Mixed commuting varieties over nilpotent matrices (Q2154625)

Let \(k\) be an algebraically closed field and \(M_d(k)\) the space of \(d\times d\) matrices over \(k\). Commuting varieties are extensively studied in literature, while in this work, the authors generalize the so-called mixed commuting variety over some subvarieties \(S_1, \ldots , S_r\) of \(M_d(k)\) defined as follows, \[ \mathfrak{C}\left(S_{1} \times \cdots \times S_{r}\right)=\left\{\left(v_{1}, \ldots, v_{r}\right) \in S_{1} \times \cdots \times S_{r}: v_{i} v_{j}=v_{j} v_{i}\right\}. \] The authors study mixed commuting varieties over nilpotent \(3 \times 3\) matrices. Irreducible components of these mixed commuting varieties are explicitly described and their dimensions are calculated. This description is also used as an application to understand the structures of the module varieties of specific local algebras and the support varieties for modules over Frobenius kernels of \(\mathrm{SL}_3\).