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Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
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Effect on homology of decorating vertices of a simplicial complex
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 $ …