By using representation theory and explicit idempotents in the group ring $F_p[GL_n(Z/p)]$, we give a new splitting of $B(Z/p)^n_+$ into $p-1$ stable wedge summands in which the numbers of occurrences of the indecomposable stable wedge summands are known. As a consequence, we find an information on the Cartan matrices of $F_p[GL_n(Z/p)]$ and $F_p[M_n(Z/p)]$. Moreover we point out the occurrence of some index-composable stable wedge summands of $B(Z/p)^n_+$ in the Campbell-Selick summands.