On maximal rank properties for symmetric polynomials in an equigenerated monomial complete intersection

TL;DR

This paper explores the maximal rank properties of symmetric polynomials within equigenerated monomial complete intersections, extending known results about the strong Lefschetz property to broader classes of symmetric polynomials.

Contribution

It provides a complete characterization for power sum symmetric polynomials and two-variable Schur polynomials, with partial results and open questions for other symmetric polynomials in multiple variables.

Findings

Power sum symmetric polynomial has maximal rank in this setting.

Two-variable Schur polynomials exhibit maximal rank, with explicit determinant formulas.

Partial results suggest similar properties for elementary and complete homogeneous symmetric polynomials.

Abstract

It is well known that a monomial complete intersection has the strong Lefschetz property in characteristic zero. This property is equivalent to the statement that any power of the sum of the variables is a maximal rank element on the complete intersection. In this paper, we investigate what happens when this element is replaced by another symmetric polynomial, in an equigenerated complete intersection. We answer the question completely for the power sum symmetric polynomial using a grading technique, and for any Schur polynomial in the case of two variables by deriving a closed formula for the determinants of a family of Toeplitz matrices. Further, we obtain partial results in three or more variables for the elementary and the complete homogeneous symmetric polynomials and pose several open questions.