2008/03/03 by Céline Righi, Righi, Celine
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.0803.0267
For an ad-nilpotent ideal i of a Borel subalgebra of sll+1(C), we denote by Ii the maximal subset I of the set of simple roots such that i is an ad-nilpotent ideal of the standard parabolic subalgebra pI. 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 sll+1(C) and the set of Dyck paths of length 2l+2, to explicit a bijection between ad-nilpotent ideals i of the Borel subalgebra such that the cardinality of Ii is equal to r and the Dyck paths of length 2l+2 having r occurence "udu". We obtain also a duality between antichains of cardinality p and l-p in the set of positive roots.