A representation-theoretic computation of the rank of $1$-intersection incidence matrices: $2$-subsets vs. $n$-subsets

Abstract

Let W i k,n (m) denote a matrix with rows and columns indexed by the k-subsets and n-subsets, respectively, of an m-element set. The row S, column T entry ofand is 0 otherwise. We compute the rank of the matrix W 1 2,n (m) over any field by making use of the representation theory of the symmetric group. We also give a simple condition under which W i k,n (m) has large p-rank.