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

Filip Jonsson Kling, Samuel Lundqvist

TL;DR

This work addresses when symmetric polynomials act as max-rank elements on equigenerated monomial complete intersections, extending the Lefschetz paradigm beyond the classical linear form. Via a refined $k$-degree grading, Macaulay inverse systems, and determinant analysis of Toeplitz matrices, the authors achieve a complete classification for power-sum polynomials and a detailed two-variable Schur-polynomial criterion, plus partial results and conjectures for elementary and complete homogeneous symmetric polynomials. The results illuminate intricate interactions between graded Gorenstein structure, invariant theory, and combinatorics, and expose several open questions for higher variables and broader partitions. The findings have potential implications for related topics in algebraic combinatorics, Hilbert-series symmetry, and representation theory, offering a solid foundation for future work on Lefschetz-type phenomena in symmetric-invariant algebras.

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.

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

TL;DR

This work addresses when symmetric polynomials act as max-rank elements on equigenerated monomial complete intersections, extending the Lefschetz paradigm beyond the classical linear form. Via a refined -degree grading, Macaulay inverse systems, and determinant analysis of Toeplitz matrices, the authors achieve a complete classification for power-sum polynomials and a detailed two-variable Schur-polynomial criterion, plus partial results and conjectures for elementary and complete homogeneous symmetric polynomials. The results illuminate intricate interactions between graded Gorenstein structure, invariant theory, and combinatorics, and expose several open questions for higher variables and broader partitions. The findings have potential implications for related topics in algebraic combinatorics, Hilbert-series symmetry, and representation theory, offering a solid foundation for future work on Lefschetz-type phenomena in symmetric-invariant algebras.

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.
Paper Structure (10 sections, 18 theorems, 92 equations, 4 figures)

This paper contains 10 sections, 18 theorems, 92 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Algebraic preliminaries
  3. The power sum symmetric polynomial
  4. Schur polynomials
  5. The case $n=2$
  6. The case $n \geq 3$
  7. The elementary symmetric polynomial
  8. The complete homogeneous symmetric polynomial
  9. Outlook and further observations
  10. Artinian Gorenstein algebras

Key Result

Lemma 3.2

An element $x_1^{a_1}\cdots x_n^{a_n}\cdot f(x_1^k,\dots, x_n^k)$ is a non-zero element in the kernel of multiplication by $p_{n,k}$ in $A=\mathbf{k}[x_1,\dots, x_n]/(x_1^d,\dots, x_n^d)$ if and only if $f(x_1^k,\dots, x_n^k)$ is a non-zero element in the kernel of multiplication by $p_{n,k}$ in $B=

Figures (4)

  • Figure 1: Failures of $p_{n,k}$ to be a max-rank element on $A$ given $d=km+i$ for some values of $n$ and $k$.
  • Figure 2: All $d>k$ where $p_{n,k}$ is a max-rank element on $A$.
  • Figure 3: Multiplication matrix when $n=4, k=4$ and $d=6$.
  • Figure 4: Multiplication matrix when $n=7, k=6$ and $d=3$.

Theorems & Definitions (43)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Lemma 3.6
  • proof
  • ...and 33 more