-nilpotent -ideals in having a fixed class of nilpotence: combinatorics and enumeration

Authors:
George E. Andrews, Christian Krattenthaler, Luigi Orsina and Paolo Papi

Journal:
Trans. Amer. Math. Soc. **354** (2002), 3835-3853

MSC (2000):
Primary 17B20; Secondary 05A15, 05A19, 05E15, 17B30

Published electronically:
June 10, 2002

MathSciNet review:
1926854

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the combinatorics of -nilpotent ideals of a Borel subalgebra of . 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 -nilpotent ideals and Dyck paths. Finally, we propose a -analogue of the Catalan number . These -Catalan numbers count, on the one hand, -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.

**1.**Paolo Papi,*Inversion tables and minimal left coset representatives for Weyl groups of classical type*, J. Pure Appl. Algebra**161**(2001), no. 1-2, 219–234. MR**1834087**, 10.1016/S0022-4049(00)00101-8**2.**Henrik Eriksson and Kimmo Eriksson,*Affine Weyl groups as infinite permutations*, Electron. J. Combin.**5**(1998), Research Paper 18, 32 pp. (electronic). MR**1611984****3.**P. Flajolet,*Combinatorial aspects of continued fractions*, Discrete Math.**32**(1980), no. 2, 125–161. MR**592851**, 10.1016/0012-365X(80)90050-3**4.**A. M. Garsia and M. Haiman,*A remarkable 𝑞,𝑡-Catalan sequence and 𝑞-Lagrange inversion*, J. Algebraic Combin.**5**(1996), no. 3, 191–244. MR**1394305**, 10.1023/A:1022476211638**5.**Victor G. Kac,*Infinite-dimensional Lie algebras*, 3rd ed., Cambridge University Press, Cambridge, 1990. MR**1104219****6.**Bertram Kostant,*Eigenvalues of the Laplacian and commutative Lie subalgebras*, Topology**3**(1965), no. suppl. 2, 147–159 (German). MR**0167567****7.**Bertram Kostant,*The set of abelian ideals of a Borel subalgebra, Cartan decompositions, and discrete series representations*, Internat. Math. Res. Notices**5**(1998), 225–252. MR**1616913**, 10.1155/S107379289800018X**8.**C. Krattenthaler, L. Orsina and P. Papi,*Enumeration of**-nilpotent**-ideals for simple Lie algebras*, Adv. Appl. Math. (to appear).**9.**George Lusztig,*Some examples of square integrable representations of semisimple 𝑝-adic groups*, Trans. Amer. Math. Soc.**277**(1983), no. 2, 623–653. MR**694380**, 10.1090/S0002-9947-1983-0694380-4**10.**Sri Gopal Mohanty,*Lattice path counting and applications*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1979. Probability and Mathematical Statistics. MR**554084****11.**Luigi Orsina and Paolo Papi,*Enumeration of ad-nilpotent ideals of a Borel subalgebra in type A by class of nilpotence*, C. R. Acad. Sci. Paris Sér. I Math.**330**(2000), no. 8, 651–655 (English, with English and French summaries). MR**1763905**, 10.1016/S0764-4442(00)00253-6**12.**Paolo Papi,*Inversion tables and minimal left coset representatives for Weyl groups of classical type*, J. Pure Appl. Algebra**161**(2001), no. 1-2, 219–234. MR**1834087**, 10.1016/S0022-4049(00)00101-8**13.**Jian-Yi Shi,*The number of ⊕-sign types*, Quart. J. Math. Oxford Ser. (2)**48**(1997), no. 189, 93–105. MR**1439701**, 10.1093/qmath/48.1.93**14.**Jian-yi Shi,*On two presentations of the affine Weyl groups of classical types*, J. Algebra**221**(1999), no. 1, 360–383. MR**1722917**, 10.1006/jabr.1999.8000**15.**Richard P. Stanley,*Enumerative combinatorics. Vol. 1*, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. MR**1442260**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
17B20,
05A15,
05A19,
05E15,
17B30

Retrieve articles in all journals with MSC (2000): 17B20, 05A15, 05A19, 05E15, 17B30

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:
http://dx.doi.org/10.1090/S0002-9947-02-03064-7

Keywords:
\emph{ad}-nilpotent ideal,
Lie algebra,
order ideal,
Dyck path,
Catalan number,
Chebyshev polynomial

Received by editor(s):
April 25, 2000

Published electronically:
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

Article copyright:
© Copyright 2002
American Mathematical Society