Let $W$ be a finite reflection group with length function $l$ and let $I$ be a set of simple reflections that generate $W$. Let $\phi$ be an automorphism of $W$ permuting $I$. Consider the orbits of $\phi$ on the set $I$. For each orbit $J$ consider the longest element $s_J$ of the parabolic subgroup $W_J$. Let $W_\phi$ be the subgroup of $W$ generated by the elements $s_J$.

My question is the following: is there a distingushed coset representatives of the group $W_\phi$?.

  • $\begingroup$ Yes, there is a typo in the title. OP should correct 'cosed' to 'coset'. But the question text itself is formulated clearly and definitely shows the author knows what he is talking about. I'd like to the question to be active, and vote to keep it. $\endgroup$ – P Vanchinathan Dec 11 '13 at 0:05
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    $\begingroup$ In this set-up, what is the definition of "distinguished"? (Maybe a small example would help to clarify what the question is.) $\endgroup$ – Jim Humphreys Dec 11 '13 at 1:34
  • $\begingroup$ I guess the intended word was 'distinct' and not 'distinguished'. $\endgroup$ – P Vanchinathan Dec 11 '13 at 2:30
  • $\begingroup$ "Unique shortest" is the usual way to distinguish a coset representative, in similar contexts. $\endgroup$ – Allen Knutson Dec 11 '13 at 6:07
  • $\begingroup$ @Jim Distingushed means minimal coset representatives of $W_\phi$. Maybe this is a better question: is $W_\phi$ a parabolic subgroup of $W$? $\endgroup$ – user35957 Dec 11 '13 at 8:05

The comments suggest that your notational choices may be obscuring the question, which I and others have found difficult to untangle. If stated more precisely, the question might answer itself.

There is some variation of notation and terminology in the liteature, since twisted groups of Lie type are treated a little differently in each of Carter's 1972, 1985 books as well as in Steinberg's work, etc. The main point here seems to be that for a given Chevalley group with Weyl group $W$, you may have a subgroup $W^F$ of $W$ consisting of fixed points under a Frobenius-type map involving a symmetry of the Coxeter graph as in types $A_n$ for $n \geq 2$, $D_n, E_6$. (There are more elaborate versions leading to the Suzuki and Ree groups.) Here $W^F$ is again a finite Coxeter group, though possibly not crystallographic. So the basic theory of reflection groups applies to $W^F$ and its parabolic subgroups. But this doesn't involve finding coxet representatives for $W^F$ in $W$. To clarify such matters it would help to focus more on specific cases, which at some point you have to do anyway in the study of these finite groups.

[ADDED] It's worth emphasizing that the Coxeter generators of $W^F$ need not be reflections in $W$ (though they are involutions), so $W^F$ won't be a "reflection subgroup" of $W$. In particular, the length functions of these two Coxeter groups aren't directly comparable.

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