2019/04/30 by Yuki Kanakubo, Toshiki Nakashima · 2 citations
Mathematics · #math.QA #math.CO #math.RT
published as Communications in Algebra, Volume 48, (2020), no. 11
arxiv created 2021/01/20 · arxiv updated 2021/01/21
The polyhedral realization of crystal base has been introduced by A.Zelevinsky and the second author([T.Nakashima, A.Zelevinsky, Adv. Math. 131, no. 1 (1997)]), which describe the crystal base B(∞) as a polyhedral convex cone in the infinite ℤ-lattice ℤ∞. To construct the polyhedral realization, we need to fix an infinite sequence ι from the indices of the simple roots. According to this ι, one has certain set of linear functions defining a polyhedral convex cone and under the `positivity condition' on ι, it has been shown that the polyhedral convex cone is isomorphic to the crystal base B(∞). To confirm the positivity condition for a given ι, we need to obtain the whole feature of the set of linear functions, which requires, in general, a bunch of explicit calculations. In this article, we introduce the notion of the adapted sequence and show that if ι is an adapted sequence then the positivity condition holds for classical Lie algebras. Furthermore, we reveal the explicit forms of the polyhedral realizations associated with arbitrary adapted sequences ι in terms of column tableaux.