Bases and determinants in the discrete Grassmanian incidence matrix

Characterize the subsets S of Gr_{r+1}(n) over the finite field F_q for which the columns of the inclusion-incidence matrix B (with entries B_{v,u}=1 if v⊆u and 0 otherwise, where V=Gr_r(n) and U=Gr_{r+1}(n)) form a basis of the column space of B; and determine explicit structural properties or formulas for determinants of restricted submatrices of B formed by selecting rows and columns indexed by subsets of Gr_r(n) and Gr_{r+1}(n), respectively.

Background

In the discrete Grassmanian example, the authors consider the bipartite graph with V=Gr_r(n) and U=Gr_{r+1}(n) over a fixed finite field F_q, with edges representing subspace inclusion. The corresponding incidence matrix B has full row rank when r+1<n/2. They paper local limits of determinantal processes defined via the row space of B.

Despite establishing distributional limits via their main theorems, the combinatorial structure of bases (i.e., which column sets in U yield a basis of the column space of B) and the behavior of determinants of restricted submatrices of B remain unknown in this setting, and resolving these would clarify the algebraic and combinatorial underpinnings of the determinantal measures considered.