This applet implements the Littelmann path model for the root system A2. Let 1, 2, and 3, denote the usual generators for the weight lattice of SL_3, so 1 + 2 + 3 = 0. The roots of SL_3 are 1-2, 1-3, 2-3, 2-1, 3-1, and 3-2. If we choose positive roots 1-2, 1-3, 2-3, then the simple roots are 1-2 and 2-3.

The path model begins with a continuous path in the dominant cone which starts at the origin and ends at a dominant weight. For simplicity, we restrict ourselves to lattice paths in the dominant cone. We can represent such a path by a finite string of 1's followed by a finite string of 2's (or more generally, a finite string of 1's and 2's so that any set of 2's has at least as many 1's to the left).

We now describe the crystal lowering operators. Given a simple root r and a lattice path P as above, consider the set of minima of the path with respect to the hyperplane H orthogonal to the simple root. Find the time t of the last such minimum. If t = 1 (i.e. the last minimum is the final endpoint of the path), then the operator sends the path to the null path. If t < 1, then the operator reflects the path over the hyperplane H+P(t), but only the "step" of the path that starts at P(t).

In our situation, the operator corresponding to 1-2 (drawn in red) simply changes the appropriate 1 to 2 (in the spot corresponding to the last minimum of the path relative to 1-2). Similarly, the operator corresponding to 2-3 (drawn in blue) changes the appropriate 2 to 3 (in the spot corresponding to the last minimum of the path relative to 2-3).

Type in a finite string of 1's followed by a finite string of 2's (make sure there are less 2's than 1's) and hit Enter. If you identify all the vertices with the same label, you get the crystal graph.