Changes between Initial Version and Version 1 of Ticket #20477, comment 3
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 Apr 21, 2016, 1:57:46 PM (6 years ago)
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Ticket #20477, comment 3
initial v1 1 Just wondering: is that still the case for a complex reflection group W that, if you take a parabolic subgroup W_I and its coset representatives W^I the factorization of an element w=w_I w^Iis reduced?2 (that is len(w)=len(w_I)+len(w^I))?1 Just wondering: is that still the case for a complex reflection group W that, if you take a parabolic subgroup `W_I` and its coset representatives `W^I` the factorization of an element `w=w_I w^I` is reduced? 2 (that is `len(w)=len(w_I)+len(w^I)`)? 3 3 4 If yes, would there be a way to compute this decomposition? Then one could use induction to compute the reduced word for w_I and only do the depth first search on W^I?4 If yes, would there be a way to compute this decomposition? Then one could use induction to compute the reduced word for `w_I` and only do the depth first search on `W^I`? 5 5 6 I would assume that, if one chooses carefully the base for the permutation group (e.g. by starting with the simple roots s_i for i not in I, and then the others), then W_I would be one of the groups in the stabilizer chain. So expressing w in terms of the strong generators would give thedecomposition.6 I would assume that, if one chooses carefully the base for the permutation group (e.g. by starting with the simple roots `s_i` for i not in I, and then the others), then `W_I` would be one of the groups in the stabilizer chain. So expressing w in terms of the strong generators would give the desired decomposition. 7 7 8 It's probably best to take I a maximal parabolic subgroup. Then, W_I should be the first subgroup in the stabilizer chain, and the strong generators for this last inclusion W_I \subset W should be exactly the coset representatives. Andthe Shreier tree computed by GAP might actually just be the depth first search tree on then (I don't know if GAP uses a depth first search though).8 It's probably best to take a maximal parabolic subgroup `W_I` for `I={1,...,n1}`. Then, `W_I` should be the first subgroup in the stabilizer chain (fixing the first element of the base `alpha_n`), and the strong generators for this last inclusion W_I \subset W should be exactly the coset representatives. Even better, the Shreier tree computed by GAP might actually just be the depth first search tree on then (I don't know if GAP uses a depth first search though). 9 9 10 All of this to be taken with a grain of salt ...10 All of this to be taken with a big grain of salt ...