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

2022/05/10 by Joshua E. Ducey, Ducey, Joshua E., Colby J. Sherwood +1
Computer Science · Engineering · #05E18 (primary) #20C30 (secondary) #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Matrix Theory and Algorithms #Representation Theory (math.RT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2205.04660

openalex publication_date 2022/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let W_k,nⁱ(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,nⁱ(m) is 1 if |S ∩ T| = i, and is 0 otherwise. We compute the rank of the matrix W_2,n¹(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,nⁱ(m) has large p-rank.

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