# $q,t$-Catalan numbers and generators for the radical ideal defining the diagonal locus of $C^2^n$ - Mathematics > Combinatorics

$q,t$-Catalan numbers and generators for the radical ideal defining the diagonal locus of $C^2^n$ - Mathematics > Combinatorics - Download this document for free, or read online. Document in PDF available to download.

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 ofthe graded vector space $M=\bigoplus {d 1,d 2}M {d 1,d 2}$ spanned by aminimal 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 suchbi-degrees, we also find explicit bases for $M {d 1, d 2}$. The main idea is todefine and study a nontrivial linear map from $M$ to a polynomial ring$\C ho 1, ho 2,

.$.

Author: ** Kyungyong Lee, Li Li**

Source: https://arxiv.org/