ad-nilpotent \frak b-ideals in sl(n) having a fixed class of nilpotence: combinatorics and enumeration

2000/04/17 by George E. Andrews, Christian Krattenthaler, Luigi Orsina +1
Mathematics · #math.RA #math.CO #msc:17B20 #msc:05A15 #msc:05A19 #msc:05E15 #msc:17B30

paper · pdf

published as Trans. Amer. Math. Soc. 354 (2002), 3835-3853. · 19 pages, AmS-TeX

arxiv created 2000/04/17 · arxiv updated 2009/11/30

Abstract

We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n+1,\Bbb C). We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals and Dyck paths. Finally, we propose a (q,t)-analogue of the Catalan number C_n. These (q,t)-Catalan numbers count on the one hand ad-nilpotent ideals with respect to dimension and class of nilpotence, and on the other hand admit interpretations in terms of natural statistics on Dyck paths.

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