ad-nilpotent ideals containing a fixed number of simple root spaces

Details ad-nilpotent ideals containing a fixed number of simple root spaces ad-nilpotent ideals containing a fixed number of simple root spaces

2004-01-09

arxiv.org/abs/math/0401090v3

Rendiconti di Matematica, vol. 29, serie 7; p. 153-162, (2009)

Mathematics

Representation Theory

Latex, 9 page, revised and expanded version: a uniform formula for the statistics in the abelian case is provided

Scientific paper

We give formulas for the number of ad-nilpotent ideals of a Borel subalgebra of a Lie algebra of type B or D containing a fixed number of root spaces attached to simple roots. This result solves positively a conjecture of Panyushev (cf. D. Panyushev, ad-nilpotent ideals: generators and duality, J. of Alg., to appear) and affords a complete knowledge of the above statistics for any simple Lie algebra. We also study the restriction of the above statistics to the abelian ideals of a Borel subalgebra, obtaining uniform results for any simple Lie algebra.

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