Faces of polyhedra associated with relation modules

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.

Keywords

tableaux realization, relation modules, Gelfand–Tsetlin polytopes

2010 Mathematics Subject Classification

Primary 17B10. Secondary 52B05.

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G.B. was partially supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brasil (CAPES), Finance Code 001. L.E.R. is supported in part by CNPq (316163/2021-0) and FAPESP (2018/17955-7).

Received 2 July 2021

Received revised 7 April 2022

Accepted 24 April 2022

Published 26 October 2022