Faces of weight polytopes and a generalization of a theorem of Vinberg

Details Faces of weight polytopes and a generalization of a theorem of Vinberg Faces of weight polytopes and a generalization of a theorem of Vinberg

2010-05-07

arxiv.org/abs/1005.1114v2

Mathematics

Representation Theory

Statement changed in Section 4. Typos fixed and some proofs updated. Submitted to "Algebra and Representation Theory." 18 page

Scientific paper

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 $\lie g$. 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 ask 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 $\F$ of the real numbers, and introduce the idea of a weak $\F$-face of any subset of Euclidean space. We classify the weak $\F$-faces of all lattice polytopes, as well as of the set of lattice points in them. We show that a weak $\F$-face of the weights of a finite-dimensional $\lie g$-module is precisely the set of weights lying on a face of the convex hull.

Affiliated with

Also associated with

No associations

Rating

Faces of weight polytopes and a generalization of a theorem of Vinberg does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Faces of weight polytopes and a generalization of a theorem of Vinberg, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Faces of weight polytopes and a generalization of a theorem of Vinberg will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-221974