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Bootstrapping the Kashiwara crystal

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Speaker: 
Anthony Joseph
Affiliation: 
Weizmann Institute
Date: 
Sat, 2018-11-24 09:00 - 10:00
Location: 
MPIM Lecture Hall

The Kashiwara $B(\infty)$ crystal gives a uniform combinatorial description of each integrable highest weight module $V$ for every Kac-Moody Lie algebra $g$ using only the Cartan matrix as data.  More than this it realizes Bott's dream (motivated by the Bott-Samelson resolution of the flag variety) of a multiplicity-free character formula. Just as in the Bott-Samelson resolution, one fixes a (family) of reduced decompositions $J$ and constructs a realization of $B(\infty)$ as a subset $B_J(\infty)$, of integer points in an affine space $B_J$. If the Weyl group $W$ of $g$ is finite, then after Berenstein and Zelevinsky, $B_J(\infty)$ is given by inequalities defined by linear functions constructed from "trails", themselves dependent on $J$, in the fundamental modules (of the Langlands dual).  Yet trails are not combinatorially defined and moreover the above result is not yet known to hold in general. A crystal is itself a union of "strings" for each simple root, which are linearly ordered.

Here it is proposed that the set of trails decomposes into $S$-sets for each simple root. This would both give a precise description of the set of all trails and extend the Berenstein-Zelevinsky result to all $g$. $S$-sets are now fully understood but instead of being linearly ordered, are described by the corners of a hypercube.  Again the existence of the required simultaneous decomposition into $S$-sets remains elusive.
When $J$ is "bipartite" one can expect that the set of trails identifies with the crystal $B_J(\varpi_t)$ of an appropriate fundamental module.
This is shown when $g$ is classical.  Thus $B_J(\infty)$ is described in terms of part of itself. This is the "bootstrapping" referred to in the title.
$S$-sets are now fully understood but instead of being linearly ordered, are described by the corners of a hypercube.  Again the existence of the required simultaneous decomposition into $S$-sets remains elusive.

%The relation of $S$-sets to trails comes about through the Chevalley-Serre relations in a Demazure module (as used to describe the global functions on a Schubert variety). In principle this allows one to prove independence of the decomposition on the simple root chosen.

When $J$ is "bipartite'' one can expect that the set of trails identifies with the crystal $B_J(\varpi_t)$ of an appropriate fundamental module.

This is shown when $\eufm g$ is classical.  Thus $B_J(\infty)$ is described in terms of part of itself. This is the "bootstrapping'' referred to in the title.
 

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