# Koszul sign convention and symmetric group action on the graded n-th tensor product

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) ?

• What do you mean by a formula for the sign? It can be readily computed. There isn't really even a formula for the sign of a permutation in the usual sense. You can compute it more quickly than finding a reduced word, even asymptotically, but it's $n\log n$ instead of $n^2$. It's an algorithm; no one would call it a formula. – Matt Samuel Sep 2 '15 at 23:42
• OK. $n\log n$ is for finding the number of inversions. I just thought of a $O(n)$ method for finding the sign, but it's still not a formula. – Matt Samuel Sep 3 '15 at 0:00
• I thaught about a formula where the degrees of the elements $v_1, \ldots, v_n$ and the sign of the permutation apeared but it seems hopeless :( – jeanmfischer Sep 3 '15 at 14:55
• Really it's only odd and even that matter. Perhaps it's something like the sign of the permutation you get when you restrict to the elements of odd degree. (I also never understood why people would multiply in an order so that the final result is applying the inverse. I used the opposite convention in my dissertation and I didn't encounter any problems. It's also the most natural convention when you compose functions as though they act on the left, which is what you do in virtually anything except arrow diagrams.) – Matt Samuel Sep 3 '15 at 15:04
• oh! the thing is to have a left action, I'm working with operads and collections (or $\Sigma_*$-modules) and they use that convention in the literature (think it's because they like to see it as a presheaf in vector spaces i.e. contravariant functor from the permutation groupoid mathoverflow.net/questions/117533/…). Don't think it's essential for the question :) – jeanmfischer Sep 3 '15 at 15:31