Littelmann path model

Definition

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and simple roots $\alpha_1, \dots, \alpha_r$. Let $X \subset \mathfrak{h}^*$ be the weight lattice, let $X_\R$ be its real span, and let $\alpha^\vee$ be the coroot of a simple root $\alpha$. A path is a piecewise linear map $\pi \colon [0, 1] \to X_\R$ with $\pi(0) = 0$, taken up to reparametrisation. For a path $\pi$ and a simple root $\alpha$, put $h(t) = \ip{\pi(t)}{\alpha^\vee}$ and let $m$ be the minimum of $h$ on $[0, 1]$. The root operator $f_\alpha$ is defined as follows.

1. If $h(1) - m < 1$, then $f_\alpha \pi = 0$.
2. Otherwise let $t_0$ be the largest $t$ with $h(t) = m$, and $t_1$ the smallest $t \ge t_0$ with $h(t) = m + 1$.
3. Then $f_\alpha \pi(t) = \pi(t)$ for $t \le t_0$, $f_\alpha \pi(t) = \pi(t_0) + s_\alpha(\pi(t) - \pi(t_0))$ for $t_0 \le t \le t_1$, and $f_\alpha \pi(t) = \pi(t) - \alpha$ for $t \ge t_1$.

Here $s_\alpha(x) = x - \ip{x}{\alpha^\vee} \alpha$ is the simple reflection. For a dominant integral weight $\lambda$, let $\pi_\lambda(t) = t\lambda$. The path model of $\lambda$ is the set $B(\lambda)$ of nonzero paths $f_{\alpha_{i_1}} \cdots f_{\alpha_{i_k}} \pi_\lambda$, over all $k \ge 0$ and all sequences of simple roots.