Representation variety of an associative algebra (${\mathrm{Rep}}_A(\mathbf{n})$)

For a finitely generated associative algebra, choose matrices for its generators and impose its polynomial relations. This defines an affine variety. With prescribed orthogonal idempotents $1={\sum }_i{e}_i$, require their images to be the standard projections on ${⨁}_i{k}^{n_i}$. The resulting variety is ${\mathrm{Rep}}_A(\mathbf{n})$, acted on by ${\prod }_i{\mathrm{GL}}_{n_i}(k)$. Without a chosen idempotent decomposition, the usual variety is ${\mathrm{Rep}}_A(r)={\mathrm{Hom}}_{k\text{-alg}}(A,M_r(k))$. Its orbits are module isomorphism classes, and their stabilizers are automorphism groups.