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mathematics-physics-wiki/docs/en/mathematics/linear-algebra/tensors/tensor-symmetries.md
2024-05-13 22:27:25 +02:00

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Tensor symmetries

We have a n \in \mathbb{N} finite dimensional vector space V such that \dim V = n, with a basis \{\mathbf{e}_i\}_{i=1}^n, a pseudo inner product \bm{g} on V and a corresponding dual space V^* with a basis \{\mathbf{\hat e}^i\}.

Symmetric tensors

Definition 1: let \pi = [\pi(1), \dots, \pi(k)] be any permutation of labels \{1, \dots, k\}, then \mathbf{T} \in \mathscr{T}^0_q(V) is a symmetric covariant tensor if for all \mathbf{v}_1, \dots, \mathbf{v}_q \in V we have

\mathbf{T}(\mathbf{v}{\pi(1)}, \dots, \mathbf{v}{\pi(q)}) = \mathbf{T}(\mathbf{v}_1, \dots, \mathbf{v}_q),

with k = q.

Likewise \mathbf{T} \in \mathscr{T}^p_0(V) is called a symmetric contravariant tensor if for all \mathbf{\hat u}_1, \dots, \mathbf{\hat u}_p \in V^* we have

\mathbf{T}(\mathbf{\hat u}{\pi(1)}, \dots, \mathbf{\hat u}{\pi(p)}) = \mathbf{T}(\mathbf{\hat u}_1, \dots, \mathbf{\hat u}_p),

with k = p.

This symmetry implies that the ordering of the (co)vector arguments in a tensor evaluation do not affect the outcome.

Definition 2: the vector space of symmetric covariant $q$-tensors is denoted by \bigvee_q(V) \subset \mathscr{T}^0_q(V) and the vector space of symmetric contravariant $p$-tensors is denoted by \bigwedge^p(V) \subset \mathscr{T}^p_0(V).

Alternatively one may write \bigvee_q(V) = V^* \otimes_s \cdots \otimes_s V^* and \bigwedge^p(V) = V \otimes_s \cdots \otimes_s V.

Antisymmetric tensors