Plücker Relations

Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Plücker relations are quadratic forms on $\wedge^d V$ whose zero set is exactly the set of decomposable vectors in $\wedge^d V$ (i.e. which are of the form $v_1 \wedge ... \wedge v_d$), thus describing the ideal corresponding to the Plücker embedding $\text{Gr}_d(V) \to \mathbb{P}(\wedge^d V)$. But in every book I've read so far, these Plücker relations are constructed by means of many identifications between duals, exterior powers, etc. so that I am not able to write them down explicitely. Although I've tried it, many signs and sums confuse me.

Question. Is it possible to write down these Plücker relations explicitely as a set of polynomials in the ring $K[\{x_H\}]$, where $H$ runs through the subsets of $\{1,...,n\}$ with $d$ elements? (Of course it is possible, but I wonder how do this in general)

Edit: Following the answer below, here is the

Answer: Instead of using these subsets $H$, use indices $1 \leq i_1 < ... < i_d \leq n$, and extend the definition of $x_{i_1,...,i_d}$ to all $d$-tuples in such a way that $x_{i_1,...,i_d}=0$ if these $i_j$ are not pairwise distinct, and otherwise $x_{i_1,....,i_d} = sign(\sigma) \cdot x_{i_{\sigma(1)},...,i_{\sigma(d)}}$, where $\sigma$ is the unique permutation of $1,...,d$ which makes $i_{\sigma(1)} < ... < i_{\sigma(d)}$. Then the Plücker relations are

$\sum\limits_{j=0}^{d} (-1)^j x_{i_1,...,i_{d-1},k_j} * x_{k_0,...,\hat{k_j},...,k_d} = 0$

for integers $i_1,...,i_{d-1},k_0,...,k_d$ between $1,...,n$.

Solution 1:

Yes, the Plücker relations are written down totally explicitly in terms of the polynomials you require on page 110, equation (3.4.10), of Jacobson's book Finite-Dimensional Algebras over Fields. The proof, attributed by the author to Faulkner (a student of his?), is completely down-to-earth: no identifications, no duality,...

Edit Since Martin doesn't have access to the book, I'm adding an online presentation, with the relevant equations on page 21. It is very elementary, with concrete examples, and might appeal to readers whose interest has been whetted by Martin's question.
And the bibliography contains a reference to a masterful article by Kleiman and Laksov, which also contains the Plücker relations handled with minors of determinants and nothing else.

Solution 2:

I think the answer provided above could be improved a bit. (Not logically, just made a bit clearer.) My method avoids having extra variables and setting repeating sequences equal to zero.

Take a field $F$, and let $$X = \{x_H : H \subset \{1,2,...n\} \text{ and } |H| = d\}$$ So we can think of variables as being indexed by length $d$ increasing sequences. For $I,K \subset \{1,2,...,n\}$ of size $d-1$ and $d+1$, respectively, we can define the $(I,K)$ Plucker relation $Pl_{I,K}(X)$ by the formula:$$ Pl_{I,K}(X) = \sum_{k \in K - I} (-1)^{S_{I,K}(k)} x_{I \cup \{k\}} x_{K - \{k\}} $$ And the sign $S_{I,K}(k) = \#\{i \in I : k < i\} + \#\{\ell \in K: k < \ell\} $