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

Kyungyong Lee; Li Li

paper

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

arXiv:0909.1612

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 $M (= ⨁_{d_{1}, d_{2}} M_{d_{1}, d_{2}})$ spanned by a minimal set of generators for $I$ . In this paper we give simple upper bounds on $dim M_{d_{1}, d_{2}}$ 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 $\C[ρ_1, ρ_2,...]$.

29 pages

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

Abstract

References in corpus (2)

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