Number of ``\(udu\)s of a Dyck path and \(ad\)-nilpotent ideals of parabolic subalgebras of \({\mathrm{sl}}_{\ell+1}(C)\)

zbMATH Open1179.17012arXiv0803.0267MaRDI QIDQ2654620

Publication date: 19 January 2010

Published in: Séminaire Lotharingien de Combinatoire (Search for Journal in Brave)

Abstract: For an ad-nilpotent ideal

i

of a Borel subalgebra of

s l_{l + 1} (C)

, we denote by

I_{i}

the maximal subset

I

of the set of simple roots such that

i

is an ad-nilpotent ideal of the standard parabolic subalgebra

p_{I}

. We use the bijection given by G.E. Andrews, C. Krattenthaler, L. Orsina and P. Papi between the set of ad-nilpotent ideals of a Borel subalgebra in

s l_{l + 1} (C)

and the set of Dyck paths of length

2 l + 2

, to explicit a bijection between ad-nilpotent ideals

i

of the Borel subalgebra such that the cardinality of

I_{i}

is equal to

r

and the Dyck paths of length

2 l + 2

having

r

occurence "udu". We obtain also a duality between antichains of cardinality

p

and

l - p

in the set of positive roots.

Full work available at URL: https://arxiv.org/abs/0803.0267

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Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) Simple, semisimple, reductive (super)algebras (17B20)

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