Faces of polyhedra associated with relation modules

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 certain tilings and matrices related to the corresponding graph.