\(q,t\)-Catalan numbers and generators for the radical ideal defining the diagonal locus of \((\mathbb C^2)^n\)
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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 bi-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 number of partitions, 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 $M_{d_1, d_2}$.
@article{10_37236_645,
author = {Kyungyong Lee and Li Li},
title = {\(q,t\)-Catalan numbers and generators for the radical ideal defining the diagonal locus of \((\mathbb {C^2)^n\)}},
journal = {The electronic journal of combinatorics},
year = {2011},
volume = {18},
number = {1},
doi = {10.37236/645},
zbl = {1235.05013},
url = {http://geodesic.mathdoc.fr/articles/10.37236/645/}
}
TY - JOUR AU - Kyungyong Lee AU - Li Li TI - \(q,t\)-Catalan numbers and generators for the radical ideal defining the diagonal locus of \((\mathbb C^2)^n\) JO - The electronic journal of combinatorics PY - 2011 VL - 18 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.37236/645/ DO - 10.37236/645 ID - 10_37236_645 ER -
%0 Journal Article %A Kyungyong Lee %A Li Li %T \(q,t\)-Catalan numbers and generators for the radical ideal defining the diagonal locus of \((\mathbb C^2)^n\) %J The electronic journal of combinatorics %D 2011 %V 18 %N 1 %U http://geodesic.mathdoc.fr/articles/10.37236/645/ %R 10.37236/645 %F 10_37236_645
Kyungyong Lee; Li Li. \(q,t\)-Catalan numbers and generators for the radical ideal defining the diagonal locus of \((\mathbb C^2)^n\). The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/645
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