A Preparation Theorem for the Kashiwara $$B(\infty )$$ B ( ∞ ) crystal

Abstract

The Kashiwara \(B(\infty )\) crystal pertains to a Verma module for a Kac–Moody algebra. It has a rich and important combinatorial structure. A Preparation Theorem is established as a preliminary step in an explicit description \(B(\infty )\). It is noted that \(B(\infty )\) 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 and to describe explicitly the linear inequalities involved. Here an approach to this question is initiated based on constructing dual Kashiwara functions. Our basic hypothesis is that dual Kashiwara functions exist and can be expressed as differences of successive Kashiwara functions with non-negative integer coefficients. This last requirement is called the positivity condition. Here one starts from an explicit linear function called the “initial driving function”. Then in terms of the reduced decomposition one seeks an algorithm to provide further dual Kashiwara functions which in particular will provide “driving functions” for the next induction step. The Preparation Theorem resolves a main difficulty in this step-wise construction, namely that the resulting functions must exhibit an invariance with respect to the Kashiwara operators. This is expressed through a sum, or simply S, condition. It depends very subtly on inequalities between the coefficients occurring in the driving function obtained from the previous step. The proof of the Preparation Theorem depends on the construction of S-graphs constructed by a process of “binary fusion” based on the structure of the driving function.