Nested Hilbert Schemes and the nested $q,t$-Catalan series

Abstract

In this paper we study the tangent spaces of the smooth nested Hilbert scheme $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\mathbb{T}^2$-equivariant locally free sheaf on $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients. We call this conjecturally positive polynomial as the "nested $q,t$-Catalan series,'' for it has many conjectural properties similar to that of the $q,t$-Catalan series.

Keywords

• atiyah-bott lefschetz formula
• (nested) hilbert scheme of points
• tangent spaces
• diagonal coinvariants
• [math.math-co] mathematics [math]/combinatorics [math.co]
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]