Algebras and Representation Theory

, Volume 15, Issue 3, pp 593-611

First online:

Faces of Weight Polytopes and a Generalization of a Theorem of Vinberg

  • Apoorva KhareAffiliated withDepartment of Mathematics, Yale University Email author 
  • , Tim RidenourAffiliated withDepartment of Mathematics, Northwestern University

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The paper is motivated by the study of graded representations of Takiff algebras, cominuscule parabolics, and their generalizations. We study certain special subsets of the set of weights (and of their convex hull) of the generalized Verma modules (or GVM’s) of a semisimple Lie algebra \(\mathfrak{g}\). In particular, we extend a result of Vinberg and classify the faces of the convex hull of the weights of a GVM. When the GVM is finite-dimensional, we answer a natural question that arises out of Vinberg’s result: when are two faces the same? We also extend the notion of interiors and faces to an arbitrary subfield \(\mathbb{F}\) of the real numbers, and introduce the idea of a weak \(\mathbb{F}\)–face of any subset of Euclidean space. We classify the weak \(\mathbb{F}\)–faces of all lattice polytopes, as well as of the set of lattice points in them. We show that a weak \(\mathbb{F}\)–face of the weights of a finite-dimensional \(\mathfrak{g} \)–module is precisely the set of weights lying on a face of the convex hull.


Weak \(\mathbb{F}\)-face Positive weak \(\mathbb{F}\)-face Generalized Verma module Polyhedron

Mathematics Subject Classifications (2010)

Primary 17B20 Secondary 17B10