Trails for Minuscule Modules and Dual Kashiwara Functions for the $B(\infty )$ Crystal

Abstract

Let $B(\infty )$ be the Kashiwara crystal associated to a Kac-Moody algebra $\mathfrak g$. It may be presented (in generally infinitely many ways) as an infinite tuple $B_J(\infty )$ of non-negative integers. Any such presentation has been shown to be polyhedral in type A [O. Gleizer and A. Postnikov, Littlewood-Richardson coefficients via Yang-Baxter equation. Internat. Math. Res. Notices 2000, no. 14, 741–774] and in finite type [A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties. Invent. Math. 143 (2001), no. 1, 77–128]. The latter theory has the difficulty that it involves trails in the fundamental modules. These are not combinatorially defined and almost impossible to determine. A general method was introduced in [A. Joseph, Dual Kashiwara functions for the $B(\infty )$ crystal, Transf. Groups] involving notably a new combinatorial object - namely S-graphs. This theory has not yet been completed, a difficulty being in the understanding of relations in Demazure modules [A. Joseph, Trails S-graphs and identities in Demazure modules, arXiv: 1702.00243]. Nevertheless it is suggested there that trails admit an additional additive structure which can be used to compute them. Here this additive structure is illustrated in the relatively easy case when the fundamental module is minuscule. Proofs are detailed and mainly self-contained.