Algebras and Representation Theory
, Volume 15, Issue 3, pp 593611
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 finitedimensional, 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 finitedimensional \(\mathfrak{g} \)–module is precisely the set of weights lying on a face of the convex hull.
Keywords
Weak \(\mathbb{F}\)face Positive weak \(\mathbb{F}\)face Generalized Verma module PolyhedronMathematics Subject Classifications (2010)
Primary 17B20 Secondary 17B10 Title
 Faces of Weight Polytopes and a Generalization of a Theorem of Vinberg
 Journal

Algebras and Representation Theory
Volume 15, Issue 3 , pp 593611
 Cover Date
 201206
 DOI
 10.1007/s1046801092613
 Print ISSN
 1386923X
 Online ISSN
 15729079
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Weak $\mathbb{F}$ face
 Positive weak $\mathbb{F}$ face
 Generalized Verma module
 Polyhedron
 Primary 17B20
 Secondary 17B10
 Authors

 Apoorva Khare ^{(1)}
 Tim Ridenour ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Yale University, New Haven, CT, USA
 2. Department of Mathematics, Northwestern University, Evanston, IL, USA