Symmetric polynomials and exterior power of a polynomial ring in one\n variable

In this article we consider the exterior power and the symmetric tensors of\nthe polynomial ring in one variable. The structure of an associative semigraded\nalgebra of this polynomial ring induces on the symmetric tensors the structure\nof an associative semigraded algebra, and on the exterior power induces\nstructure of a semigraded module over semigraded algebra of symmetric tensors.\nThe algebra of symmetric polynomials is isomorphic to the algebra of the\nsymmetric tensors of polynomial ring in one variables. We obtained the explicit\nexpression for symmetric polynomials via elementary symmetric polynomials and\nthe explicit expression for elements of the exterior power via elementary\nsymmetric polynomials and elements of the exterior power of the lower\npolynomial degree.\n