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In this paper, we use semidefinite programming and representation theory tocompute new lower bounds on the crossing number of the complete bipartite graph$K_{m,n}$, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20(2006), 189--202] and the subsequent reduction by De Klerk, Pasechnik andSchrijver [Math. Prog. Ser. A and B, 109 (2007) 613--624]. We exploit the fullsymmetry of the problem using a novel decomposition technique. This results ina full block-diagonalization of the underlying matrix algebra, which we use toimprove bounds on several concrete instances. Our results imply that$\text{cr}(K_{10,n}) \geq 4.87057 n^2 - 10n$, $\text{cr}(K_{11,n}) \geq 5.99939n^2-12.5n$, $\text{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\text{cr}(K_{13,n})\geq 8.65675 n^2-18n$ for all $n$. The latter three bounds are computed using anew and well-performing relaxation of the original semidefinite programmingbound. This new relaxation is obtained by only requiring one small matrix blockto be positive semidefinite.

New lower bounds on crossing numbers of $K_{m,n}$ from semidefinite programming