A preparation theorem for the kashiwara $b(\infty)$ crystal

TL;DR

This paper introduces a new approach to proving the polyhedral structure of the Kashiwara $B()$ crystal by constructing dual Kashiwara functions and addressing key obstacles in the step-wise construction process.

Contribution

It develops a novel method using dual Kashiwara functions to establish the polyhedral nature of $B()$, advancing understanding of its combinatorial structure.

Findings

Resolved one major obstacle in the step-wise construction of the crystal

Demonstrated linearity of dual Kashiwara functions in entries

Connected the structure to Catalan numbers in combinatorial enumeration

Abstract

The Kashiwara $B(\infty)$ crystal pertains to a Verma module for a Kac- Moody Lie algebra. Ostensibly it provides only a parametrisation of the global/canonical basis for the latter. Yet it is much more having a rich combinatorial structure from which one may read of a parametrisation of the corresponding basis for any integrable highest weight module, describe the decomposition of the tensor products of highest weight modules, the Demazure submodules of integrable highest weight modules and Demazure flags for translates of Demazure modules. $B(\infty)$ has in general infinitely many presentations as subsets of countably many copies of the natural numbers each given by successive reduced decompositions of Weyl group elements. In each presentation there is an action of Kashiwara operators determined by Kashiwara functions. These functions are linear in the entries. Thus a natural question is to show that in each presentation the subset $B(\infty)$ is polyhedral. Here a new approach to this question is initiated based on constructing dual Kashiwara functions and in this it is enough to show that the latter are also linear in the entries. The present work resolves one of the two very difficult obstacles in a step-wise construction, namely that the resulting functions must satisfy a sum, or simply S, condition. It depends very subtly on inequalities between the coefficients occurring in functions obtained from the previous step. The only remaining obstacle, that sufficiently many functions are obtained, can at least be verified in many families of cases, though this is to be postponed to a subsequent paper. This theory has some intriguing numerology which involves the Catalan numbers in two different ways.