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Let V {\displaystyle V} be a vector space. We define a linear map σ {\displaystyle \sigma } from the tensor product V ⊗ V {\displaystyle V\otimes V} to itself by
σ ( v ⊗ w ) = w ⊗ v {\displaystyle \sigma (v\otimes w)=w\otimes v}
for all v ⊗ w ∈ V ⊗ V {\displaystyle v\otimes w\in V\otimes V} .
Then the symmetric-square of V {\displaystyle V} is
S y m 2 ( V ) = S 2 ( V ) := { a ∈ V ⊗ V : σ a = a } {\displaystyle \mathrm {Sym} ^{2}(V)=S^{2}(V):=\{a\in V\otimes V:\sigma a=a\}}
and the alternating-square or exterior-square of V {\displaystyle V} is
A l t 2 ( V ) = Λ 2 ( V ) := { a ∈ V ⊗ V : σ a = − a } {\displaystyle \mathrm {Alt} ^{2}(V)=\Lambda ^{2}(V):=\{a\in V\otimes V:\sigma a=-a\}} .
These are both eigenspaces of σ {\displaystyle \sigma } .
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