We assume again that we have at our disposal \( n \) different single-particle orbits \( \alpha_0,\alpha_2,\dots,\alpha_{n-1} \) and that we can distribute among these orbits \( N\le n \) particles. The ordering among these states is important as it defines the order of the creation operators. We will write the determinant $$ \Phi_{\Lambda} = a_{\alpha_3}^{\dagger} a_{\alpha_6}^{\dagger} a_{\alpha_{10}}^{\dagger} a_{\alpha_{13}}^{\dagger} |0\rangle, $$ in a more compact way as $$ \Phi_{3,6,10,13} = |0001001000100100\rangle. $$ The action of a creation operator is thus $$ a^{\dagger}_{\alpha_4}\Phi_{3,6,10,13} = a^{\dagger}_{\alpha_4}|0001001000100100\rangle=a^{\dagger}_{\alpha_4}a_{\alpha_3}^{\dagger} a_{\alpha_6}^{\dagger} a_{\alpha_{10}}^{\dagger} a_{\alpha_{13}}^{\dagger} |0\rangle, $$ which becomes $$ -a_{\alpha_3}^{\dagger} a^{\dagger}_{\alpha_4} a_{\alpha_6}^{\dagger} a_{\alpha_{10}}^{\dagger} a_{\alpha_{13}}^{\dagger} |0\rangle=-|0001101000100100\rangle. $$