Bipartite determinantal ideals and concurrent vertex maps

Abstract

Bipartite determinantal ideals are introduced in Illian and Li (Gröbner basis for the double determinantal ideals, http://arxiv.org/abs/2305.01724) as a vast generalization of the classical determinantal ideals intensively studied in commutative algebra, algebraic geometry, representation theory, and combinatorics. We introduce a combinatorial model called concurrent vertex maps to describe the Stanley–Reisner complex of the initial ideal of any bipartite determinantal ideal, and study properties and applications of this model including vertex decomposability, shelling orders, formulas of the Hilbert series, and h-polynomials.