1014: Symmetrization of Tensor w.r.t. Some Arguments

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definition of symmetrization of tensor w.r.t. some arguments

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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: Note

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Starting Context

• The reader knows a definition of tensors space with respect to field and k vectors spaces and vectors space over field.

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Target Context

• The reader will have a definition of symmetrization of tensor with respect to some arguments.

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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:
$F$: $\in \{\text{ the fields }\}$
$\{V_1,...,V_k,W\}$: $\subseteq \{\text{ the }F\text{ vectors spaces }\}$, where $V_{j_1}=...=V_{j_l}:=V$ for some $\{V_{j_1},...,V_{j_l}\}\subseteq \{V_1,...,V_k\}$
$L(V_1,...,V_k:W)$: $=\text{ the tensors space }$
$P_{\{j_1,...,j_l\}}$: $=\text{ the group of the permutations of }(1,...,k)\text{ that move only }(j_1,...,j_l)$
$\ast Sy{m}_{\{j_1,...,j_l\}}$: $:L(V_1,...,V_k:W)\to L(V_1,...,V_k:W)$
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Conditions:
$\mathrm{\forall }f\in L(V_1,...,V_k:W),\mathrm{\forall }(v_1,...,v_k)\in V_1\times ...\times V_k(Sy{m}_{\{j_1,...,j_l\}}(f)(v_1,...,v_k)=1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}f(v_{{\sigma }_1},...,v_{{\sigma }_k}))$
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When $\{V_{j_1},...,V_{j_l}\}=\{V_1,...,V_k\}$, $Sy{m}_{\{j_1,...,j_l\}}$ is denoted also as just $Sym$.

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2: Note

$V_{j_1}=...=V_{j_l}:=V$ is required because otherwise, putting $v_{{\sigma }_{j_m}}$, which was in $\{v_{j_1},...,v_{j_l}\}$, into the $j_m$-th argument would not make sense.

While it requires that $V_{j_1}=...=V_{j_l}:=V$, another $V_j$ is allowed to equal $V$: we do not necessarily need to do the symmetrization with respect to all the vectors spaces that equal $V$.

Let us see that $Sy{m}_{\{j_1,...,j_l\}}$ is indeed into $L(V_1,...,V_k:W)$.

For each $f\in L(V_1,...,V_k:W)$, $Sy{m}_{\{j_1,...,j_l\}}(f)$ is obviously $:V_1\times ...\times V_k\to F$.

Let us see the multi-linearity of $Sy{m}_{\{j_1,...,j_l\}}(f)$: for $(v_1,...,v_k)=(v_1,...,r^{\prime }{v}_j^{\prime }+r^{″}{v}_j^{″},...,v_k)$, $Sy{m}_{\{j_1,...,j_l\}}(f)(...,r^{\prime }{v}_j^{\prime }+r^{″}{v}_j^{″},...)=r^{\prime }Sy{m}_{\{j_1,...,j_l\}}(f)(...,v_j^{\prime },...)+r^{″}Sy{m}_{\{j_1,...,j_l\}}(f)(...,v_j^{″},...)$?

There are 2 cases: 1) $j\notin \{j_1,...,j_l\}$; 2) $j\in \{j_1,...,j_l\}$.

Let $j\notin \{j_1,...,j_l\}$.

$Sy{m}_{\{j_1,...,j_l\}}(f)(v_1,...,v_k)=1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}f(v_{{\sigma }_1},...,v_{{\sigma }_k}))=1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}f(v_{{\sigma }_1},...,r^{\prime }{v}_j^{\prime }+r^{″}{v}_j^{″},...,v_{{\sigma }_k}))$, where $r^{\prime }{v}_j^{\prime }+r^{″}{v}_j^{″}$ is not moved by any $\sigma$, $=1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}(r^{\prime }f(v_{{\sigma }_1},...,v_j^{\prime },...,v_{{\sigma }_k}))+r^{″}f(v_{{\sigma }_1},...,v_j^{″},...,v_{{\sigma }_k})))=r^{\prime }1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}f(v_{{\sigma }_1},...,v_j^{\prime },...,v_{{\sigma }_k})+r^{″}1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}f(v_{{\sigma }_1},...,v_j^{″},...,v_{{\sigma }_k})=r^{\prime }Sy{m}_{\{j_1,...,j_l\}}(f)(...,v_j^{\prime },...)+r^{″}Sy{m}_{\{j_1,...,j_l\}}(f)(...,v_j^{″},...)$.

