Correspondence between Young walls and Young tableaux and its application
For generic \(q\), the representation theory of \(U_q({\mathfrak g})\) is analogous of that of \(U({\mathfrak g})\). For integrable modules over \(U_q({\mathfrak g})\), the crystal bases introduced by \textit{M. Kashiwara} [Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009); Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005)] can be viewed as bases at \(q= 0\), and they have the structure of colored oriented graphs. For a classical Lie algebra \({\mathfrak g}\), \textit{M. Kashiwara} and \textit{T. Nakashima} gave an explict realization of crystal bases for finite-dimensional irreducbile modules as sets of semistandard Young tableaux [J. Algebra 165, No. 2, 295--345 (1994; Zbl 0808.17005)]. Crystal bases for basic representations (highest weight ones of level one) of quantum affine-algebras were given in terms of affine combinatorial objects called Young walls by \textit{J. Hong} and \textit{S.-J. Kang} [Introduction to quantum groups and crystal bases. Graduate Studies in Mathematics 42. Providence, RI: American Mathematical Society (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)]. Since any classical \({\mathfrak g}\) is embedded in an affine \(\widehat{\mathfrak g}\), Kang, Kim and the two authors of the paper under review gave a new realization of crystal bases for finite-dimensional \({\mathfrak g}\)-modules for a classical \({\mathfrak g}\), in terms of Young walls related to the connected component of the colored oriented graph for \(\widehat{\mathfrak g}\) [\textit{S.-J. Kang}, \textit{J.-A. Kim}, \textit{H. Lee} and \textit{D.-U. Shin}, Trans. Am. Math. Soc. 356, No. 6, 2349--2378 (2004; Zbl 1052.17004)]. In the paper under review, the authors use this to give a new, realization of crystal bases for finite-dimensional irreducible modules over classical \({\mathfrak g}\). The basis vector is parametrized by certain tableaux with given shape different from that of Kashiwara and Nakashima. They give a crystal isomorphism from their realization to that of Kashiwara and Nakashima. As an application, they give an insertion scheme for the reduced proper Young walls for \({\mathfrak g}\) of type \(A\) and \(C\).
scientific article; zbMATH DE number 2153382
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