2017/02/01 by Anthony Joseph, Joseph, Anthony
Mathematics · #17B35 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1702.00243
openalex publication_date 2017/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Kashiwara crystal B(∞) parametrizes a basis for the Verma module of a Kac-Moody algebra. It has a deep combinatorial structure which one seeks to understand. For each sequence J of reduced decompositions of elements of the Weyl group W, it has a realization as a subset BJ(∞) of a crystal BJ which as a set is just J copies of the natural numbers. The goal is to determine BJ(∞) and in particular to show that it is a polyhedral subset of BJ. In earlier work this led to the notion of an S-graph associated to a given simple root α. Here the notion of a giant S-graph depending on a fixed simple root is introduced. It is essentially a union of S-graphs for each simple root with one distinguished vertex depending on α. Its vertices, which forms a giant S-set, determine a set of dual Kashiwara functions. These are linear functions on BJ, whose common maximum determines the dual Kashiwara parameter with respect to α. From these parameters one may compute BJ(∞) as an explicit polyhedral subset of BJ. For W finite, Berenstein and Zelevinsky had studied this problem by introducing the notion of a trail in a fundamental module. The functions they define may also be viewed as a set of dual Kashiwara functions. The goal is to relate these two approaches and without restriction on W. It is shown under the hypothesis that no "false" trails exist, that the set of trails determines the "Z-convex envelope" of a giant S-set. The proof involves the study of identities in Demazure modules.