1.5 KiB
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 Vwe 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.