# Reference request: Reduced reflection length in Coxeter groups

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 $G = S_n$.

• Note that MathSciNet lists half a dozen or so articles relevant to "reflection length" (in an arbitrary Coxeter group your $T$ is the set of "reflections", not the set of "transpositions"). The earliest mention (involving finite Coxeter groups) is apparently this one: ams.org/mathscinet-getitem?mr=1712091, but other references including those of the paper you link to may be more relevant to your question. Or you may be on your own. – Jim Humphreys May 19 '17 at 13:14
• The problem is that the term I am looking for is "reduced reflection length" rather than "reflection length". The later is much more common and thus it is hard to find results for the first one. – Dirk May 22 '17 at 13:58
• Yes, I understand your concern about the added condition "reduced", but I wanted to emphasize the very limited literature out there even on "reflection length". I strongly suspect that the alternative mentioned at the end of my comment will prove to be the correct oner. Maybe it would be helpful to contact one of the authors of the papers mentioned? – Jim Humphreys May 22 '17 at 16:52
• That might be an idea, yes. I will try and in case that I find out something I will post it here for others to find, should they ever stumble upon this question. – Dirk May 23 '17 at 9:26
• For the record: we only show the relation between depth and reduced reflection length for the infinite families of finite Coxeter groups. As far as I can tell, figuring out if this relation holds for E8 with reasonable computational resources requires both mathematical insight and programming skill. – Alexander Woo Jun 13 '17 at 21:47

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):

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.

• Kyle Petersen and I exchanged a few emails on the joint distribution of length and depth (for permutations) several years ago. Send me an email and I'll try to dig them up for you. – Alexander Woo Jun 13 '17 at 22:02
• @AlexanderWoo Ok, I sent an email to your adress at the university of Idaho, I hope this is the correct one. Thanks for your help. – Dirk Jun 14 '17 at 8:19