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Unshuffles are used in two places but we do not seem to have any description of them as such. (I do not know anything about them so ….)
Unshuffles are used in two places
Please don’t tell anyone which places! ;-)
Ah, I found one. At L-infinity-algebra. That’s just used synonymously there to shuffle. (I think some authors just use different words.) I have changed the wording there.
Where is the other one?
I deliberately did not tell people… there is a search facility that works very well;-)
I deliberately did not tell people…
I know, you like to pose little online detective problems to us.
There are still two entries with it in. (There was a second one on L-infinity algebra but it also occurs in A-infinity algebra. I fixed a typo whilst I was at it!
I think “unshuffle” is always synonymous to “shuffle”. I learned this terminology from Jim. I can check with him, he will recall the history, if any.
Here is what Jim says:
shuffles appear in making the Bar construction BA on a commutative algebra A into a Hopf algebra at least as far back as Seminaire Cartan 54-55 i.e. on the tensor COalgebra T^c(sA) althoug Cartan said they did not have coalgebras in those days
the ‘dual’ making the tensor algebra TV into a Hopf algebra uses what are clearly UNshuffles since a deck/string is split into two pieces keeping the relative order within the pieces
calling the latter shuffles is an abuse of language which has caught on - linguistic laziness?
That looks as if ’unshuffle’ may be what is needed BUT that a gloss needs to be put somewhere explaining the meaning. (Co-shuffle might be better.) I have some things to do before I can reword those two places– I think rewording rather than simply rolling back is needed. what do you think? I need also to think about exactly how to explain what is going on here.
It seems to me that what Jim says are “clearly UNshuffles” is precisely what our entry on shuffles calls shuffles. I still think there are just two different ways of thinking about one and the same mathematical concept in terms of decks of cards. No?
Tell me what you would want to reword how, so that I know what you are worried about.
Have a look at what I have rewritten on shuffle. I looked at one or two papers on L-infty algebras and extracted a definition. I think it is just a shuffle but viewed from the end result rather than from the input. I have no difficulty with this. I have reworked the earlier entries but have put that shuffles redirects unshuffles as it does not seem necessary to have a separate entry.
There seemed to be a typo in L-infinity algebra in the same part of the page. The formula above the mention of unshuffles is hard to get right. (I looked just now and there is still a glitch. Please check. I will do what I think is correct but I am not sure.
Edit: I think the formula is wrong. It should be i+j =n+1 for a start since otherwise the $l_i$ cannot handle the list it is to process. I checked in the old paper of Lada and Markl, and their definition is more like I suggest. Can someone debug or check this?
From homotopy coherence POV are not these instances of (un)shuffles ’really’ the products of simplices aspect once again. Whether a shuffle viewpoint or an un… viewpoint is then something like ’covariance’ or ’contravariance’ of the construction???
I like the improved wording.
One thing I wonder is whether the products of simplices bit should be thought of as an ’application’ as it is really what makes the rest ’tick’. This is not important however.
Have a look at what I have rewritten on shuffle.
You need to help me: I still don’t see a difference between the two notions. Your definition of unshuffle looks to me to be equivalent to the definition of shuffle. Up to a slight change of symbol use, it is almost verbatim the same!
It should be $i+j =n+1$
Yes, I’ve fixed it.
I think there is no difference between the two notions, except possibly a bottom-up as against a top down, and even that is unsure. I think Jim’s point is that in one it is a product and the other a coproduct that is the idea. Effectively the two notions coincide but in one you are taking things and building a product in the other you are showing how to decompose things. I am not sure what is best to do. Lada and Markl do use the term unshuffle.
I think for the plain notion of shuffle/unshuffle there is no difference in the definition and the page should not pretend there is one. There are different applications of the definition, though, that make some people say shuffle and some unshuffle for the same thing.
I have changed the bit on ’unshuffle’, but I am not 100% happy with the wording.
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