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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?

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    $\begingroup$ 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.) $\endgroup$ Apr 18 '16 at 20:52
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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).

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  • $\begingroup$ 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. $\endgroup$ Apr 23 '16 at 16:33
  • $\begingroup$ @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 $W$. $\endgroup$ Apr 23 '16 at 16:49
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    $\begingroup$ And there is an edge $ww'\to ww''$ for $w',w''\in W_A$ iff there is an edge $w'\to w''$. $\endgroup$ Apr 23 '16 at 16:51

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