Generic Generalized Diagonal Matrices

TL;DR

This paper studies the algebraic and combinatorial properties of generic generalized diagonal matrices, focusing on their minors, Stanley-Reisner complexes, and Cohen-Macaulay conditions, providing new characterizations and computations.

Contribution

It characterizes the facets of Stanley-Reisner complexes for these matrices and identifies ladder configurations that produce Cohen-Macaulay ideals, including the vertex decomposability case.

Findings

Minors have square-free initial ideals.

Facets of the Stanley-Reisner complex are explicitly described.

Height and multiplicity are computed for special cases.

Abstract

Generalized diagonal matrices are matrices that have two ladders of entries that are zero in the upper right and bottom left corners. The minors of generic generalized diagonal matrices have square-free initial ideals. We give a description of the facets of their Stanley-Reisner complex. With this description, we characterize the configuration of ladders that yield Cohen-Macaulay ideals. In the special case where both ladders are triangles, we show that the corresponding complex is vertex decomposable. Also in this case, we compute the height and multiplicity of the ideals.