definition of face of orientated affine simplex
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of orientated affine simplex.
Target Context
- The reader will have a definition of face of orientated affine simplex.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\( (p_0, ..., p_n)\): \(= \text{ the orientated affine simplex }\)
\(*face_{\{j_1, ..., j_l\}} ((p_0, ..., p_n))\): \(= (-1)^{j_1} ... (-1)^{j_l} (p_0, ..., \hat{p_{j_1}}, ..., \hat{p_{j_l}}, ..., p_n)\), where \(\{j_1, ..., j_l\} \subseteq \{0, ..., n\}\) where \(j_1 \lt ... \lt j_l\) and the hat mark denotes that the element is missing
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Conditions:
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\(face_{\{j_1, ..., j_l\}} ((p_0, ..., p_n))\) is called \((n - l)\)-face of \((p_0, ..., p_n)\) (there are \((n + 1)! / (l! (n + 1 - l)!)\) \((n - l)\)-faces).
When \(0 = l\), \(face_{\{\}} ([p_0, ..., p_n]) = [p_0, ..., p_n]\) is a kind of face of \([p_0, ..., p_n]\).
When \(0 \lt l\), \(face_{\{j_1, ..., j_l\}} ([p_0, ..., p_n])\) is called proper face of \([p_0, ..., p_n]\).
\(face_j ((p_0, ..., p_n)) := face_{\{j\}} ((p_0, ..., p_n))\) is called \(j\)-th face of \((p_0, ..., p_n)\).
As \((p_0, ..., p_n) = (\sigma_0, ..., \sigma_n)\) for any even-parity permutation, \(\sigma\), of \((p_0, ..., p_n)\), the \(j\)-th face of \((p_0, ..., p_n)\) is not necessarily the \(j\)-th face of \((\sigma_0, ..., \sigma_n)\) (it cannot be when \(p_j \neq \sigma_j\)), but \(p_j = \sigma_k\) for a \(k\), and in fact, \(face_j ((p_0, ..., p_n)) = face_k ((\sigma_0, ..., \sigma_n))\) (proved in Note). So, although the term, "\(j\)-th face", requires the specification of the representation which \(j\) refers to, the face with the removed base point specified does not depend on the representation.
However, for general \(1 \lt l\) cases, an \((n - l)\)-face with the removed base points specified may depend on the representation (proved in Note).
\(face_{\{j_1, ..., j_l\}} ((p_0, ..., p_n)) = face_{j_1} (... face_{j_l} ((p_0, ..., p_n)))\), but the order in the right hand side cannot be arbitrarily changed (proved in Note).
2: Natural Language Description
For any real vectors space, \(V\), and any affine-independent set of base points, \(\{p_0, ..., p_n\} \subseteq V\), and the orientated affine simplex, \((p_0, ..., p_n)\), any \((n - k)\)-face of \((p_0, ..., p_n)\), \(face_{\{j_1, ..., j_l\}} ((p_0, ..., p_n))\), is \((-1)^{j_1} ... (-1)^{j_l} (p_0, ..., \hat{p_{j_1}}, ..., \hat{p_{j_l}}, ..., p_n)\), where \(\{j_1, ..., j_l\} \subseteq \{0, ..., n\}\) where \(j_1 \lt ... \lt j_l\) and the hat mark denotes that the element is missing
There are \((n + 1)! / (l! (n + 1 - l)!)\) \((n - l)\)-faces.
When \(0 = l\), \(face_{\{\}} ([p_0, ..., p_n]) = [p_0, ..., p_n]\) is a kind of face of \([p_0, ..., p_n]\).
When \(0 \lt l\), \(face_{\{j_1, ..., j_l\}} ([p_0, ..., p_n])\) is called proper face of \([p_0, ..., p_n]\).
\(face_j ((p_0, ..., p_n))\) is called \(j\)-th face of \((p_0, ..., p_n)\).
