Opposite convex order on the set of positive roots of a semisimple Lie algebra

Question

Let $\mathfrak{g}$ be a semisimple Lie algebra of rank l and let $\Delta^+$ be its set of positive roots. Denote by $s_1,...,s_l$ the simple generators of its Weyl group and let $w_0$ be the longest element of the Weyl group.

We have a bijective correspondence between convex orderings on $\Delta^+$ and reduced decompositions of $w_0.$

My question is: given a reduced decomposition $w_0=s_{i_i}\dots s_{i_n},$ with corresponding convex order $\alpha_{i_1}< \dots < \alpha_{i_n}$ on $\Delta^+$ what is the reduced decomposition that corresponds to the opposite convex order $\alpha_{i_n}< \dots < \alpha_{i_1}$?

Thank you so much in advance

Answer 1

I will reformulate everything in terms of inversions; replacing $\beta_i$ by the corresponding reflection $t_i$, then your sequence $t_1 < t_2 < \dots < t_n$ is just the set of left inversions of $N(w_0)$ (in fact, all the reflections) and the order is obtained from the word $s_{i_1} \cdots s_{i_n}$ as $$t_1= s_{i_1},\ t_2= s_{i_1} s_{i_2} s_{i_1}, \ \dots, \ t_n = s_{i_1} s_{i_2} \cdots s_{i_n} \cdots s_{i_2} s_{i_1}.$$ (you can do everything with right inversions and from the right if you prefer).

What I claim is that the reduced expression of $w_0$ you are looking for is given by $$\theta( s_{i_n}) \theta( s_{i_{n-1}} )\cdots \theta( s_{i_1}) ,$$ where $\theta$ is the automorphism of your Weyl group (preserving the simple reflections) defined as $\theta(s)=w_0 s w_0$. (in particular it is an automorphism of the Dynkin diagram, but it might be trivial even if the diagram has nontrivial automorphisms like in $D_4$).

Indeed, what we need to show is that $$s_{i_1} s_{i_2} \cdots s_{i_{n+1-j}} \cdots s_{i_2} s_{i_1} = \theta(s_{i_{n}}) \theta(s_{i_{n-1}}) \cdots \theta(s_{i_{n+1-j}}) \cdots \theta(s_{i_n}),$$ which is true since \begin{align}\theta(s_{i_{n}}) \theta(s_{i_{n-1}}) \cdots \theta(s_{i_{n+1-j}}) \cdots \theta(s_{i_n})&=\theta( s_{i_n} s_{i_{n-1}} \cdots s_{i_{n+1-j}} \cdots s_{i_{n-1}} s_{i_n})\\ &= w_0 s_{i_n} s_{i_{n-1}} \cdots s_{i_{n+1-j}} \cdots s_{i_{n-1}} s_{i_n} w_0\\ &= s_{i_1} \cdots s_{i_n} (s_{i_n} s_{i_{n-1}} \cdots s_{i_{n+1-j}} \cdots s_{i_{n-1}} s_{i_n}) s_{i_n} \cdots s_{i_1}\\ &=s_{i_1} s_{i_2} \cdots s_{i_{n+1-j}} \cdots s_{i_2} s_{i_1}. \end{align}

(it works for an abritrary finite Coxeter group, not only for Weyl groups)