I am working through a book (Random Fields and Geometry) which contains the following definitions.
A covariant tensor $\gamma$ of order $k$ (a $(k,0)$-tensor) is said to be alternating if
$$\gamma(v_{\sigma(1)},...v_{\sigma(k)})=\epsilon_{\sigma}\gamma(v_{1},...v_{k}) \text{ for all } \sigma\in S(k)$$
where $S(k)$ is the symmetric group of permutations of $k$ letters and $\epsilon_\sigma$ is the sign of the permutation $\sigma$.
I think I understand this so far. Please could someone confirm my understanding up until this point? As I understand it, $\epsilon_\sigma$ is like the Levi-Civata symbol and an alternating tensor is a tensor which satisfies the condition that if we permute the basis vectors using an odd permutation the sign of the vector components are flipped. Is this correct so far? I am hesitant as the book never defines the $v_i$ notation used above and I am unsure if this represents basis vectors or vector components. Any clarity would be appreciated.
1 Answer
The symbols $v_i$ and $v_{\sigma(i)}$ represent vectors, so if you permute the input vectors {$v_i$} given to the tensor $\gamma$ then you get the same result for the unpermuted vectors multiplied by the ${\rm sgn}(\sigma)$ (or $\epsilon_\sigma$). The latter gives $+1$ or $-1$ depending on the parity of the permutation used on the vectors
