Gelfand–Kirillov Dimensions and Associated Varieties of Highest Weight Modules
Zhanqiang Bai, Wei Xiao, Xun Xie
Abstract
Abstract In this paper, we present a uniform formula of Lusztig’s $ \textbf {a}$-functions on classical Weyl groups. Then we obtain an efficient algorithm for the Gelfand–Kirillov dimensions of simple highest weight modules of classical Lie algebras, whose highest weight is not necessarily regular or integral. To deal with type $ D $, we prove an interesting property about domino tableaux associated with Weyl group elements by introducing an invariant, called the hollow tableau. As an application, the associated varieties of all the simple highest weight Harish–Chandra modules are explicitly determined, including the exceptional cases.
Topics & Concepts
MathematicsInvariant (physics)DominoPure mathematicsWeyl groupSimple (philosophy)Type (biology)Lie algebraProperty (philosophy)Algebra over a fieldChemistryBiologyCatalysisMathematical physicsEpistemologyEcologyPhilosophyBiochemistryAdvanced Algebra and GeometryAlgebraic structures and combinatorial modelsAdvanced Combinatorial Mathematics