TITLE: Signs in the natural map $\Lambda^k V \otimes \Lambda^k V^* \to \Bbbk$
QUESTION [15 upvotes]: Let $V$ be a finite-dimensional vector space over a field $\Bbbk$.  Let $V^*$ denote its dual.  I strongly suspect that there is a natural map
$$\Lambda^k V \otimes \Lambda^k V^* \to \Bbbk$$
that looks something like
$$v_1 \wedge \dotsb \wedge v_k \otimes \alpha_1 \wedge \dotsb \wedge \alpha_k \mapsto \sum_{\sigma} {\operatorname{sgn} \, \sigma}\prod_i \alpha_i(v_{\sigma(i)}).$$
What does the correct, natural formula look like?  In particular, what is the correct sign convention?

REPLY [5 votes]: As I remarked in one of the comments, a nice write-up about tensor and exterior pairings is this one by Brian Conrad (as part of a series of handouts to be found at his website).
His approach is to let a general bilinear pairing $B:V\times W\to k$ yield a pairing $V^{\otimes n}\times W^{\otimes n}\to k$ given by $(\otimes v_i,\otimes l_j)\mapsto \prod_{i=1}^n B(v_i,l_i)$. Under the natural conditions (invariance under swaps, or vanishing if a sequence of inputs `stammers'), this induces parings on the symmetric, or exterior algebras.
In particular, applied to the evaluation pairing $B:V\times V^*\to k$ this induces the desired pairing $\bigwedge^n(V)\times \bigwedge(V^*)\to k$ given by $(\wedge v_i,\wedge l_j)\mapsto \det(l_i(v_j))$.
(This is a very short summary; his explanation is much better and extensive so read it yourself :))