If I understand your question correctly the chain complex for $X[m]$ has $C^i(X[m])=\oplus_\lambda C^i(X)\otimes U[i,m]_\lambda\otimes V_\lambda$ where $U[i,m]_\lambda$ has dimension $d[i,m]_\lambda$ which is the number of semistandard fillings of the shape $\lambda$ with $0, 1, 2, \ldots, i, \ldots i$ and $V_\lambda$ is an irreducible representation of $S_m$.
For some representations this is easy to compute. For instance if $\lambda_1=m-d-1$ then $d[d-1,m]_\lambda =0\not=d[d,m]_\lambda$ so $H^d_\lambda (X[m])=C^d_\lambda(X)\otimes U[d,m]_\lambda$ where $H^i(X[m])=\oplus_\lambda H^i_\lambda(X[m])\otimes V_\lambda$.