Proposition 16

*For an element $z \neq 0$ of $\bigwedge^p(E)$ to be pure, it is necessary and sufficient that, for all $u^* \in \bigwedge^{p-1}(E^*)$,

The case $p = 0$ is trivial and we assume $p \geq 1$. If $z = x_1 \wedge \cdots \wedge x_p$, formula (68) (no. 9) with $n = p - 1$ shows that $u^* \perp z$ is a linear combination of the $x_i$ ($1 \leq i \leq p$), whence (81). If on the other hand the subspace $M_z$ associated with $z$ is of dimension $> p$, consider a basis $(e_j)_{1 \leq j \leq n}$ of this subspace with $n > p$. It follows from no. 11, Proposition 13 that each of the $e_j$ is of the form $u^* \perp z$ for some $u^* \in \bigwedge^{p-1}(E^*)$ and relation (81) therefore implies $e_j \wedge z = 0$ for $1 \leq j \leq n$. It follows that in the expression $z = \sum_H a_H e_H$ (where H runs through the set of subsets of $\{1, n\}$ with $p$ elements) all the coefficients $a_H$ are zero, whence $z = 0$, contrary to the hypothesis.

The criterion of Proposition 16 is equivalent to writing conditions (81) when $u^*$ runs through a basis of $\bigwedge^{p-1}(E^*)$. In particular, suppose that E is of finite dimension $n$ and let $(e_i)_{1 \leq i \leq n}$ be a basis of E. Conditions (81) are then equivalent to the conditions

for all subsets J, H of $\{1, n\}$ such that $\mathrm{Card}(J) = p + 1$ and $\mathrm{Card}(H) = p - 1$. Now, if I and I' are two subsets of $\{1, n\}$ with $p$ elements, formulae (76) of no. 10 and multiplication table (20) of § 7, no. 8 show that

unless there exists an $i \in \{1, n\}$ such that $I - H = \{i\}$ and $J - I' = \{i\}$, in which case

where $\varepsilon_{i, J, H} = \rho_{\{i\}, H} \rho_{\{i\}, I'}$; it can then be said that for $i \in J \cap C H$, $\varepsilon_{i, J, H}$ is equal to $+1$ if the number of element of J which are $< i$ and the number of elements of H which are $< i$ have the same parity, and $-1$ otherwise.

It follows immediately that if we write $z = \sum_I a_I e_I$, where I runs through the set of subsets of $\{1, n\}$ with $p$ elements, relation (82–(J, H)) is equivalent to the relation

Relations (84) are called Grassman’s relations; these are therefore necessary and sufficient conditions (when J describes the set of subsets with $p + 1$ elements and H the set of subsets with $p - 1$ elements of $\{1, n\}$) for an element $z \neq 0$ of $\bigwedge^p(E)$ to be pure.

Note that relations (84) are not independent. For example, for $n = 4$ and $p = 2$, Grassmann’s relations reduce to the single relation

Let $D_p(E)$ be the subset of $\bigwedge^p(E)$ consisting of the pure $p$-vectors; clearly $D_p(E)$ is saturated with respect to the equivalence relation between $u$ and $v$: “there exists $\lambda \in K^*$ such that $v = \lambda u$” and two elements $u, v$ of $D_p(E)$ are equivalent under this relation if and only if the subspaces $M_u$ and $M_v$ of E which are associated with them are the same. Therefore we thus obtain a canonical bijection of the set of p-dimensional vector subspaces of E onto the image $G_p(E)$ of $D_p(E)$ in the projective space $\mathbf{P}(\bigwedge^p(E))$ associated with $\bigwedge^p(E)$. The subset $G_p(E)$ of $\mathbf{P}(\bigwedge^p(E))$ is called the Grassmannian of index $p$ of the vector space E. When E is finite-dimensional and $(e_i)_{1 \leq i \leq n}$ is a basis of E, the Grassmannian of index $p$ is the set of points of $\mathbf{P}(\bigwedge^p(E))$ for which a system of homogeneous coordinates $(a_I)$ (relative to the basis $(e_I)$ of $\bigwedge^p(E)$) satisfies Grassmann’s relations (84).

When $E = K^n$, we sometimes write $G_{n,p}(K)$ instead of $G_p(K^n)$, so that $G_{n,1}(K) = P_{n-1}(K)$. The mapping $M \mapsto M^0$, which associates with every $p$-dimensional subspace of $K^n$ the orthogonal subspace in $E^*$ (identified with $K^n$ under the choice of basis dual to the canonical basis of $K^n$) therefore defines a canonical bijection of $G_{n,p}(K)$ onto $G_{n,n-p}(K)$; Proposition 15 shows that this bijection is the restriction to $G_{n,p}(K)$ of a canonical isomorphism of the projective space $\mathbf{P}(\bigwedge^p(K^n))$ onto the projective space $\mathbf{P}(\bigwedge^{n-p}(K^n))$.