Trivariate monomial complete intersections and plane partitions

TL;DR

This paper explores the algebraic structure of certain polynomial quotient rings, relating Smith normal forms to Toeplitz matrices and Schur polynomials, and establishes connections between determinants and plane partitions.

Contribution

It introduces new relationships between Smith normal forms of matrices, provides a bijective proof linking algebraic invariants to plane partitions, and extends identities to symmetry classes.

Findings

Smith normal forms relate to Toeplitz matrices via Schur polynomials

Determinants of specific components count plane partitions in a box

Identities extend to symmetry classes of plane partitions

Abstract

We consider the homogeneous components U_r of the map on R = k[x,y,z]/(x^A, y^B, z^C) that multiplies by x + y + z. We prove a relationship between the Smith normal forms of submatrices of an arbitrary Toeplitz matrix using Schur polynomials, and use this to give a relationship between Smith normal form entries of U_r. We also give a bijective proof of an identity proven by J. Li and F. Zanello equating the determinant of the middle homogeneous component U_r when (A, B, C) = (a + b, a + c, b + c) to the number of plane partitions in an a by b by c box. Finally, we prove that, for certain vector subspaces of R, similar identities hold relating determinants to symmetry classes of plane partitions, in particular classes 3, 6, and 8.