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 $H^i(X[m])=\oplus_\lambda H^i_\lambda(X^[m])\otimes V_\lambda$.