Solutions to the reflection equation: A bijection between lattice configurations and marked shifted tableaux

This thesis relates Young tableaux and marked shifted tableaux with non-intersecting lattice paths. These lattice paths are generated by certain exactly solvable statistical mechanics models, including the vicious and osculating walkers. These models arise from solutions to the Yang-Baxter and Reflection equations. The Yang-Baxter Equation is a consistency condition in integrable systems; the Reflection Equation is a generalisation of the Yang-Baxter equation to systems which have a boundary. We further establish a bijection between two types of marked shifted tableaux