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

Abstract

Let $W_{k,n}^i(m)$ 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 of $W_{k,n}^i(m)$ is $1$ if $|S\cap T|=i$, and is $0$ otherwise. We compute the rank of the matrix $W_{2,n}^1(m)$ over any field by making use of the representation theory of the symmetric group. We also give a simple condition under which $W_{k,n}^i(m)$ has large $p$-rank.

arXiv comment13 pages. Corrected the statement of Theorem 2.1, part (v), and consequently needed to modify the proof of the case "char(F)=2, m odd"