Equality of Decomposable Symmetrized Tensors

Abstract

Let V be an n-dimensional vector space over the field F. Let ꕕm V be the rath tensor power of V. If ᓂ ∈ Sm, the symmetric group, there exists a linear operator P (ᓂ1) on ꕕm V such thatfor all x1, … , xm ∈ V. (Here, x1 ꕕ … ꕕ xm denotes the decomposable tensor product of the indicated vectors.) If c is any function of Sm taking its values in F, we define