Let $j\in \{j_1,...,j_l\}$.

$Sy{m}_{\{j_1,...,j_l\}}(f)(v_1,...,v_k)=1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}f(v_{{\sigma }_1},...,v_{{\sigma }_k}))$, where $v_j=r^{\prime }{v}_j^{\prime }+r^{″}{v}_j^{″}$ is moved to the $j_m$-th argument as $v_{{\sigma }_{j_m}}$, which means that ${\sigma }_{j_m}=j$, $=1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}f(v_{{\sigma }_1},...,v_{{\sigma }_{j_m}},...,v_{{\sigma }_k}))=1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}f(v_{{\sigma }_1},...,r^{\prime }{v}_j^{\prime }+r^{″}{v}_j^{″},...,v_{{\sigma }_k}))=1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}(r^{\prime }f(v_{{\sigma }_1},...,v_j^{\prime },...,v_{{\sigma }_k}))+r^{″}f(v_{{\sigma }_1},...,v_j^{″},...,v_{{\sigma }_k})))$, but $v_j^{\prime }$ and $v_j^{″}$ can be denoted as $v_{{\sigma }_{j_m}}^{\prime }$ and $v_{{\sigma }_{j_m}}^{″}$, $=r^{\prime }1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}f(v_{{\sigma }_1},...,v_{{\sigma }_{j_m}}^{\prime },...,v_{{\sigma }_k}))+r^{″}1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}f(v_{{\sigma }_1},...,v_{{\sigma }_{j_m}}^{″},...,v_{{\sigma }_k})))=r^{\prime }Sy{m}_{\{j_1,...,j_l\}}(f)(...,v_j^{\prime },...)+r^{″}Sy{m}_{\{j_1,...,j_l\}}(f)(...,v_j^{″},...)$.

So, yes, $Sy{m}_{\{j_1,...,j_l\}}(f)$ is indeed multi-linear, and $Sy{m}_{\{j_1,...,j_l\}}(f)\in L(V_1,...,V_k:W)$.

Let us see that $Sy{m}_{\{j_1,...,j_l\}}(f)$ is symmetric with respect to the $\{j_1,...,j_l\}$ arguments, which is the reason why $Sy{m}_{\{j_1,...,j_l\}}$ is called "symmetrization".

Let ${\sigma }^{\prime }\in P_{\{j_1,...,j_l\}}$ be any. What we need to see is that $Sy{m}_{\{j_1,...,j_l\}}(f)(v_{{\sigma }_1^{\prime }},...,v_{{\sigma }_k^{\prime }})=Sy{m}_{\{j_1,...,j_l\}}(f)(v_1,...,v_k)$.

$Sy{m}_{\{j_1,...,j_l\}}(f)(v_{{\sigma }_1^{\prime }},...,v_{{\sigma }_k^{\prime }})=1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}f(v_{(\sigma \circ {\sigma }^{\prime }{)}_1},...,v_{(\sigma \circ {\sigma }^{\prime }{)}_k}))$, but by the proposition that for any group, the multiplication map with any fixed element from left or right is a bijection, $\sigma \circ {\sigma }^{\prime }$ visits each element of $P_{\{j_1,...,j_l\}}$ once, so, $=1/l!\sum _{\sigma \circ {\sigma }^{\prime }\in P_{\{j_1,...,j_l\}}}f(v_{(\sigma \circ {\sigma }^{\prime }{)}_1},...,v_{(\sigma \circ {\sigma }^{\prime }{)}_k}))=Sy{m}_{\{j_1,...,j_l\}}(f)(v_1,...,v_k)$.

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References

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