  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.
