Monomial Complete Intersections, The Weak Lefschetz Property and Plane Partitions
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).
Cite
@article{arxiv.1002.4400,
title = {Monomial Complete Intersections, The Weak Lefschetz Property and Plane Partitions},
author = {Jizhou Li and Fabrizio Zanello},
journal= {arXiv preprint arXiv:1002.4400},
year = {2010}
}
Comments
16 pages. Minor revisions, mainly to keep track of two interesting developments following the original posting. Final version to appear in Discrete Math