Faces of polyhedra associated with relation modules

TL;DR

This paper introduces polyhedra linked to relation Gelfand-Tsetlin modules, generalizing classical Gelfand-Tsetlin polytopes, and characterizes their faces using graph-related matrices.

Contribution

It constructs new polyhedra for relation modules and provides a face characterization in terms of associated graphs, extending previous Gelfand-Tsetlin polytope theory.

Findings

Polyhedra associated with relation modules are constructed.

Faces of these polyhedra are characterized via graph-related matrices.

Includes classical Gelfand-Tsetlin polytopes as special cases.

Abstract

Relation Gelfand-Tsetlin $\mathfrak{gl}_n$-modules were introduced in [FRZ19], and are determined by some special directed graphs and Gelfand-Tsetlin characters. In this work we constructed polyhedra associated with the class of relation modules, which includes as a particular case, any classical Gelfand-Tsetlin polytope. Following the ideas presented in [LM04] we give a characterization of $d$-faces of the associated polyhedra in terms of a matrix related to the corresponding graph.