Trails, $S$-graphs and Identities in Demazure Modules

TL;DR

This paper explores the combinatorial and polyhedral structure of Demazure modules using $S$-graphs and trails, establishing connections between different approaches and providing explicit descriptions of crystal subsets.

Contribution

It introduces the concept of a giant $S$-graph and relates trail-based functions to dual Kashiwara functions, extending previous finite Weyl group results to general cases.

Findings

Defined giant $S$-graphs and $S$-sets for simple roots.

Connected trail functions with dual Kashiwara functions.

Provided conditions under which trails determine the $Z$-convex envelope.

Abstract

The Kashiwara crystal $B(\infty)$ 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 $B_J(\infty)$ of a crystal $B_J$ which as a set is just $J$ copies of the natural numbers. The goal is to determine $B_J(\infty)$ and in particular to show that it is a polyhedral subset of $B_J$. In earlier work this led to the notion of an $S$-graph associated to a given simple root $\alpha$. 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 $\alpha$. Its vertices, which forms a giant $S$-set, determine a set of dual Kashiwara functions. These are linear functions on $B_J$, whose common maximum determines the dual Kashiwara parameter with respect to $\alpha$. From these parameters one may compute $B_J(\infty)$ as an explicit polyhedral subset of $B_J$. 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.