In this paper, we shall construct a vector space, called the (G, σ) space, which generalizes the tensor space, the Grassman space, and the symmetric space. Then we shall determine a necessary and sufficient condition that the (G, σ) product of the vectors x
1, x
2, …, xn
is zero.
1. Let G be a permutation group on I = {1, 2, …, n} and F, an arbitrary field. Let σ be a linear character of G, i.e., σ is a homomorphism of G into the multiplicative group F
* of F.
For each i ∈ I, let Vi
be a finite-dimensional vector space over F. Consider the Cartesian product W = V
1 × V
2 × … × Vn
.
1.1. Definition. W is called a G-set if and only if Vi
= Vg(i)
for all i ∊ I, and for all g ∊ G.