The crystal bases are powerful tools for studying the representation theory of Lie algebras and quantum groups. By realizing crystal bases as combinatorial objects, one can reveal skeleton structures of representations.

Nakashima and Zelevinsky invented `polyhedral realizations', which are realizations of crystal bases

as integer points in some polyhedral convex cones or polytopes. It is a natural problem to find an explicit form of inequalities that define the polyhedral convex cones and polytopes.

In this talk, we will explain an outline of representation theory of Lie algebras and quantum groups over the complex field and give an explicit form of inequalities in terms of extended Young diagrams in the case the associated Lie algebra is of affine type A^{(1)}_{n-1}.

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