812: Same-Length Multi-Dimensional Array Antisymmetrized with Respect to Set of Indexes

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definition of same-length multi-dimensional array antisymmetrized with respect to set of indexes

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Topics

About: set

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The table of contents of this article

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

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Starting Context

• The reader knows a definition of same-length multi-dimensional array.
• The reader admits the proposition that any permutation bijectively maps the set of all the permutations onto the set of all the permutations by composition from left or right.

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Target Context

• The reader will have a definition of same-length multi-dimensional array antisymmetrized with respect to set of indexes.

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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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0: Note 1

Any $n$-length 2-dimensional array is an $n\times n$ matrix.

The components of any $(p-q)$ tensor with respect to any bases is a vectors-space-dimension-length (p + q)-dimensional array, but a same-length multi-dimensional array is not necessarily the components of a tensor: a same-length multi-dimensional array is in general just a collection of numbers not necessarily related to any tensor.

The dimensions of the same-length multi-dimensional array have to have the same length for our purpose, because otherwise, a permutation of indexes would not make sense.

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$M$: $\in \{\text{ the same-length }n\text{ -dimensional arrays }\}$, $=\left(\begin{array}{c}{M}_{j_1,...,j_n}\end{array}\right)$
$S$: $\subseteq \{1,...,n\}$, $=\{l_1,...,l_k\}$
$\ast M^{\prime }$: $\in \{\text{ the same-length }n\text{ -dimensional arrays }\}$, $=\left(\begin{array}{c}{M}_{j_1,...,j_n}^{\prime }\end{array}\right)=\left(\begin{array}{c}1/k!\sum _{\sigma }sgn\sigma M_{\sigma ((j_1,...,j_n){)}_1,...,\sigma ((j_1,...,j_n){)}_n}\end{array}\right)$, where each $\sigma$ is a permutation of $(j_1,...,j_n)$ such that only $(j_{l_1},...,j_{l_k})$ is permutated
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Conditions:
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2: Natural Language Description

For any same-length $n$-dimensional array, $M=\left(\begin{array}{c}{M}_{j_1,...,j_n}\end{array}\right)$, and any subset, $S=\{l_1,...,l_k\}\subseteq \{1,...,n\}$, the array, $M^{\prime }=\left(\begin{array}{c}{M}_{j_1,...,j_n}^{\prime }\end{array}\right)=\left(\begin{array}{c}1/k!\sum _{\sigma }sgn\sigma M_{\sigma ((j_1,...,j_n){)}_1,...,\sigma ((j_1,...,j_n){)}_n}\end{array}\right)$, where each $\sigma$ is a permutation of $(j_1,...,j_n)$ such that only $(j_{l_1},...,j_{l_k})$ is permutated

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3: Note 2

When, for example, $S=\{1,...,k\}$, the antisymmetrized array is denoted as $\left(\begin{array}{c}{M}_{[j_1,...,j_k],j_{k+1},...,j_n}\end{array}\right)$.

For any permutation, ${\sigma }^{\prime }$, that permutates only $(j_{l_1},...,j_{l_k})$, $M_{{\sigma }^{\prime }((j_1,...,j_n){)}_1,...,{\sigma }^{\prime }((j_1,...,j_n){)}_n}^{\prime }=sgn{\sigma }^{\prime }{M}_{j_1,...,j_n}^{\prime }$, which is indeed the purpose of antisymmetrizing, because $M_{{\sigma }^{\prime }((j_1,...,j_n){)}_1,...,{\sigma }^{\prime }((j_1,...,j_n){)}_n}^{\prime }=1/k!\sum _{\sigma }sgn\sigma M_{\sigma \circ {\sigma }^{\prime }((j_1,...,j_n){)}_1,...,\sigma \circ {\sigma }^{\prime }((j_1,...,j_n){)}_n}=1/k!\sum _{\sigma }sgn\sigma sgn{\sigma }^{\prime }sgn{\sigma }^{\prime }{M}_{\sigma \circ {\sigma }^{\prime }((j_1,...,j_n){)}_1,...,\sigma \circ {\sigma }^{\prime }((j_1,...,j_n){)}_n}=sgn{\sigma }^{\prime }1/k!\sum _{\sigma }sgn(\sigma \circ {\sigma }^{\prime })M_{\sigma \circ {\sigma }^{\prime }((j_1,...,j_n){)}_1,...,\sigma \circ {\sigma }^{\prime }((j_1,...,j_n){)}_n}=sgn{\sigma }^{\prime }1/k!\sum _{\sigma \circ {\sigma }^{\prime }}sgn(\sigma \circ {\sigma }^{\prime })M_{\sigma \circ {\sigma }^{\prime }((j_1,...,j_n){)}_1,...,\sigma \circ {\sigma }^{\prime }((j_1,...,j_n){)}_n}$, because as $\sigma$ iterates all the permutations of $S$, also $\sigma \circ {\sigma }^{\prime }$ iterates all the permutations of $S$, by the proposition that any permutation bijectively maps the set of all the permutations onto the set of all the permutations by composition from left or right, $=sgn{\sigma }^{\prime }1/k!\sum _{{\sigma }^{″}}sgn({\sigma }^{″})M_{{\sigma }^{″}((j_1,...,j_n){)}_1,...,{\sigma }^{″}((j_1,...,j_n){)}_n}$, with $\sigma \circ {\sigma }^{\prime }$ just renamed as ${\sigma }^{″}$, $=sgn{\sigma }^{\prime }{M}_{j_1,...,j_n}^{\prime }$.

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References

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