Quadratic ideals and Rogers-Ramanujan recursions

TL;DR

This paper provides a recursive framework for understanding quadratic ideals related to jet schemes of a double point, connecting these recursions to Rogers-Ramanujan identities and confirming a prior conjecture.

Contribution

It introduces an explicit recursive description of Hilbert series and Gr"obner bases for quadratic ideals, linking algebraic geometry with Rogers-Ramanujan identities.

Findings

Recursive formulas for Hilbert series and Gr"obner bases

Connection between quadratic ideals and Rogers-Ramanujan identities

Proof of a conjecture by Oblomkov and Rasmussen

Abstract

We give an explicit recursive description of the Hilbert series and Gr\"obner bases for the family of quadratic ideals defining the jet schemes of a double point. We relate these recursions to the Rogers-Ramanujan identity and prove a conjecture of the second author, Oblomkov and Rasmussen.