A diagonal form for the incidence matrices of \(t\)-subsets vs. \(k\)- subsets (Q1814091)

Let \(X\) be a set with \(v\) elements. Denote by \(W_{tk}(v)\) the \({v\choose t}\times{v\choose k}\) matrix whose rows are indexed by the \(t\)-subsets \(T\) of \(X\) and whose columns are indexed by the \(k\)-subsets \(K\) of \(X\), where the entry in row \(T\) and column \(K\) is 1 if \(K\supseteq T\) and 0 otherwise. In a recent paper \textit{N. Linial} and \textit{B. L. Rothschild} [SIAM J. Algebraic Discrete Methods 2, 333-340 (1981; Zbl 0499.05017)] obtained a formula for the rank of \(W_{tk}(v)\) over the field \(\mathbb{Z}_ 2\) (the integers modulo 2), and gave a formula for the rank over \(\mathbb{Z}_ 3\) in case \(k=t+1\). In the present paper the author obtains a formula for the rank of \(W_{tk}(v)\) over the field \(\mathbb{Z}_ p\) in case \(t\leq\min(k,v-k)\). He also obtains a diagonal form (similar to a Smith normal form) for \(W_{tk}(v)\).