Koszul sign rule for graded tensor products and permutations

Let $k$ be a field (Field), and let $V_1,\dots ,V_n$ be graded $k$-vector spaces. A $k$-vector space is a unital left $k$-module as in Unital left and right modules over a ring; unqualified module means left module. Tensor powers are formed using the tensor products of Tensor algebra of a vector space. If $f_i:V_i\to W_i$ are homogeneous linear maps of degrees $d_i$, and $x_i\in V_i$ are homogeneous of degrees $p_i$, the graded tensor map is defined on pure tensors by

$(f_1\otimes \cdots \otimes f_n)(x_1\otimes \cdots \otimes x_n)=(-1{)}^{{\sum }_{i>j}{d}_i{p}_j}{f}_1(x_1)\otimes \cdots \otimes f_n(x_n)\mathrm{.}$

It is extended linearly. In particular, for two maps, $(f\otimes g)(x\otimes y)=(-1{)}^{\mathrm{\mid }g\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }}f(x)\otimes g(y)$.

For homogeneous $x_1,\dots ,x_n$ and a permutation $\sigma \in S_n$, the Koszul sign of reordering them to $x_{\sigma (1)}\otimes \cdots \otimes x_{\sigma (n)}$ is

$ϵ(\sigma ;x_1,\dots ,x_n)=(-1{)}^{{\sum }_{\begin{array}{c}i<j\\ \sigma (i)>\sigma (j)\end{array}}\mathrm{\mid }{x}_{\sigma (i)}\mathrm{\mid }\mathrm{\mid }{x}_{\sigma (j)}\mathrm{\mid }}\mathrm{.}$

Thus swapping two adjacent homogeneous factors $x,y$ contributes $(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}$. Every tensor of maps, permutation, unshuffle, and symmetric power below uses this same convention. A sign is a scalar in $k$ determined only by the parity of the displayed integer exponent.