# Bruhat order of reflection subgroups

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 and Deodhar, 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?

• It would be worth asking Matthew Dyer at Notre Dame, since he is still active in this area. Besides proving (as did Deodhar) that a reflection subgroup of a Coxeter group is again a Coxeter group, he looked in an old paper at the Bruhat order as well: ams.org/mathscinet-getitem?mr=1104786 (This paper has been frequently cited, so probably some of its ideas have been followed up further; but I haven't kept track of all the research.) Apr 18, 2016 at 20:52

• Thank you! I am getting convinced that the answer should be yes, however, in my question, the element of the large group $W$ may not be in the reflection subgroup. Following Prof. Humphrey's suggestion, I have contacted prof Dyer, and he said that there is a similar statement for cosets of reflection groups, which may answer my question. I'll post it when I hear from him. Apr 23, 2016 at 16:33
• @Balazs the result is simple to describe. The Bruhat graph on $wW_A$, where $w$ is the minimal element of the coset in Bruhat order of $W$, is an induced subgraph of the Bruhat graph of $W$. Apr 23, 2016 at 16:49
• And there is an edge $ww'\to ww''$ for $w',w''\in W_A$ iff there is an edge $w'\to w''$. Apr 23, 2016 at 16:51