Timeline for $S_n$ action on the sequences of transpositions
Current License: CC BY-SA 3.0
9 events
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| Dec 24, 2016 at 10:14 | comment | added | user79456 | @darijgrinberg Returning again to the "local straightening rules". The most convenient description of the action is the following. Conjugate a strictly monotone factorization termwise by a chosen element of $S_n$. The resulting factorization, in general, fails to be strictly monotone. But its orbit with respect to the action of the braid group on the set of factorization contains a unique strictly monotone representative. | |
| Dec 23, 2016 at 10:28 | history | edited | Fedor Petrov | CC BY-SA 3.0 |
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| Dec 23, 2016 at 9:25 | comment | added | user79456 | @darijgrinberg For the second one -- conjugate a sequence termwise by a permutation. The monotonicity condition generally fails, but there exists a set of "local straightening rules" admitting a description in terms of the braid group action on the set of sequences of permutations. The question was motivated by Hurwitz numbers computation problem, which can be stated as a count of numbers of sequences of transpositions, without any additional restriction. The $S_n$-action on Hurwitz factorizations is obvious, so it is natural to ask, is there a natural action in the monotone case as well. | |
| Dec 23, 2016 at 9:17 | comment | added | user79456 | @darijgrinberg For your first comment -- the change of monotonicity direction really changes nothing, there is a natural 1-1 correspondence between monotonically increasing and monotonically decreasing sequences, which corresponds to reversing the order of elements in the sequence. | |
| Dec 23, 2016 at 7:47 | comment | added | darij grinberg | When you say "This action can be completely described in the terms of stictly monotone factorizations only", do you have a specific description in mind? I see how to describe the action of a simple transposition $s_k$: Namely, it replaces all $k$'s in the transpositions by $k+1$'s and vice versa. Then, it checks if the result is still a monotone factorization. There are three possible cases in which it isn't, and they can be dealt with separately (by "local straightening rules"). But this doesn't sound like a very useful description :( | |
| Dec 23, 2016 at 7:30 | comment | added | darij grinberg | Barring that, what if we replace $b_i \leq b_{i+1}$ by $b_i \geq b_{i+1}$ ? | |
| Dec 23, 2016 at 6:37 | history | edited | user79456 |
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| Dec 23, 2016 at 4:26 | history | edited | YCor |
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| Dec 22, 2016 at 20:20 | history | asked | user79456 | CC BY-SA 3.0 |