Dual near-factorizations and Schur products in association schemes

Source abstract

Let $G$ and $H$ be Gram matrices of spherical $2$-designs on the same number of indices. We study when their Schur product is again the Gram matrix of a spherical design. We give a criterion in terms of orthogonal projections with constant diagonal, and an explicit $15$-point spherical $3$-design whose Gram matrix belongs to no symmetric Bose-Mesner algebra. For a symmetric association scheme, the factorization problem takes the form $v(E_S\circ E_T)=μE_U$. This is a counterpart of ordinary matrix product factorizations in the adjacency basis. The case $E_U=I-E_0$ gives a dual near-factorization and has a regular simplex as its normalized output. We characterize complementary input supports by a strongly regular fusion, give a reciprocal criterion, and classify the case in which one factor is the last Hamming idempotent.