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Journal of Algebraic Combinatorics

, Volume 44, Issue 2, pp 325–344 | Cite as

Weight posets associated with gradings of simple Lie algebras, Weyl groups, and arrangements of hyperplanes

  • Dmitri I. Panyushev
Article

Abstract

The set of weights of a finite-dimensional representation of a reductive Lie algebra has a natural poset structure (“weight poset”). Studying certain combinatorial problems related to antichains in weight posets, we realised that the best setting is provided by the representations associated with \({\mathbb {Z}}\)-gradings of simple Lie algebras (arXiv:1411.7683 [math.CO]). If \({\mathfrak {g}}\) is a simple Lie algebra, then a \({\mathbb {Z}}\)-grading of \({\mathfrak {g}}\) induces a \({\mathbb {Z}}\)-grading of the corresponding root system \(\varDelta \). In this article, we elaborate on a general theory of lower ideals (or antichains) in the corresponding weight posets \(\varDelta (1)\). In particular, we provide a bijection between the lower ideals in \(\varDelta (1)\) and certain elements of the Weyl group of \({\mathfrak {g}}\). An inspiring observation is that, to a great extent, the theory of lower ideals in \(\varDelta (1)\) is similar to the theory of upper (=ad-nilpotent) ideals in the whole poset of positive roots \(\varDelta ^+\).

Keywords

Root system Graded Lie algebra Lower ideal Coxeter arrangement 

Mathematics Subject Classification

06A07 17B20 20F55 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute for Information Transmission Problems of the R.A.S.MoscowRussia
  2. 2.Independent University of MoscowMoscowRussia

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