Plücker coordinates: Difference between revisions

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A rational point <math>P</math> on <math>\mathrm{Gr}(k, n)</math> is a k-dimensional subspace such that <math>P \cap \mathbb{Z}^n</math> is a rank k sublattice of <math>\mathbb{Z}^n</math>.

The same relations as above can be derived, where we represent M as integer matrix <math>M \in \mathbb{Z} ^ {k \times n}</math> and the projective coordinates similarly have entries in <math>\mathbb{Z}</math> instead.

The same relations as above can be derived, where we represent P as integer matrix <math>M \in \mathbb{Z} ^ {k \times n}</math> and the projective coordinates similarly have entries in <math>\mathbb{Z}</math> instead.

Because the Plücker coordinates are homogeneous, we can always put them in a 'canonical' form by dividing all entries by their GCD and ensuring the first element is non-negative.

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A height function is a way to measure the 'arithmetic complexity' of a rational point. For example, the rational numbers <math>\frac{3}{2}</math> and <math>\frac{3001}{2001}</math> are close to eachother, but intuitively the second is much more complicated.

We can define the height of a rational point simply as the Euclidean norm on ~~the~~ Plücker coordinates.

We can define the height of a rational point simply as the Euclidean norm on its Plücker coordinates <math>X = \iota (P)</math>.

$$

H(P) = \left\| P \right\| = \left\| ~~m_1~~ \wedge \ldots \wedge ~~m_n~~ \right\| \\

H(P) = \left\| X \right\| = \left\| p_1 \wedge \ldots \wedge p_n \right\| \\

$$

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$$

\begin{align}

\mathrm{G}_{ij} &= \left\langle ~~m_i~~, ~~m_j~~ \right\rangle \\

\mathrm{G}_{ij} &= \left\langle p_i, p_j \right\rangle \\

~~\det(\mathrm{G}) &= \det(M^{\mathsf{T}} M)~~ \\

\sqrt{\det(\mathrm{G})} &= \left\| p_1 \wedge \ldots \wedge p_n \right\| = \left\| X \right\| \, .

\sqrt{\det(\mathrm{G})} &= \left\| ~~m_1~~ \wedge \ldots \wedge ~~m_n~~ \right\| = \left\| P \right\| \, .

\end{align}

$$

@@ Line 72: / Line 72: @@
 A rational point <math>P</math> on <math>\mathrm{Gr}(k, n)</math> is a k-dimensional subspace such that <math>P \cap \mathbb{Z}^n</math> is a rank k sublattice of <math>\mathbb{Z}^n</math>.
-The same relations as above can be derived, where we represent M as integer matrix <math>M \in \mathbb{Z} ^ {k \times n}</math> and the projective coordinates similarly have entries in <math>\mathbb{Z}</math> instead.
+The same relations as above can be derived, where we represent P as integer matrix <math>M \in \mathbb{Z} ^ {k \times n}</math> and the projective coordinates similarly have entries in <math>\mathbb{Z}</math> instead.
 Because the Plücker coordinates are homogeneous, we can always put them in a 'canonical' form by dividing all entries by their GCD and ensuring the first element is non-negative.
@@ Line 79: / Line 79: @@
 A height function is a way to measure the 'arithmetic complexity' of a rational point. For example, the rational numbers <math>\frac{3}{2}</math> and <math>\frac{3001}{2001}</math> are close to eachother, but intuitively the second is much more complicated.
-We can define the height of a rational point simply as the Euclidean norm on the Plücker coordinates.
+We can define the height of a rational point simply as the Euclidean norm on its Plücker coordinates <math>X = \iota (P)</math>.
 $$
-	H(P) = \left\| P \right\| = \left\| m_1 \wedge \ldots \wedge m_n \right\| \\
+	H(P) = \left\| X \right\| = \left\| p_1 \wedge \ldots \wedge p_n \right\| \\
 $$
@@ Line 87: / Line 87: @@
 $$
 	\begin{align}
-	\mathrm{G}_{ij} &= \left\langle m_i, m_j \right\rangle \\
+	\mathrm{G}_{ij} &= \left\langle p_i, p_j \right\rangle \\
-	\det(\mathrm{G}) &= \det(M^{\mathsf{T}} M) \\
+	\sqrt{\det(\mathrm{G})} &= \left\| p_1 \wedge \ldots \wedge p_n \right\| = \left\| X \right\| \, .
-	\sqrt{\det(\mathrm{G})} &= \left\| m_1 \wedge \ldots \wedge m_n \right\| = \left\| P \right\| \, .
 	\end{align}
 $$