Young tableau descriptions for the polyhedral realizations of crystal bases in type $A_n$

By utilizing the combinatorial properties of various tableau models, we establish an explicit correspondence between the polyhedral realizations of the crystal bases $\mathcal{B}(\lambda )$ (resp. $\mathcal{B}(\mathrm{\infty })$) of type $A_n$ and the reverse semi-standard Young tableaux (resp. reverse marginally large tableaux), thereby providing a combinatorial description of the corresponding polyhedral realizations. Furthermore, a crystal structure on the set of Gelfand-Tsetlin patterns is obtained via the correspondence between the polyhedral realization of $\mathcal{B}(\lambda )$ and the reverse tableaux. As applications of our framework, we present concrete combinatorial realizations of the crystal embedding of $\mathcal{B}(\lambda )$ into $\mathcal{B}(\mathrm{\infty })$ and the set of Lusztig data.