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If I understand your question correctly $C_i(X[m])=\oplus_P C_i(X)\otimes U^{i,m}_P\otimes V_P$ where $U^{i,m}_P$ has dimension $d^{i,m}_P$ which is the number of semistandard fillings of the shape $P$ with $0, 1, 2, \ldots, i, \ldots i$ and $V_P$ is an irreducible representation of $S_m$.

For some representations this is easy to compute. For instance if $P_1=m-d-1$ then $d^{d-1,m}_P= 0$, but $d^{d,m}_P > 0$, so $H_{d,P}(X[m])=H_d(X)\otimes U^{d,m}_P\otimes V_P$.

If I understand your question correctly $C_i(X[m])=\oplus_P C_i(X)\otimes U^{i,m}_P\otimes V_P$ where $U^{i,m}_P$ has dimension $d^{i,m}_P$ which is the number of semistandard fillings of the shape $P$ with $0, 1, 2, \ldots, i, \ldots i$ and $V_P$ is an irreducible representation of $S_m$.

For some representations this is easy to compute. For instance if $P_1=m-d-1$ then $d^{d-1,m}_P= 0$, but $d^{d,m}_P > 0$, so $H_{d,P}(X[m])=C_d(X)\otimes U^{d,m}_P\otimes V_P$.

If I understand your question correctly $C_i(X[m])=\oplus_P C_i(X)\otimes U^{i,m}_P\otimes V_P$ where $U^{i,m}_P$ has dimension $d^{i,m}_P$ which is the number of semistandard fillings of the shape $P$ with $0, 1, 2, \ldots, i, \ldots i$ and $V_P$ is an irreducible representation of $S_m$.

For some representations this is easy to compute. For instance if $P_1=m-d-1$ then $d^{d-1,m}_P= 0$, but $d^{d,m}_P> 0$, so $H_{d,P}(X[m])=H_d(X)\otimes U^{d,m}_P\otimes V_P$.

If I understand your question correctly $C_i(X[m])=\oplus_P C_i(X)\otimes U^{i,m}_P\otimes V_P$ where $U^{i,m}_P$ has dimension $d^{i,m}_P$ which is the number of semistandard fillings of the shape $P$ with $0, 1, 2, \ldots, i, \ldots i$ and $V_P$ is an irreducible representation of $S_m$.

For some representations this is easy to compute. For instance if $P_1=m-d-1$ then $d^{d-1,m}_P= 0$, but $d^{d,m}_P > 0$, so $H_{d,P}(X[m])=H_d(X)\otimes U^{d,m}_P\otimes V_P$.

If I understand your question correctly the chain complex for $X[m]$ has i-chains the direct sum over partitions $P$ of (i-chains for $X$) tensor $U(i,m)(P)$ tensor $V(P)$ so that $C^i(X[m])=\oplus_P C^i(X)\otimes U^{i,m}_P\otimes V_P$ where $U(i,m)(P)$ has dimension $d(i,m)(P)$ which is the number of semistandard fillings of the shape $P$ with $0, 1, 2, \ldots, i, \ldots i$ and $V(P)$ is an irreducible representation of $S_m$.

For some representations this is easy to compute. For instance if $P(1)=m-d-1$ then $d(d-1,m)(P)$ is 0, but $d(d,m)(P)$ is nonzero so the $P$ representation in the top homology of $X[m]$ consists of (the d-chains of $X$) tensor $U(d,m)(P)$.

If I understand your question correctly $C_i(X[m])=\oplus_P C_i(X)\otimes U^{i,m}_P\otimes V_P$ where $U^{i,m}_P$ has dimension $d^{i,m}_P$ which is the number of semistandard fillings of the shape $P$ with $0, 1, 2, \ldots, i, \ldots i$ and $V_P$ is an irreducible representation of $S_m$.

For some representations this is easy to compute. For instance if $P_1=m-d-1$ then $d^{d-1,m}_P= 0$, but $d^{d,m}_P> 0$, so $H_{d,P}(X[m])=H_d(X)\otimes U^{d,m}_P\otimes V_P$.