Reduction of symmetric semidefinite programs using the regular *-representation

Abstract. We consider semidefinite programming problems on which a permutation group is acting. We describe a general technique to reduce the size of such problems, exploiting the symmetry. The technique is based on a low-order matrix ∗-representation of the commutant (centralizer ring) of the matrix algebra generated by the permutation matrices. We apply it to extending a method of de Klerk et al. that gives a semidefinite programming lower bound to the crossing number of complete bipartite graphs. It implies that cr(K8,n) ≥ 2.9299n −6n, cr(K9,n) ≥ 3.8676n 2 − 8n, and (for any m ≥ 9)