The commuting varieties associated with a group scheme are extensively explored in existing literature.
In this talk, I will introduce a derived version of a commuting variety - representation homology, and relate its computation with some specific geometric property of the classical commuting variety. In particular, using representation homology, I will present a theorem linking the vanishing result with the property being a complete intersection, which provides a numerical criterion for determining whether commuting varieties form a complete intersection, for groups consisting of upper triangular unipotent matrices.