Crystal Bases as Tuples of Integer Sequences

We describe a set consisting of tuples of integer sequences and provide certain explicit maps on it. We show that this defines a semiregular crystal for and , respectively. Furthermore, we define for any dominant integral weight a connected subcrystal in , such that this crystal is isomorphic to the crystal graph . Finally, we provide an explicit description of these connected crystals . 1. Introduction Let be a symmetrizable Kac-Moody algebra and let be the corresponding quantum algebra. For these quantum algebras, Kashiwara developed the crystal bases theory for integrable modules in [1] and thus provided a remarkable combinatorial tool for studying these modules. In particular crystal bases can be viewed as bases at and they contain structures of edge-colored oriented graphs satisfying a set of axioms, called the crystal graphs. These crystal graphs have certain nice properties; for instance, characters of -modules can be computed and the decomposition of tensor products of modules into irreducible ones can also be determined from the crystal graphs, to name but a few. It is thus an important problem to have explicit realizations of crystal graphs. There are many such realizations, combinatorial and geometrical, worked out during the last decades; for instance, we refer to [2–5]. In [2], the authors give a tableaux realization of crystal graphs for irreducible modules over the quantum algebra for all classical Lie algebras, which is a purely combinatorial model. Another significant combinatorial model for any symmetrizable Kac-Moody algebra is provided in [3], called Littelmann's path model. The underlying set here is a set of piecewise linear maps, and the crystal graph of an irreducible module of any dominant integral highest weight can be generated by an algorithm using the straight path connecting 0 and . A geometrical realization of crystals is also known and is provided by Nakajima [5] by showing that there exists a crystal structure on the set of irreducible components of a lagrangian subvariety of the quiver variety . This realization can be translated into a purely combinatorial model, the set of Nakajima monomials, where the action of the Kashiwara operators can be understood as a multiplication with monomials. Moreover, it is shown in [6] that the connected component of any highest weight monomial of highest weight is isomorphic to the crystal graph obtained from Kashiwara's crystal bases theory. For special highest weight monomials these connected components are explicitly characterized for in [7] and for the other classical Lie algebras