Demazure crystals of type \(A_{n}\) and Young walls
A crystal basis is roughly a basis of the representation of a quantum group \(U_q({\mathfrak g})\) when \(q= 0\). It has a structure of a colored oriented graph, called the crystal graph. An explicit realization for finite-dimensional classical simple Lie algebras \({\mathfrak g}\) in terms of generalized Young tableaux was given by \textit{M. Kashiwara} and \textit{T. Nakashima} [J. Algebra 165, No. 2, 295--345 (1994; Zbl 0808.17005)]. The Demazure module is characterized by its highest weight and Weyl group element pair. \textit{M. Kashiwara} obtained crystal bases of Demazure modules as subsets of crystal bases for the corresponding highest weight modules [Duke Math. J. 71, No. 3, 839--858 (1993; Zbl 0794.17008)]. These sets are called Demazure crystals. Young walls were introduced by \textit{J. Hong} and \textit{S.-J. Kang} [Introduction to quantum groups and crystal bases (Grad. Stud. Math. 42, Providence, RI, AMS) (2002; Zbl 1134.17007)] and by \textit{S. J. Kang} [Proc. Lond. Math. Soc., III. Ser. 86, No. 1, 29--69 (2003; Zbl 1030.17013)]. They are generalizations of the Young diagrams, consisting of colored blocks of various shapes. A description of crystal bases for quantum classical simple Lie algebras was realized by \textit{S.-J. Kang}, \textit{J.-A. Kim}, \textit{D.-U. Shin} and the author of the paper under review using Young walls [Trans. Am. Math. Soc. 356, No. 6, 2349--2378 (2004; Zbl 1052.17004)]. In the paper under review, the author gives another realization of crystal bases for quantum classical simple Lie algebras which is different from the one of Kang, Kim, Shin and himself. His realization is chosen so that there is a natural correspondence between Young walls and semistandard tableaux. Using this, he gives a new realization of the Demazure crystal over \({\mathfrak {sl}}_{n+1}\). In fact, he characterizes the Demazure crystals of type \(A_n\) for any Weyl group element and for the dominant integral weight satisfying one of several conditions.
scientific article; zbMATH DE number 2153351
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