Study on 1-dimensional sums and special polynomials in mathematical physics in terms of path models

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In the theory of Mirkovic-Vilonen (MV for short) polytopes, we obtain the following three results.(1) Let P and Q be MV polytopes. We have proved that the MV polytope corresponding to the tensor product of P and Q is contained in the Minkowski sum of P and Q.(2) We have described the crystal bases of Demazure modules and opposite Demazure modules in terms of MV polytopes.(3) We have extended the theory of MV polytopes to the case of the quantum group associated to the nontwisted affine Lie algebra of type A.