3: Note
Let us prove that when \((p_0, ..., p_n) = (\sigma_0, ..., \sigma_n)\), \(face_j ((p_0, ..., p_n)) = face_k ((\sigma_0, ..., \sigma_n))\) where \(p_j = \sigma_k\).
Let us suppose that \(j \le k\) without loss of generality.
\((p_0, ..., p_n)\) is permutated by the \(k - j\) switches such that \(p_j\) is at the \(k\)-th position: \(p_j\) and \(p_{j + 1}\) are switched; then, \(p_j\) and \(p_{j + 2}\) are switched; ...; then, \(p_j\) and \(p_{k}\) are switched. The result is permutated by some \(x\) switches to become \((\sigma_0, ..., \sigma_n)\): as \(p_j\) was already at the \(k\)-th position, it is permutating only the other points. \(k - j + x\) is even, \(2 m\), because \((p_0, ..., p_n)\) and \((\sigma_0, ..., \sigma_n)\) have the same parity. But \((p_0, ..., \hat{p_j}, ..., p_n)\) is permutated to become \((\sigma_0, ..., \hat{\sigma_k}, ..., \sigma_n)\) by the \(x\) switches, obviously, which means that \((\sigma_0, ..., \hat{\sigma_k}, ..., \sigma_n) = (-1)^x (p_0, ..., \hat{p_j}, ..., p_n)\), which means that \((-1)^j (-1)^x (\sigma_0, ..., \hat{\sigma_k}, ..., \sigma_n) = (-1)^j (-1)^x (-1)^x (p_0, ..., \hat{p_j}, ..., p_n)\), which means that \((-1)^{j + x} (\sigma_0, ..., \hat{\sigma_k}, ..., \sigma_n) = (-1)^j (p_0, ..., \hat{p_j}, ..., p_n)\), but \(k + j + x = k - j + x + 2 j = 2 m + 2 j\), which means that when \(j + x\) is even, \(k\) is even; when \(j + x\) is odd, \(k\) is odd, which means that \((-1)^{j + x} = (-1)^k\), and so, \((-1)^k (\sigma_0, ..., \hat{\sigma_k}, ..., \sigma_n) = (-1)^j (p_0, ..., \hat{p_j}, ..., p_n)\).
We said that \(j \le k\) is without loss of generality, because otherwise, \((\sigma_0, ..., \sigma_n)\) can be permutated to \((p_0, ..., p_n)\) instead.
For any general \((n - l)\)-face case with \(1 \lt l\), let us see a counterexample. While \((p_0, p_1, p_2, p_3) = (p_3, p_2, p_1, p_0)\), the face with \(p_0, p_3\) removed for the former representation is \(face_{\{0, 3\}} ((p_0, p_1, p_2, p_3)) = (-1)^0 (-1)^3 (p_1, p_2)\), but the face with \(p_0, p_3\) removed for the latter representation is \(face_{\{0, 3\}} ((p_3, p_2, p_1, p_0)) = (-1)^0 (-1)^3 (p_2, p_1)\), and they are different.
\(face_{\{j_1, ..., j_l\}} ((p_0, ..., p_n)) = face_{j_1} (... face_{j_l} ((p_0, ..., p_n)))\) is obvious, because \(face_{j_k}\) does not influence the indexes of the elements before \(j_k\). However, for \(face_{\{0, 3\}} ((p_0, p_1, p_2, p_3))\), \(face_3 (face_0 (p_0, p_1, p_2, p_3))\) is even invalid (\(face_3 ((-1)^0 (p_1, p_2, p_3))\) is invalid because \(p_3\) has the index \(2\) now), and even if we mean \(face_2 (face_0 (p_0, p_1, p_2, p_3))\) with the removed points specified, \(face_2 (face_0 (p_0, p_1, p_2, p_3)) = face_2 ((-1)^0 (p_1, p_2, p_3)) = (-1)^2 (-1)^0 (p_1, p_2) \neq face_{\{0, 3\}} ((p_0, p_1, p_2, p_3)) = (-1)^0 (-1)^3 (p_1, p_2)\).
