Monomial Complete Intersections, The Weak Lefschetz Property and Plane Partitions

TL;DR

This paper characterizes when certain monomial complete intersections in three variables satisfy the Weak Lefschetz Property, revealing a surprising link with the enumeration of plane partitions and the primes dividing their counts.

Contribution

It establishes a novel connection between the Weak Lefschetz Property in algebra and the enumeration of plane partitions, providing explicit criteria based on prime divisors.

Findings

Primes dividing the number of plane partitions correspond to failures of WLP in certain monomial complete intersections.

A bijective construction relates the parameters of the algebraic objects to the enumeration of plane partitions.

Initial results show how algebraic techniques can inform number-theoretic properties of combinatorial counts.

Abstract

We characterize the monomial complete intersections in three variables satisfying the Weak Lefschetz Property (WLP), as a function of the characteristic of the base field. Our result presents a surprising, and still combinatorially obscure, connection with the enumeration of plane partitions. It turns out that the rational primes p dividing the number, M(a,b,c), of plane partitions contained inside an arbitrary box of given sides a,b,c are precisely those for which a suitable monomial complete intersection (explicitly constructed as a bijective function of a,b,c) fails to have the WLP in characteristic p. We wonder how powerful can be this connection between combinatorial commutative algebra and partition theory. We present a first result in this direction, by deducing, using our algebraic techniques for the WLP, some explicit information on the rational primes dividing M(a,b,c).