$q,t$-Catalan numbers and generators for the radical ideal defining the diagonal locus of $(\C^2)^n$

TL;DR

This paper investigates the structure of generators for the ideal of alternating polynomials in two variable sets, providing bounds, explicit bases, and a linear map approach related to the $q,t$-Catalan numbers.

Contribution

It introduces simple upper bounds on the dimensions of graded components and explicitly constructs bases for certain bi-degrees, advancing understanding of the ideal's generators.

Findings

Derived bounds on the dimensions of graded components.

Identified all bi-degrees where the bounds are tight.

Constructed explicit bases for these bi-degrees.

Abstract

Let $I$ be the ideal generated by alternating polynomials in two sets of $n$ variables. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the graded vector space $M(=\bigoplus_{d_1,d_2}M_{d_1,d_2})$ spanned by a minimal set of generators for $I$. In this paper we give simple upper bounds on $\text{dim}M_{d_1, d_2}$ in terms of partition numbers, and find all bi-degrees $(d_1,d_2)$ such that $\dim M_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $M_{d_1, d_2}$. The main idea is to define and study a nontrivial linear map from $M$ to a polynomial ring $\C[\rho_1, \rho_2,...]$.