Kyungyong Lee ; Li Li - On the diagonal ideal of $(\mathbb{C}^2)^n$ and $q,t$-Catalan numbers

dmtcs:2838 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) - https://doi.org/10.46298/dmtcs.2838
On the diagonal ideal of $(\mathbb{C}^2)^n$ and $q,t$-Catalan numbersArticle

Authors: Kyungyong Lee 1; Li Li ORCID2

  • 1 Department of mathematics Purdue University
  • 2 Department of Mathematics [Urbana]

Let $I_n$ be the (big) diagonal ideal of $(\mathbb{C}^2)^n$. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the graded vector space $M_n=\bigoplus_{d_1,d_2}(M_n)_{d_1,d_2}$ spanned by a minimal set of generators for $I_n$. We give simple upper bounds on $\textrm{dim} (M_n)_{d_1, d_2}$ in terms of partition numbers, and find all bi-degrees $(d_1,d_2)$ such that $\textrm{dim} (M_n)_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $(M_n)_{d_1, d_2}$.


Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: diagonal ideal,$q,t$-Catalan number,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Commutative Algebra of Alternating Polynomials; Funder: National Science Foundation; Code: 0901367

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