Bases and determinants for incidence matroids of l–r set inclusion

Characterize the bases of the matroid defined by the columns of the unsigned incidence matrix B between l-subsets and r-subsets of [n] (with entries B_{v,u}=1 if v⊆u and 0 otherwise, for fixed integers l<r≤n), and determine general formulas or properties for determinants of submatrices of B associated with these bases.

Background

The incidence matroid setting takes V=\binom{[n]}{l} and U=\binom{[n]}{r} with B the unsigned inclusion-incidence matrix. It is known that B has full row rank, and in certain special cases (e.g., l=1, r=2) bases and determinant-related weights are understood; however, a general characterization is lacking.

Understanding which column sets form bases and how the determinants of corresponding submatrices behave would illuminate the structure of the matroid and enable deeper analysis of determinantal processes and their local limits beyond currently solved special cases.