Semi-Riemannian Geometry

Chapter 7 Multicovectors

It was remarked in Section 5.1 that the determinant function is the classic example of a multilinear function. In addition to its multilinearity, the determinant function has another characteristic feature—it is alternating. This chapter is devoted to an examination of tensors that have a corresponding property.

7.1 Multicovectors

Let V be a vector space of dimension m, and let s ≥ 1 be an integer. Following Section B.2, we denote by the group of permutations on {1, … , s}. For each permutation σ in , consider the linear map

defined by

(7.1.1)

for all tensors in and all vectors v ₁, ..., v _s in V. By saying that σ is linear, we mean that for all tensors , ℬ in and all real numbers c,

There is potential confusion arising from (7.1.1), ...