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Symmetric Tensor Product (·)_s

The symmetrization of a pure tensor, obtained by averaging over all permutations of its factors.

Let HH be a complex and let x1,…,xn∈Hx_1,\dots,x_n\in H. Their symmetric tensor is the symmetrization

(x1⊗⋯⊗xn)s=1n!∑π∈Snxπ(1)⊗⋯⊗xπ(n).(x_1\otimes\cdots\otimes x_n)_s=\frac1{n!}\sum_{\pi\in S_n} x_{\pi(1)}\otimes\cdots\otimes x_{\pi(n)}.

It belongs to the subspace of H⊗nH^{\otimes n} fixed by permutations of the tensor factors.

Remarks

Symmetric tensors span the nn-particle symmetric tensor power; these powers form the used for .

Examples
  • For n=2n=2, (x⊗y)s=12(x⊗y+y⊗x)(x\otimes y)_s=\tfrac12(x\otimes y+y\otimes x).