ as defined in http://ncatlab.org/nlab/show/symmetric+algebra and http://ncatlab.org/nlab/show/exterior+algebra. Let also $(s^{-1}V)_{\bullet}$ be de desuspension of $V_\bullet$.
Is there any relation between $S(V)$ and $\Lambda(s^{-1}V)$ (isomorphism modulo suspension maybe) ?