The Yangian Scaffold

Hyperquot schemes parameterize sequences of quotient sheaves on a curve — cascading filtrations of a vector bundle. Their cohomology rings are large, complicated, and hard to describe by generators and relations alone.

Kaushik shows that a shifted Yangian algebra acts on this cohomology, producing a natural basis. The algebra — a quantum group associated to type A Lie algebras — doesn’t emerge from the geometry by accident. It’s built through “skew-nested Quot schemes,” new correspondence varieties that generalize the skew-nested Hilbert schemes of refined Donaldson-Thomas theory.

The through-claim: the cohomology of these moduli spaces is not just a vector space to be computed — it is a representation of a known algebra, and the algebraic structure organizes the geometry. The basis elements aren’t chosen; they’re forced by the Yangian action.

The extension from rank 1 (where Hilbert schemes suffice) to higher rank (where Hyperquot schemes live) requires iterated commutator constructions — building the algebra’s generators layer by layer. Each layer corresponds to a step in the quotient filtration. The algebraic depth mirrors the geometric depth.

When a representation theory exists for a geometric object, it means the geometry is more structured than it appears. The Yangian is the scaffold that was always there.