Reference request: Reduced reflection length in Coxeter groups

Question

I recently read this paper, where the authors define on page 26 what they call the reduced reflection length. For that we take a Coxeter group $G$ with Coxeter generators $S$ and transpositions $T$. The function $\ell$ gives the classical Coxeter length on $G$, that is $\ell(w)$ is the length of a shortest word in $S$ that yields $w$. The said reduced length is then defined as $$\ell_R(w) := \min \{r \mid w = t_1t_2\ldots t_r, \,\, t_i \in T, \,\, \ell(w) = \sum_{i=1}^r \ell(t_i)\}$$ and the authors show that the depth statistic discussed in the given paper satisfies $$dp = \frac{\ell + \ell_R}{2}$$ in many cases of interest.

I tried to find more information on $\ell_R$, but couldn't find the term in any other work. Has anyone already heard of the term and knows where to look?

What I am most interested in is the joint distribution of length and reduced length or length and depth (by the above formula, both problems are equivalent), that is the polynomials $$\sum_{g \in G} x^{\ell(g)}y^{\ell_R(g)} \text{ and } \sum_{g \in G} x^{\ell(g)}y^{dp(g)}$$ where we assume $G$ to be finite. I also couldn't find any paper discussing the length with either one of these functions together. While I would love to find answers for all finite Coxeter groups, it would already be great to get information on the special case $G = S_n$.

Answer 1

After contacting the authors of the paper mentioned above, I want to share the information they gave me with anyone who might ever stumble upon this question. For that, define an ordering on $G$ by saying $g \leq h$ if and only if there are reflections $t_1,t_2,\ldots, t_k$ such that

1. $g = h t_1t_2\ldots, t_k.$
2. $\ell(g) = \ell(h) + \sum_{i=1}^k \ell(t_i).$

Then the reduced reflection length is the rank function for this order. Now this order appears in some works under different names (some only defined for symmetric groups):

• The Grassmannian order in Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Nantel Bergeron, Frank Sottile, 1998.
• The $T_G$ order in Partial orders generalizing the weak order on Coxeter groups, Curtis D Bennetta, Rieuwert J Blok, 2003.
• The BeSo order in A simple definition for the universal Grassmannian order, Curtis D. Bennetta, Lakshmi Evanib, David Grabinerc, 2003

Note that I don't claim this list to be complete.

If anyone happens to find this question and is interested in discussing the problems I mentioned above or related topics, feel free to leave me a message.