Skew-Symmetry and Symmetry in the Tensor Algebra

Abstract

Given a vector space E consider the pth tensorial power ⊗^pE and let S _p denote the permutation group on p letters. Then every permutation σ ∊ S _p determines a linear automorphism of ⊗^pE (also denoted by σ) defined by

$$\sigma (x_1 \otimes \cdot \cdot \cdot \otimes x_p ) = x_{\sigma ^{ - 1} (1)} \otimes \cdot \cdot \cdot \otimes {\text{ }}x_{\sigma ^{ - 1} (p)} {\text{ }}x_v \in E.$$

All vector spaces in this chapter are defined over a field of characteristic zero.