I am trying to understand, explicitly, the commutation relation between $X_\vartheta$ and $Y_\vartheta = T_0T_{s_\vartheta}$ in the (twisted) DAHA for a root system $R$, where $\vartheta$ is the highest short root for $R$. Moreover, I am trying to express this commutation relation uniformly for any $R$.
For those unfamiliar with DAHA:
Is there some canonical (reduced) expression (in terms of simple reflections) for the reflection associated to the highest (short) root?
For example, write the highest (short) root as: $$\vartheta = \displaystyle\sum_{i=1}^n m_i\alpha_i$$ for simple roots $\{\alpha_i\}$. Then can one, generally, write a reduced expression for $s_{\vartheta}$ in terms of $s_i$ and $m_i$? An algorithm for writing such an expression will also suffice.
Also useful: how many occurences of $s_i$ such that $(\vartheta,\alpha_i)\neq 0$ are in a reduced expression for $s_\vartheta$?
I am mostly interested in the case of simply-laced $R$, but a general answer is better.
