Bounds of Gelfand-Tsetlin multiplicities and tableaux realizations of Verma modules

We introduce the notion of essential support of a simple Gelfand-Tsetlin gln-module as an attempt to understand the character formula of such module. This support detects the weights having maximal possible Gelfand-Tsetlin multiplicities. Using combinatorial tools we describe the essential supports of the simple socles of the universal tableaux modules. We also prove that every simple Verma module appears as the socle of a universal tableaux module. As a consequence, we prove the Strong Futorny-Ovsienko Conjecture on the sharpness of the upper bounds of the Gelfand-Tsetlin multiplicities. We also give a very explicit description of the support and essential support of the simple singular Verma module M(−ρ).Fil: Futorny, Vyacheslav. Universidade de Sao Paulo; BrasilFil: Grantcharov, Dimitar. University of Texas; Estados UnidosFil: Ramirez, Luis Enrique. Universidad Federal Do Abc; BrasilFil: Zadunaisky Bustillos, Pablo Mauricio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin