#####
Séminaire Lotharingien de Combinatoire, B73c (2015), 21 pp.

# John Claxton and Peter Tingley

# Young Tableaux, Multisegments, and PBW Bases

**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 multisegments and PBW bases seems to be new.

Received: April 17, 2015.
Revised: May 13, 2015.
Accepted: May 23, 2015.

The following versions are available: