546: Face of Orientated Affine Simplex

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definition of face of orientated affine simplex

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of orientated affine simplex.

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Target Context

• The reader will have a definition of face of orientated affine simplex.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$V$: $\in \{\text{ the real vectors spaces }\}$
$\{p_0,...,p_n\}$: $\subseteq V$, $\in \{\text{ the affine-independent sets of base points on }V\}$
$(p_0,...,p_n)$: $=\text{ the orientated affine simplex }$
$\ast fac{e}_{\{j_1,...,j_l\}}((p_0,...,p_n))$: $=(-1{)}^{j_1}...(-1{)}^{j_l}(p_0,...,\widehat{p_{j_1}},...,\widehat{p_{j_l}},...,p_n)$, where $\{j_1,...,j_l\}\subseteq \{0,...,n\}$ where $j_1<...<j_l$ and the hat mark denotes that the element is missing
//

Conditions:
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$fac{e}_{\{j_1,...,j_l\}}((p_0,...,p_n))$ is called $(n-l)$-face of $(p_0,...,p_n)$ (there are $(n+1)!/(l!(n+1-l)!)$ $(n-l)$-faces).

When $0=l$, $fac{e}_{\{\}}([p_0,...,p_n])=[p_0,...,p_n]$ is a kind of face of $[p_0,...,p_n]$.

When $0<l$, $fac{e}_{\{j_1,...,j_l\}}([p_0,...,p_n])$ is called proper face of $[p_0,...,p_n]$.

$fac{e}_j((p_0,...,p_n)):=fac{e}_{\{j\}}((p_0,...,p_n))$ 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 $p_j\ne {\sigma }_j$), but $p_j={\sigma }_k$ for a $k$, and in fact, $fac{e}_j((p_0,...,p_n))=fac{e}_k(({\sigma }_0,...,{\sigma }_n))$ (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 $1<l$ cases, an $(n-l)$-face with the removed base points specified may depend on the representation (proved in Note).

$fac{e}_{\{j_1,...,j_l\}}((p_0,...,p_n))=fac{e}_{j_1}(...fac{e}_{j_l}((p_0,...,p_n)))$, but the order in the right hand side cannot be arbitrarily changed (proved in Note).

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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 $(p_0,...,p_n)$, $fac{e}_{\{j_1,...,j_l\}}((p_0,...,p_n))$, is $(-1{)}^{j_1}...(-1{)}^{j_l}(p_0,...,\widehat{p_{j_1}},...,\widehat{p_{j_l}},...,p_n)$, where $\{j_1,...,j_l\}\subseteq \{0,...,n\}$ where $j_1<...<j_l$ and the hat mark denotes that the element is missing

There are $(n+1)!/(l!(n+1-l)!)$ $(n-l)$-faces.

When $0=l$, $fac{e}_{\{\}}([p_0,...,p_n])=[p_0,...,p_n]$ is a kind of face of $[p_0,...,p_n]$.

When $0<l$, $fac{e}_{\{j_1,...,j_l\}}([p_0,...,p_n])$ is called proper face of $[p_0,...,p_n]$.

$fac{e}_j((p_0,...,p_n))$ is called $j$-th face of $(p_0,...,p_n)$.

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3: Note

Let us prove that when $(p_0,...,p_n)=({\sigma }_0,...,{\sigma }_n)$, $fac{e}_j((p_0,...,p_n))=fac{e}_k(({\sigma }_0,...,{\sigma }_n))$ 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, $2m$, because $(p_0,...,p_n)$ and $({\sigma }_0,...,{\sigma }_n)$ have the same parity. But $(p_0,...,\widehat{p_j},...,p_n)$ is permutated to become $({\sigma }_0,...,\widehat{{\sigma }_k},...,{\sigma }_n)$ by the $x$ switches, obviously, which means that $({\sigma }_0,...,\widehat{{\sigma }_k},...,{\sigma }_n)=(-1{)}^x(p_0,...,\widehat{p_j},...,p_n)$, which means that $(-1{)}^j(-1{)}^x({\sigma }_0,...,\widehat{{\sigma }_k},...,{\sigma }_n)=(-1{)}^j(-1{)}^x(-1{)}^x(p_0,...,\widehat{p_j},...,p_n)$, which means that $(-1{)}^{j+x}({\sigma }_0,...,\widehat{{\sigma }_k},...,{\sigma }_n)=(-1{)}^j(p_0,...,\widehat{p_j},...,p_n)$, but $k+j+x=k-j+x+2j=2m+2j$, 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, $(-1{)}^k({\sigma }_0,...,\widehat{{\sigma }_k},...,{\sigma }_n)=(-1{)}^j(p_0,...,\widehat{p_j},...,p_n)$.

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 $1<l$, 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 $fac{e}_{\{0,3\}}((p_0,p_1,p_2,p_3))=(-1{)}^0(-1{)}^3(p_1,p_2)$, but the face with $p_0,p_3$ removed for the latter representation is $fac{e}_{\{0,3\}}((p_3,p_2,p_1,p_0))=(-1{)}^0(-1{)}^3(p_2,p_1)$, and they are different.

$fac{e}_{\{j_1,...,j_l\}}((p_0,...,p_n))=fac{e}_{j_1}(...fac{e}_{j_l}((p_0,...,p_n)))$ is obvious, because $fac{e}_{j_k}$ does not influence the indexes of the elements before $j_k$. However, for $fac{e}_{\{0,3\}}((p_0,p_1,p_2,p_3))$, $fac{e}_3(fac{e}_0(p_0,p_1,p_2,p_3))$ is even invalid ($fac{e}_3((-1{)}^0(p_1,p_2,p_3))$ is invalid because $p_3$ has the index $2$ now), and even if we mean $fac{e}_2(fac{e}_0(p_0,p_1,p_2,p_3))$ with the removed points specified, $fac{e}_2(fac{e}_0(p_0,p_1,p_2,p_3))=fac{e}_2((-1{)}^0(p_1,p_2,p_3))=(-1{)}^2(-1{)}^0(p_1,p_2)\ne fac{e}_{\{0,3\}}((p_0,p_1,p_2,p_3))=(-1{)}^0(-1{)}^3(p_1,p_2)$.

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References

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