In this paper, we use semidefinite programming and representation theory to compute 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 and Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613--624]. We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that $\text{cr}(K_{10,n}) \geq 4.87057 n^2 - 10n$, $\text{cr}(K_{11,n}) \geq 5.99939 n^2-12.5n$, $\text{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\text{cr}(K_{13,n}) \geq 8.65675 n^2-18n$ for all $n$. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.

翻译：本文利用半定规划与表示理论，为完全二分图 $K_{m,n}$ 的交叉数计算了新的下界，这是对 de Klerk 等人 [SIAM J. Discrete Math. 20 (2006), 189--202] 的方法及后续由 De Klerk、Pasechnik 和 Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613--624] 提出的约化方法的扩展。我们采用一种新颖的分解技术，充分挖掘了问题的对称性。该技术实现了底层矩阵代数的完全块对角化，并用于改进多个具体实例的界。我们的结果表明：对所有 $n$，有 $\text{cr}(K_{10,n}) \geq 4.87057 n^2 - 10n$, $\text{cr}(K_{11,n}) \geq 5.99939 n^2-12.5n$, $\text{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\text{cr}(K_{13,n}) \geq 8.65675 n^2-18n$。后三个界是通过对原始半定规划界进行一种新颖且高效的松弛计算得到的。该新松弛仅需一个小的矩阵块保持半正定即可实现。