Bipartite Determinantal Ideals and concurrent vertex maps

Abstract

Bipartite determinantal ideals are introduced by Illian and the author 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.