Regular sequences of symmetric polynomials

TL;DR

This paper investigates conditions under which certain sets of symmetric power sum polynomials form regular sequences, proposing a conjecture and providing evidence for specific cases, advancing understanding in algebraic combinatorics.

Contribution

It introduces a conjecture characterizing when three power sums form a regular sequence in three variables and offers partial proofs and evidence supporting this conjecture.

Findings

n consecutive power sums in n variables form a regular sequence

A necessary condition is that n! divides the product of degrees

Conjecture: for three variables, p_a, p_b, p_c form a regular sequence iff 6 divides abc

Abstract

Denote by p_k the k-th power sum symmetric polynomial n variables. The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular sequence. We consider then the following problem: describe the subsets n powersums forming a regular sequence. A necessary condition is that n! divides the product of the degrees of the elements. To find an easily verifiable sufficient condition turns out to be surprisingly difficult already in 3 variables. Given positive integers a<b<c with GCD(a,b,c)=1, we conjecture that p_a, p_b, p_c is a regular sequence for n=3 if and only if 6 divides abc. We provide evidence for the conjecture by proving it in several special instances.