Plücker coordinates: Difference between revisions

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[[File:Plucker_embedding.png|thumb|600px|right|Schematic illustration of the Plücker embedding. Linear subspaces of <math>\mathbb{R}^n</math> (here lines) get mapped to points on a quadric surface in projective space.]]

In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the ~~'''~~[[wedgie]]~~'''~~), are a way to assign coordinates to temperaments, by viewing them as elements of some projective space.

In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the [[wedgie]]) are a way to assign coordinates to abstract temperaments, by viewing them as elements of some projective space.

The usual way to write down an abstract temperament is via its mapping matrix, but Plücker coordinates give us a unique description that is useful for some calculations.

== Definition ==

~~We have~~ a Grassmannian variety <math>\mathrm{Gr} (k, n)</math> ~~consisting of the~~ k-dimensional subspaces of <math>\mathbb{R}^n</math>.

A temperament can be viewed as a point in what is called a Grassmannian variety, written as <math>\mathrm{Gr} (k, n)</math>.

~~The rational points on this variety can be identified with~~ rank~~-k temperaments on a JI space with~~ n primes.

This variety contains all possible k-dimensional subspaces of <math>\mathbb{R}^n</math>.

In musical terms, k represents the rank of the temperament (how many independent generators it has), and n is the number of primes we're considering in our [[just intonation subgroup]].

Let <math>M</math> be an element of <math>\mathrm{Gr} (k, n)</math>, spanned by basis vectors <math>m_1, \ldots, m_k</math>.

~~We can embed~~ the ~~Grassmannian~~ into a projective space ~~using~~ the ~~Plücker map~~:

These basis vectors are the rows of the temperament mapping matrix.

The Plücker map takes a temperament and embeds it into a projective space by taking the wedge product of the basis vectors:

$$

\begin{align}

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$$

Here, <math>\Lambda^{k} \mathbb{R}^n</math> is the k-th exterior power of ~~our~~ original space ~~<math>\mathbb{R}^n</math>. The~~ dimension ~~of <math>\mathrm{Gr} (k, n)</math> is~~ <math>k(n-k)</math>, ~~while~~ the ~~dimension~~ of ~~<math>\Lambda^{k} \mathbb{R}^n</math>~~ is <math>\binom{n}{k}</math>~~, which is typically much larger~~.

Here, <math>\Lambda^{k} \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose.

While the original space of temperaments has dimension <math>k(n-k)</math>, the space of Plücker coordinates is typically larger, with dimension <math>\binom{n}{k} - 1</math>.

== Examples ==

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$$

These are not 'proper' coordinates, as doing row operations on this matrix preserves the row-span.

~~Put another way, we can always multiply by some <math>g \in GL_k (\mathbb{R})</math>~~.

The projective coordinates can be calculated by taking the determinants of all <math>2 \times 2</math> sub-matrices

$$

p_{ij} =

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== Plücker relations ==

The coordinates must satisfy some algebraic relations called Plücker relations. Generally, the projective space is much 'larger' than the Grassmannian, and the image in the projective space is some quadric surface.

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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>.

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).

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.

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+{{Expert}}
 [[File:Plucker_embedding.png|thumb|600px|right|Schematic illustration of the Plücker embedding. Linear subspaces of <math>\mathbb{R}^n</math> (here lines) get mapped to points on a quadric surface in projective space.]]
 {{Wikipedia|Plücker embedding}}
-In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the '''[[wedgie]]'''), are a way to assign coordinates to temperaments, by viewing them as elements of some projective space.
+In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the [[wedgie]]) are a way to assign coordinates to abstract temperaments, by viewing them as elements of some projective space.
+The usual way to write down an abstract temperament is via its mapping matrix, but Plücker coordinates give us a unique description that is useful for some calculations.
 == Definition ==
-We have a Grassmannian variety <math>\mathrm{Gr} (k, n)</math> consisting of the k-dimensional subspaces of <math>\mathbb{R}^n</math>.
+A temperament can be viewed as a point in what is called a Grassmannian variety, written as <math>\mathrm{Gr} (k, n)</math>.
-The rational points on this variety can be identified with rank-k temperaments on a JI space with n primes.
+This variety contains all possible k-dimensional subspaces of <math>\mathbb{R}^n</math>.
+In musical terms, k represents the rank of the temperament (how many independent generators it has), and n is the number of primes we're considering in our [[just intonation subgroup]].
 Let <math>M</math> be an element of <math>\mathrm{Gr} (k, n)</math>, spanned by basis vectors <math>m_1, \ldots, m_k</math>.
-We can embed the Grassmannian into a projective space using the Plücker map:
+These basis vectors are the rows of the temperament mapping matrix.
+The Plücker map takes a temperament and embeds it into a projective space by taking the wedge product of the basis vectors:
 $$
 \begin{align}
@@ Line 19: / Line 25: @@
 $$
-Here, <math>\Lambda^{k} \mathbb{R}^n</math> is the k-th exterior power of our original space <math>\mathbb{R}^n</math>. The dimension of <math>\mathrm{Gr} (k, n)</math> is <math>k(n-k)</math>, while the dimension of <math>\Lambda^{k} \mathbb{R}^n</math> is <math>\binom{n}{k}</math>, which is typically much larger.
+Here, <math>\Lambda^{k} \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose.
+While the original space of temperaments has dimension <math>k(n-k)</math>, the space of Plücker coordinates is typically larger, with dimension <math>\binom{n}{k} - 1</math>.
 == Examples ==
@@ Line 36: / Line 43: @@
 $$
 These are not 'proper' coordinates, as doing row operations on this matrix preserves the row-span.
-Put another way, we can always multiply by some <math>g \in GL_k (\mathbb{R})</math>.
 The projective coordinates can be calculated by taking the determinants of all <math>2 \times 2</math> sub-matrices
 $$
 p_{ij} =
@@ Line 60: / Line 65: @@
 == Plücker relations ==
 The coordinates must satisfy some algebraic relations called Plücker relations. Generally, the projective space is much 'larger' than the Grassmannian, and the image in the projective space is some quadric surface.
@@ Line 71: / Line 77: @@
 == 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>.
+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).
 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.