Plücker coordinates: Difference between revisions

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== Rational points ==

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>. Abstract temperaments correspond exactly to these rational points (although ~~the vast majority of them will be terrible temperaments)~~.

A '''rational point''' <math>P</math> on <math>\mathrm{Gr}(k, n)</math> is a k-dimensional subspace such that <math>\mathcal{L} = P \cap \mathbb{Z}^n</math> is a rank k sublattice of <math>\mathbb{Z}^n</math>. Abstract temperaments correspond exactly to these rational points, although most have no practical musical use.

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~~.

The same relations as above can be derived, where we represent P as integer matrix <math>M \in \mathbb{Z} ^ {k \times n}</math>, whose rows span <math>\mathcal{L}</math>.

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.

The projective coordinates similarly have integer entries.

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

An advantage of studying rational points is that we do not have to worry about [[torsion]].

The quotient group <math>\mathbb{Z}^n / \mathcal{L}</math> is a finitely generated abelian group.

When the Plücker coordinates are normalized (GCD = 1), we ensure that

<math>

\mathbb{Z}^n / \mathcal{L} \cong \mathbb{Z}^{n-k},

</math>

which is torsion-free.

== Height ==

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 == Rational points ==
-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>. Abstract temperaments correspond exactly to these rational points (although the vast majority of them will be terrible temperaments).
+A '''rational point''' <math>P</math> on <math>\mathrm{Gr}(k, n)</math> is a k-dimensional subspace such that <math>\mathcal{L} = P \cap \mathbb{Z}^n</math> is a rank k sublattice of <math>\mathbb{Z}^n</math>. Abstract temperaments correspond exactly to these rational points, although most have no practical musical use.
-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.
+The same relations as above can be derived, where we represent P as integer matrix <math>M \in \mathbb{Z} ^ {k \times n}</math>, whose rows span <math>\mathcal{L}</math>.
-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.
+The projective coordinates similarly have integer entries.
+Because the Plücker coordinates are homogeneous, we can always put them in a canonical form by dividing all entries by their greatest common divisor (GCD) and ensuring the first element is non-negative.
+An advantage of studying rational points is that we do not have to worry about [[torsion]].
+The quotient group <math>\mathbb{Z}^n / \mathcal{L}</math> is a finitely generated abelian group.
+When the Plücker coordinates are normalized (GCD = 1), we ensure that
+<math>
+\mathbb{Z}^n / \mathcal{L} \cong \mathbb{Z}^{n-k},
+</math>
+which is torsion-free.
 == Height ==