Let $(W,S)$ be a Coxeter group, $T=\bigcup\_{w\in W}wSw^{-1}$ its set of reflections, and $A\subseteq T$. From results of [Dyer](http://www.sciencedirect.com/science/article/pii/002186939090149I) and [Deodhar](http://link.springer.com/article/10.1007%2FBF01199813), we know that the subgroup $W\_A$ generated by the elements of $A$ is a Coxeter group, with a canonical system of generators $S\_A$.


In the final remarks to his note, Deodhar points out that it would be interesting to compare the Bruhat order on $(W\_A,S\_A)$ to the restriction of the Bruhat order of $(W,S)$ to $W\_A$, and that he plans to take up this project.


**My question is:** Has this been done? I would be happy with any references, but I am particularly interested in the following special case: Assume that $\beta\_1,\ldots, \beta\_k$ are positive roots in $W$'s root system. Is it possible to relate elements $w\in W$ satisfying $w\lessdot wr\_{\beta\_i}$ for all $i=1,\ldots ,k$, where $\lessdot$ denotes covering in the Bruhat order of $(W,S)$ and $r\_{\beta\_i}$ is the reflection across $\beta\_i$ to elements $\widetilde{w}\in W\_A$ with the same covering relations, but now in the Bruhat order of $(W\_A,S\_A)$?


**An example:** Consider the root system of $G\_2$ with $\alpha$ the short, and $\beta$ the long simple root. The only element of $W$ satisfying $w\lessdot wr\_\alpha, wr\_{3\alpha+\beta}$ is $w=r\_\alpha r\_\beta r\_\alpha r\_\beta$, so it is true that $wr\_\beta <w$, $wr\_{\alpha+\beta} <w$, $wr\_{3\alpha+2\beta}<w$, $wr\_{2\alpha+\beta}<w$. Now let $\alpha,\beta$ be positive roots forming a sub-root system of type $G\_2$ in some large root system. Do the four relations above still hold for the Bruhat order of the large group?



Bruhat order of reflection subgroups
The Bruhat graph of a reflection subgroup is an induced subgraph of the Bruhat graph of the larger group. This is proved by Dyer in "On the Bruhat graph of a Coxeter system." Thus if two elements that happen to be in the same reflection subgroup have a covering relation, then (the same) one also covers the other when you restrict to Bruhat order of the reflection subgroup. This actually generalizes the concept of permutation patterns, which was almost the topic of my dissertation (I did something else instead).


