Standard parabolic subsets of highest weight modules

Abstract

Abstract. In this paper we study certain fundamental and distinguished subsets of weights of an arbitrary highest weight module over a complex semisimple Lie algebra. These sets wt J V λ are defined for each highest weight module V λ and each subset J of simple roots; we term them "standard parabolic subsets of weights". It is shown that for any highest weight module, the sets of simple roots whose corresponding standard parabolic subsets of weights are equal form intervals in the poset of subsets of the set of simple roots under containment. Moreover, we provide closed-form expressions for the maximum and minimum elements of the aforementioned intervals for all highest weight modules V λ over semisimple Lie algebras g. Surprisingly, these formulas only require the Dynkin diagram of g and the integrability data of V λ . As a consequence, we extend classical work by Satake, Borel-Tits, Vinberg, and Casselman, as well as recent variants by Cellini-Marietti to all highest weight modules.We further compute the dimension, stabilizer, and vertex set of standard parabolic faces of highest weight modules, and show that they are completely determined by the aforementioned closed-form expressions. We also compute the f -polynomial and a minimal half-space representation of the convex hull of the set of weights. These results were recently shown for the adjoint representation of a simple Lie algebra, but analogues remain unknown for any other finite-or infinite-dimensional highest weight module. Our analysis is uniform and type-free, across all semisimple Lie algebras and for arbitrary highest weight modules.

Faces of highest weight modules and the universal Weyl polyhedron

“…The classification of inclusion relations between faces, akin to the one in finite type [7,18], is our next main result. By Theorem 2.1, it suffices to study the fibers of the face map F V : W I V × 2 I → 2 wt V , defined as in (1.2).…”

“…The determination of the fibers for the restriction F V (1, −) is implicit in the above works of Vinberg and Casselman. A complete understanding of the fibers of F V (−, −) was achieved very recently by Cellini-Marietti [7] for the adjoint representation, and subsequently, by Khare [18] for all highest weight modules. While the fibers of the face map were understood for all highest weight modules V , it was not known whether all faces were of the above form.…”

“…For the inclusion ⊂, by assumption the simple coroots corresponding to J \ J min act by 0 on λ, and are disconnected from J min in the Dynkin diagram. In particular, [18], namely, that…”

“…As we have seen, J \ J min doesn't play a role in wt l . 1 Motivated by this, let: [7] and subsequent work by Khare [18], again in finite type. Finally, we comment on the relationship of the language in Theorems 5.1 and 5.7 to the two previous approaches: (i) that of Vinberg [26] and Khare [18], and (ii) that of Satake [25], Borel-Tits [3], Casselman [6], Cellini-Marietti [7], and Li-Cao-Li [21].…”

Coxeter submodular functions and deformations of Coxeter permutahedra

“…These polytopes are of fundamental importance in the theory of Lie algebras 2 . [18,27,28] A semisimple complex Lie algebra g has an associated root system Φ which controls its representation theory. The irreducible representations L(λ) of g are in bijection with the points λ ∈ D ∩ Λ, where D is the dominant chamber of the root system Φ and Λ is the weight lattice generated by the fundamental weights.…”