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

TL;DR

This paper explores the algebraic structures of symmetric tensors and exterior powers of a polynomial ring in one variable, establishing isomorphisms and explicit formulas involving elementary symmetric polynomials.

Contribution

It provides explicit expressions for symmetric polynomials and exterior power elements in terms of elementary symmetric polynomials, clarifying their algebraic relationships.

Findings

Isomorphism between algebra of symmetric polynomials and symmetric tensors

Explicit formulas for symmetric polynomials using elementary symmetric polynomials

Explicit expressions for exterior power elements in terms of elementary symmetric polynomials

Abstract

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