Suppose that an $A$-module homomorphism is given between two free and finite dimensional $A$ modules $M$ and $N$, with basis $e_1 \ldots e_n $ and $f_1 \ldots f_m$ respectively, by a matrix $T \in M_{m \times n, A}$.
I am wondering if there is a "classical" / concrete way to describe the matrix corresponding to $Sym^k T$, with the canonical basis $e_i \odot \ldots \odot e_j$, etc.
(The related situation I know is of course the determinant, and also the Kronecker product of matrices for the tensor product.)
What about for the alternating product? Edit: The alternating product is easy, it is just the minors of the transition matrices.