In his paper "Paths and Root Operators in Representation Theory," Littelmann gives an action of the Weyl group on the set of integral paths via $$ \tilde{s}_\alpha(\pi):= \begin{cases} f^n_\alpha(\pi), \ n:=\langle \pi(1) \alpha^\vee \rangle \geq 0 \\ e^{-n}_\alpha(\pi), \ n:= \langle \pi(1), \alpha^\vee \rangle < 0 \end{cases} $$

For a straight line path $\pi_\lambda(t)=t\lambda$, it is clear that for any $w \in W$ we have $w(\pi_\lambda)=\pi_{w\lambda}$.

Question: Is there a clear picture of how $W$ acts on more general paths? For example, what if we take a "dog leg" path $$ \pi=\pi_\lambda \ast \pi_\mu, $$ the concatenation of two straight line paths?

In his paper "Paths and Root Operators in Representation Theory," Littelmann gives an action of the Weyl group on the set of integral paths via $$ \tilde{s}_\alpha(\pi):= \begin{cases} f^n_\alpha(\pi), \ n:=\langle \pi(1) \alpha^\vee \rangle \geq 0 \\ e^{-n}_\alpha(\pi), \ n:= \langle \pi(1), \alpha^\vee \rangle < 0 \end{cases} $$

For a straight line path $\pi_\lambda(t)=t\lambda$, it is clear that for any $w \in W$ we have $w(\pi_\lambda)=\pi_{w\lambda}$.

Is there a clear picture of how $W$ acts on more general paths? For example, what if we take a "dog leg" path $$ \pi=\pi_\lambda \ast \pi_\mu, $$ the concatenation of two straight line paths?