Realization of the crystal bases of level-zero representations of quantum affine algebras as algebraic cycles

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In various areas of (mathematical) physics, such as particle physics, string theory, and statistical mechanics, affine Lie algebras appear as a natural symmetry; a quantum affine algebra is introduced as a q-deformations (or a quantum deformation) of the universal enveloping algebra of an affine Lie algebra.The study of representations (i.e., linear actions on vector spaces) of quantum affine algebras are very useful in examining the states of particles or strings.The main result of our research is an explicit combinatorial description, in terms of convex polytopes, of the crystal bases of Verma modules (i.e., the most universal highest weight modules) for type A quantum affine algebras.