Symmetric -polynomials on Cn

Authors

  • T.V. Vasylyshyn Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0001-9055-6341
https://doi.org/10.15330/cmp.10.2.395-401

Keywords:

(p,q)-polynomial, -polynomial, symmetric -polynomial
Published online: 2018-12-31

Abstract

-Polynomials are natural generalizations of usual polynomials between complex vector spaces. A -polynomial is a function between complex vector spaces X and Y, which is a sum of so-called (p,q)-polynomials. In turn, for nonnegative integers p and q, a (p,q)-polynomial is a function between X and Y, which is the restriction to the diagonal of some mapping, acting from the Cartesian power Xp+q to Y, which is linear with respect to every of its first p arguments, antilinear with respect to every of its last q arguments and invariant with respect to permutations of its first p arguments and last q arguments separately.

In this work we construct formulas for recovering of (p,q)-polynomial components of -polynomials, acting between complex vector spaces X and Y, by the values of -polynomials. We use these formulas for investigations of -polynomials, acting from the n-dimensional complex vector space Cn to C, which are symmetric, that is, invariant with respect to permutations of coordinates of its argument. We show that every symmetric -polynomial, acting from Cn to C, can be represented as an algebraic combination of some "elementary" symmetric -polynomials.

Results of the paper can be used for investigations of algebras, generated by symmetric -polynomials, acting from Cn to C.

How to Cite
(1)
Vasylyshyn, T. Symmetric -Polynomials on Cn. Carpathian Math. Publ. 2018, 10, 395-401.

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