Commuting Schemes of Upper Triangular Matrices and Representation Homology

We establish a connection between generalised commuting schemes $C_g(U_n)$ of higher genus $g$, which are associated with a group scheme $U_n$ consisting of upper triangular unipotent matrices, and the representation homology $H{R}_{\ast }({\mathrm{\Sigma }}_g,U_n)$ of a Riemann surface ${\mathrm{\Sigma }}_g$ with coefficients in the group $U_n$. As an outcome, we provide a numerical criterion for determining whether commuting schemes $C(U_n)$ form a complete intersection.