Knowlpedia

Symmetric Tensor Product (·)_s

Averaging over permutations to land in the symmetric subspace

For x1,…,xn∈Hx_1,\dots,x_n\in H, the symmetric tensor

(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)}.
Remarks

Key property (paper use):

  • These generate the symmetric Fock space used for Fock–Cook quantization.
Examples
  • For n=2n=2, (x⊗y)s=12(x⊗y+y⊗x)(x\otimes y)_s=\tfrac12(x\otimes y+y\otimes x).