$\Sigma^-$ admixture in the $^{10}_\Lambda$Li hypernucleus with a repulsive $\Sigma$-nucleus potential
$\Sigma^-$ hyperon component of the $^{10}_\Lambda$Li wave function is studied. The $\Sigma^-$ admixture is vital for production of neutron-rich $\Lambda$ hypernuclei via mesonic beams. We use a simplified shell model wave function for the $\Lambda$ channel and calculate the $\Sigma^-$ admixture directly from coupled equations. Probability of the $\Sigma^-$ admixture for realistic repulsive $\Sigma^-$-nucleus potentials is less by several times than that for attractive potentials and does not exceed $0.1\%$. We conclude that the cross sections of the $^{10}\mbox{B}(\pi^-,K^+){}^{10}_\Lambda\mbox{Li}$ reaction measured at KEK cannot be explained by production via $\Sigma^-$ admixture as a doorway state.