Summary: Let \(\mathfrak g\) be a simple algebra. Let \(r\) be the number of its positive roots. The Kashiwara \(B(\infty)\) crystal is an important combinatorial object that describes a crystal basis for the simple finite dimensional \(\mathfrak g\) modules. After [A. Berenstein and A. Zelevinsky, Invent. Math. 143, No. 1, 77–128 (2001; Zbl 1061.17006)], it is a polyhedral subset of the integer points of an \(r\)-dimensional affine space, and moreover, the linear functions describing this polyhedral subset are given by “trails” in the fundamental modules of lowest weight of the Langlands dual. In general, trails depend on a reduced decomposition of the longest element of the Weyl group and are very difficult to compute. In the present work, it is shown for \(\mathfrak g\) classical, when the reduced decomposition is given by a power of a suitably chosen Coxeter element, that the set of trails is in natural bijection with the crystal of a suitable highest weight fundamental module.
The proofs are based on a specially developed theory of \(S\)-sets and do not require any of the results in Berenstein-Zelevinsky (loc cit).
For the entire collection see [Zbl 1481.17001].