We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n + 1, 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 Cn. 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.

ad-nilpotent b-Ideals in sl(n) having a fixed class of nilpotence: Combinatorics and enumeration / G. E., Andrews; C., Krattenthaler; Orsina, Luigi; Papi, Paolo. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 354:10(2002), pp. 3835-3853. [10.1090/s0002-9947-02-03064-7]

ad-nilpotent b-Ideals in sl(n) having a fixed class of nilpotence: Combinatorics and enumeration

ORSINA, Luigi;PAPI, Paolo
2002

Abstract

We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n + 1, 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 Cn. 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.
ad-nilpotent ideal; catalan number; chebyshev polynomial; dyck path; lie algebra; order ideal
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ad-nilpotent b-Ideals in sl(n) having a fixed class of nilpotence: Combinatorics and enumeration / G. E., Andrews; C., Krattenthaler; Orsina, Luigi; Papi, Paolo. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 354:10(2002), pp. 3835-3853. [10.1090/s0002-9947-02-03064-7]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/47737
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