Crystal bases for quantum affine algebras and combinatorics of Young walls

In this paper, we give a realization of crystal bases for quantum affine algebras using some new combinatorial objects which we call the Young walls. The Young walls consist of colored blocks with various shapes that are built on the given ground-state wall and can be viewed as generalizations of Young diagrams. The rules for building Young walls and the action of Kashiwara operators are given explicitly in terms of combinatorics of Young walls. The crystal graphs for basic representations are characterized as the set of all reduced proper Young walls. The characters of basic representations can be computed easily by counting the number of colored blocks that have been added to the ground-state wall

The violation of a uniqueness theorem and an invariant in the application of Poincar\'{e}--Perron theorem to Heun's equation

The domain of convergence of a Heun function obtained through the Poincar\'{e}--Perron (P--P) theorem is not absolute convergence but conditional one [2]. We show that a uniqueness theorem is not available if we apply the P--P theorem into the Heun's equation. We verify that the uniqueness theorem is only applicable when a local Heun function is absolutely convergent.Comment: 10 pages, 5 figures. Change the title and abstract. The number of pages has been reduced. Correct some of the problems with Englis