2026/07/16 by Allen W. Herman, Bobby Miraftab · 1 citation
#math.CO
We study matrix product factorizations (MPFs) in symmetric association schemes: identities ASAT=AU where AS,AT,AU are loopless unions of basic relations and the ordinary matrix product is again a 0-1 adjacency matrix. We give equivalent structural and spectral criteria for MPFs, derive valency and rank restrictions, and analyze several standard families. For 2-class schemes, the only nontrivial loopless MPF comes from the scheme of the 5-cycle. For P-polynomial schemes, the distance-regular recurrence gives strong restrictions on products A1Ai. We also prove a universal pentagon theorem for the case ASAT=J-I, and show that extremal rank forces all non-zero eigenvalues of AU to be ± k(U), hence gives bipartiteness. Finally, in Hamming schemes we obtain rank obstructions and classify MPFs of the form A1AT=AU: in H(d,2), for d≥2, the only non-zero loopless example is A1Ad=Ad-1, which is trivial since Ad has valency 1; for q>2, no non-zero example occurs.