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→Symmetric part of a tensor: being clear about brackets |
→Symmetric product: fmt |
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:<math>v_1\odot v_2\odot\cdots\odot v_r := \frac{1}{r!}\sum_{\sigma\in\mathfrak{S}_r} v_{\sigma 1}\otimes v_{\sigma 2}\otimes\cdots\otimes v_{\sigma r}.</math>
In general we can turn Sym(''V'') into an [[algebra]] by defining the commutative and associative product
| last1 = Kostrikin | first1 = Alexei I.
| last2 = Manin | first2 = Iurii Ivanovich
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| pages = 276–279
| isbn = 9056990497
}}</ref> Given two tensors {{nowrap|''T''<sub>1</sub> ∈ Sym<sup>''k''<sub>1</sub></sup>(''V'')}} and {{nowrap|''T''<sub>2</sub> ∈ Sym<sup>''k''<sub>2</sub></sup>(''V'')}}, we use the symmetrization operator to define:
:<math>T_1\odot T_2 = \operatorname{Sym}(T_1\otimes T_2)\quad\left(\in\operatorname{Sym}^{k_1+k_2}(V)\right).</math>
It can be verified (as is done by Kostrikin and Manin<ref name="Kostrikin1997" />) that the resulting product is in fact commutative and associative. In some cases the operator is
In some cases an exponential notation is used:
:<math>v^{\odot k} = \underbrace{v \odot v \odot \cdots \odot v}_{k\text{ times}}=\underbrace{v \otimes v \otimes \cdots \otimes v}_{k\text{ times}}=v^{\otimes k}.</math>
Where ''v'' is a vector.
Again, in some cases the
:<math>v^k=\underbrace{v\,v\,\cdots\,v}_{k\text{ times}}=\underbrace{v\odot v\odot\cdots\odot v}_{k\text{ times}}.</math>
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