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# On the Vanishing of a (G, σ) Product in a (G, σ) Space

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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 iI, 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 iI, and for all gG.

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## References

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1. Bourbaki, N., Eléments de mathématique. I: Les structures fondamentales de Vanalyse, Fasc. VII, Livre II: Algèbre, Chapitre 3: Algèbre multilinêaire, Nouvelle éd., Actualités Sci. Indust., No. 1044 (Hermann, Paris, 1958).
2. Marvin, Marcus and Morris, Newman, Inequalities for the permanent function, Ann. of Math. (2) 75 (1962), 4762.
3. Mostow, G.D., Sampson, J. H., and Meyer, J.-P., Fundamental structures of algebra (McGraw- Hill, New York, 1963).
4. Hans, Schneider, Recent advances in matrix theory (Univ. Wisconsin Press, Madison, 1964).
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# On the Vanishing of a (G, σ) Product in a (G, σ) Space

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