Tableau models for semi-infinite Bruhat order and level-zero representations of quantum affine algebras.

We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application, we give new tableau models for the crystal bases of a level-zero fundamental representation and a level-zero extremal weight module over a quantum affine algebra of classical untwisted type, which we call quantum Kashiwara-Nakashima columns and semi-infinite Kashiwara-Nakashima tableaux. We give an explicit description of the crystal isomorphisms among three different realizations of the crystal basis of a level-zero fundamental representation by quantum Lakshmibai-Seshadri paths, quantum Kashiwara-Nakashima columns, and (ordinary) Kashiwara-Nakashima columns.