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