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Let $\Sigma_n\subset G$ be a set of generators of the symmetric group $S_n$. It is a well-known conjecture that the diameter of the Cayley graph $\Gamma(S_n,\Sigma_n)$ is at most $n^C$ for some absolute constant $C$. (The diameter of the Cayley graph is just the maximum of $\ell(g)$ for $g\in S_n$, where $\ell(g)$ is the length of the shortest word on $A \cup A^{-1}$ equal to $g$.)

For $\Sigma_n$ of bounded size, the diameter cannot be less than a constant times $\log |S_n|$, i.e., a constant times $n\log n$.

It is clear and well-known that, for $\Sigma_n = \{(1 2),(1 2 \dotsb n)\}$, the diameter of $\Gamma(S_n, \Sigma_n)$ is at least a constant times $n^2$. (It is also at most that.)

Are there any examples of generating sets $\Sigma_n$ for which the diameter is larger than $n^{2+\epsilon}$ for every (or infinitely many) $n$? Larger than $n^2 (\log n)^A$ for some $A>0$ and infinitely many $n$?

H A Helfgott
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    Nice question. What is the reference for "well-known conjecture" ? – Alexander Chervov Jul 20 '12 at 12:57
  • Babai and Seress (1992) call it "folklore"; to judge from some papers in conference proceedings, it was already a current question in the 80s. I don't know how much further back it goes. – H A Helfgott Jul 20 '12 at 13:03
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    How about the set $a,a^qb$ where $a$ is an $n$-cycle, $b=(1,2)$, $n$ is a prime $q\approx n/2$? It is a generating set of $S_n$. What is the diameter? –  Jul 21 '12 at 17:03
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    @Mark: For $n \le 10$, which is as far as I can go with a simple-minded brute force computer calculation, the diameter with your proposed generators is less than with $a,b$. For $n=10$ and generators $a,a^5b$, the diameter is 32, whereas with $a,b$ it is 45. (I believe it is $n(n+1)/2$ in general for $a,b$.) – Derek Holt Jul 22 '12 at 21:53
  • http://terrytao.wordpress.com/2012/02/05/254b-notes-5-product-theorems-pivot-arguments-and-the-larsen-pink-non-concentration-inequality/ http://arxiv.org/abs/1205.1596 Bounds on the diameter of Cayley graphs of the symmetric group Bamberg, Gill,Hayes, Helfgott, Seress,Spiga. Abstract:" In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of ... – Alexander Chervov Jul 24 '12 at 06:02
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    Thanks for the reference, but, again, this is an interesting paper that does not answer my questions ;) – H A Helfgott Jul 24 '12 at 08:41
  • But, while, we are at it, let me say that Babai-Kantor-Lubotzky (1989) give both (a) a proof that, for $\Sigma_n={(1 2),(1 2 \dotsc n)}$, the diameter of $\Gamma(S_n,\Sigma_n)$ is indeed at least a constant times $n^2$; (b) a construction of a set of generators $\Sigma_n$ with $|\Sigma_n|=2$ such that the diameter is at most a constant times $n \log n$. (Note (b) is not hard for $|\Sigma_n|\geq 3$.) Thanks to A. Seress for the reference. – H A Helfgott Jul 24 '12 at 08:48
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    Regarding the recent exchange, I cannot resist mentioning http://mathoverflow.net/questions/23989/ (Sorry for the off-topic.) –  Jul 24 '12 at 12:59
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    @Harold Helfgott I posted links for sake of other users. Appreciate your humbleness but I would prefer if you include links on yours papers in question.... – Alexander Chervov Jul 24 '12 at 20:20
  • In reply to Alexander Ch.'s latest comment: there's also Helfgott-Seress at http://arxiv.org/abs/1109.3550, and all the references therein to work by Babai, Pyber, Seress and many others. – H A Helfgott Sep 06 '12 at 15:26

1 Answers1

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What follows is an incomplete answer. I am in the middle of Russian woods, so would rather have somebody else trace all the refs, etc. but looking at the bounty expiration date decided that it's worth stating what is known.

The answer is NO to all, but that's a conjecture not a theorem. I have seen this conjecture stated several times in various forms, here are two I find interesting:

1) the diameter of every connected Cayley graph $\Gamma$ on $S_n$ is $O(n^2)$,

2) for every O(1) generators of $S_n$, the mixing time of the nearest neighbor r.w. on the corresponding Cayley graph $\Gamma$ is $O(n^3\log n)$.

Since the mixing time is greater than the diameter, the second implies also a bound on the diameter as well. The second conjecture was stated by Diaconis and Saloff-Coste someplace, and is also sharp for a transposition and long cycle as in the question (see Saloff-Coste's survey). The first conjecture is a dated folklore and I remember reading it in various places; it appears e.g. in this paper (p. 425) by Gamburd and me.

UPDATE
See this recent paper by Diaconis ("Some things we've learned..", 2012) where he reiterates conjecture 2) in Question 2 on p.9.

Igor Pak
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    Thank you. Could you add the references when you are back? Say hi to Ivan Susanin. – H A Helfgott Jul 30 '12 at 13:28
  • PS. What is known about the mixing time of a random pair of generators? (There are direct consequences on this from work on the diameter for a random pair of generators (Babai-Hetyei, Babai-Hayes); in particular, we do have a polynomial bound. I wonder whether there are bounds going further than that.) – H A Helfgott Jul 31 '12 at 10:27