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

Author(s): George E. Andrews; Christian Krattenthaler; Luigi Orsina; Paolo Papi
Journal: Trans. Amer. Math. Soc. 354 (2002), 3835-3853.
MSC (2000): Primary 17B20; Secondary 05A15, 05A19, 05E15, 17B30
Posted: June 10, 2002
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Abstract: We study the combinatorics of $ad$-nilpotent ideals of a Borel subalgebra of $sl(n+1,\mathbb C)$. 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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Additional Information:

George E. Andrews
Affiliation: Department of Mathematics, The Pennsylvania State University, 215 McAllister Building, University Park, Pennsylvania 16802
Email: andrews@math.psu.edu

Christian Krattenthaler
Affiliation: Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
Email: KRATT@Ap.Univie.Ac.At

Luigi Orsina
Affiliation: Dipartimento di Matematica, Istituto G. Castelnuovo, Università di Roma ``La Sapienza", Piazzale Aldo Moro 5, 00185 Roma, Italy
Email: orsina@mat.uniroma1.it

Paolo Papi
Affiliation: Dipartimento di Matematica, Istituto G. Castelnuovo, Università di Roma ``La Sapienza", Piazzale Aldo Moro 5, 00185 Rome, Italy
Email: papi@mat.uniroma1.it

DOI: 10.1090/S0002-9947-02-03064-7
PII: S 0002-9947(02)03064-7
Keywords: \emph{ad}-nilpotent ideal, Lie algebra, order ideal, Dyck path, Catalan number, Chebyshev polynomial
Received by editor(s): April 25, 2000
Posted: June 10, 2002
Additional Notes: The first author was partially supported by National Science Foundation Grant DMS 9870060.
The second author was partially supported by the Austrian Science Foundation FWF, grant P13190-MAT
The fourth author's research was partially supported by EC's IHRP Programme, grant HPRN-CT-2001-00272
Copyright of article: Copyright 2002, American Mathematical Society

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