Young tableaux, multi-segments, and PBW bases

TL;DR

This paper establishes explicit formulas connecting PBW bases and multisegments in the crystal theory of sl(n+1), providing a clear description of crystal operator actions and embedding of finite crystals into the infinity crystal.

Contribution

It offers a new, explicit description of the isomorphism between PBW bases and multisegments for a standard reduced expression, with simple formulas for crystal operators.

Findings

Explicit formulas for crystal operator actions on PBW bases.

A clear description of the embedding of finite crystals into the infinity crystal.

Proofs utilize Kashiwara's involution and characterization of the infinity crystal.

Abstract

The crystals for finite dimensional representations of sl(n+1) can be realized using Young tableaux. The infinity crystal on the other hand is naturally realized using multisegments, and there is a simple description of the embedding of each finite crystal into the infinity crystal in terms of these realizations. The infinity crystal is also parameterized by Lusztig's PBW basis with respect to any reduced expression for the longest word in the Weyl group. We give an explicit description of the unique crystal isomorphism from PBW bases to multisegments for one standard choice of reduced expression, thus obtaining simple formulas for the actions of all crystal operators on this PBW basis. Our proofs use the fact that the twists of the crystal operators by Kashiwara's involution also have simple descriptions in terms of multisegments, and a characterization of the infinity crystal due to Kashiwara and Saito. These results are to varying extents known to experts, but we do not think there is a self-contained exposition of this material in the literature, and our proof of the relationship between multi-segments and PBW bases seems to be new.