EXPLICIT ŝln CRYSTAL MAPS BETWEEN CYLINDRIC PLANE PARTITIONS, MULTI-PARTITIONS AND MULTI-SEGMENTS

There are many combinatorial realizations of the crystals BΛ for the integral highest weight representations of b sln. One of the earliest and perhaps best known has as it’s underlying set certain `-tuples of partitions, where ` is the level of the representation. Another more recent model has as it’s underlying set certain cylindric plane partitions. Here we present a simple crystal isomorphism between these two realizations. We then consider the infinity crystal B∞. This has a combinatorial realization where the underlying set consists of acyclic multi-segments. We give a simple description of the of the ei-equivariant injection of BΛ into B∞ using the cylindric plane partition model. This has been done previously using `-tuples of partitions, but using cylindric plane partitions appears to simplify things. We also note that this map to aperiodic multi-segments extends to an ei-equivariant injection from all cylindric plane partitions into all (possibly reducible) multi-segments. As discussed in [9], this means one can think of the set of all multi-segments as the infinity crystal for b gln (as opposed to b sln).