2020/05/31 by Yuki Kanakubo, Toshiki Nakashima
Mathematics · #math.QA #math.CO #math.RT
published as Journal of Algebra, Volume 574 (2021) · 38 pages. arXiv admin note: text overlap with arXiv:1904.10919
arxiv created 2021/10/27 · arxiv updated 2021/10/28
The polyhedral realizations for crystal bases of the integrable highest weight modules of Uq(\mathfrakg) have been introduced in ([T.Nakashima, J. Algebra, vol.219, no. 2, (1999)]), which describe the crystal bases as sets of lattice points in the infinite ℤ-lattice ℤ∞ given by some system of linear inequalities, where \mathfrakg is a symmetrizable Kac-Moody Lie algebra. To construct the polyhedral realization, we need to fix an infinite sequence ι from the indices of the simple roots. If the pair (ι,λ) (λ: a dominant integral weight) satisfies the `ample' condition then there are some procedure to calculate the sets of linear inequalities. In this article, we show that if ι is an adapted sequence (defined in our paper [Y.Kanakubo, T.Nakashima, arXiv:1904.10919]) then the pair (ι, λ) satisfies the ample condition for any dominant integral weight λ in the case \mathfrakg is a classical Lie algebra. Furthermore, we reveal the explicit forms of the polyhedral realizations of the crystal bases B(λ) associated with arbitrary adapted sequences ι in terms of column tableaux. As an application, we will give a combinatorial description of the function εi^* on the crystal base B(∞).