Trivariate monomial complete intersections and plane partitions

Details Trivariate monomial complete intersections and plane partitions Trivariate monomial complete intersections and plane partitions

2010-08-08

arxiv.org/abs/1008.1426v2

Mathematics

Combinatorics

21 pages, 15 figures

Scientific paper

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.

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