The symplectic plactic monoid, crystals, and MV cycles

TL;DR

This paper explores the relationship between galleries in affine buildings, MV cycles, and crystal structures in type C, providing new insights into their parametrization and morphisms.

Contribution

It introduces readable galleries in affine buildings, establishes a crystal morphism to MV cycles, and analyzes fibers using the Littelmann path model.

Findings

Closure of readable galleries corresponds to MV cycles.

The map from galleries to MV cycles is a crystal morphism.

Fibers are characterized by the Littelmann path model.

Abstract

We study cells in generalised Bott-Samelson varieties for type C. These cells are parametrised by certain galleries in the affine building. We define a set of readable galleries - we show that the closure in the affine Grassmannian associated to a gallery in this set is an MV cycle. This then defines a map from the set of readable galeries to the set of MV cycles, which we show to be a morphism of crystals. We further compute the fibres of this map in terms of the Littelmann path model.