Thursday, 14 October 2010

Word length in the Symmetric group

Previously on this blog, we saw how to think about groups abstractly via group presentations, where a group is given as a set of generators satisfying specified relations. Last time, we saw that questions about the length of reduced words in such a presentation can be visualised as questions about the length of paths in the Cayley graph of the group (relative to the generators).

This time, I want to focus on just one family of groups - the symmetric groups Sn, as generated by the adjacent transpositions {si = (i i+1)}. Here's the Haskell code defining this presentation of Sn:

newtype SGen = S Int deriving (Eq,Ord)

instance Show SGen where
    show (S i) = "s" ++ show i

_S n = (gs, r ++ s ++ t) where
    gs = map S [1..n-1]
    r = [([S i, S i],[]) | i <- [1..n-1]]
    s = [(concat $ replicate 3 [S i, S (i+1)],[]) | i <- [1..n-2]]
    t = [([S i, S j, S i, S j],[]) | i <- [1..n-1], j <- [i+2..n-1]]

The three sets of relations say: each generator si squares to the identity; if i, j are not adjacent, then si and sj commute; if i, j are adjacent, then (si*sj)^3 is the identity.

Here is the Cayley graph for S4 under this presentation:

The vertices are labeled with the reduced words in the generators si. How can we find out which permutations these correspond to? Well, that's easy:

fromTranspositions ts = product $ map (\(S i) -> p [[i,i+1]]) ts
For example:

> :load Math.Algebra.Group.CayleyGraph
> fromTranspositions [S 1, S 2, S 1, S 3, S 2, S 1]

This is the permutation that reverses the list [1..4]

> map (.^ it) [1..4]

What about the other way round? Suppose we are given a permutation in Sn. How do we find its expression as a product of the transpositions si? Well the answer is (roughly): use bubblesort!

Here's bubblesort in Haskell:

bubblesort [] = []
bubblesort xs = bubblesort' [] xs where
    bubblesort' ls (r1:r2:rs) = if r1 <= r2 then bubblesort' (r1:ls) (r2:rs) else bubblesort' (r2:ls) (r1:rs)
    bubblesort' ls [r] = bubblesort (reverse ls) ++ [r]

So we sweep through the list from front to back, swapping any pairs that we find out of order - and then repeat. At the end of each sweep, we're guaranteed that the last element (in sort order) has reached the end of the list - so for the next sweep, we can leave it at the end and only sweep through the earlier elements. Hence the list we're sweeping through is one element shorter each time, so we're guaranteed to terminate. (We could terminate early, the first time a sweep makes no swaps - but I haven't coded that.)

Just to prove that the code works:

> bubblesort [2,3,1]

How does this help for turning a permutation into a sequence of transpositions? Well it's simple - every time we swap two elements, we are performing a transposition - so just record which swaps we perform. So here's a modified version of the above code, which records the swaps:

-- given a permutation of [1..n] (as a list), return the transpositions which led to it
toTrans [] = []
toTrans xs = toTrans' 1 [] [] xs where
    toTrans' i ts ls (r1:r2:rs) =
        if r1 <= r2
        then toTrans' (i+1) ts (r1:ls) (r2:rs)         -- no swap needed
        else toTrans' (i+1) (S i : ts) (r2:ls) (r1:rs) -- swap needed
    toTrans' i ts ls [r] = toTrans (reverse ls) ++ ts

Notice that the ts are returned in reverse to the order that they were used. This is because we are using them to undo the permutation - so we are performing the inverse of the permutation we are trying to find. Since each generator is its own inverse, we can recover the permutation we are after simply by reversing. In the code, we reverse as we go along.

For example:

> toTrans [2,3,1]
> toTrans [4,3,2,1]

Now, there's only one problem. As you can see, this code takes as input a rearrangement of [1..n]. This is a permutation, yes, but considered passively. Whereas in this blog we have been more accustomed to thinking of permutations actively, as something a bit like a function, which has an action on a graph, or other combinatorial structure, or if you like, just on the set [1..n]. In other words, our Permutation type represents the action itself, not the outcome of the action. (Recall that the implementation uses a Data.Map of (from,to) pairs.)

But of course it's easy to convert from one viewpoint to the other. So here's the code to take a permutation in cycle notation and turn it into a sequence of transpositions:

-- given a permutation action on [1..n], factor it into transpositions
toTranspositions 1 = []
toTranspositions g = toTrans [i .^ (g^-1) | i <- [1..n] ] where
    n = maximum $ supp g

For example:

> toTranspositions $ p [[1,4],[2,3]]

Why does the code have [i .^ (g^-1) | i <- [1..n]], rather than [i .^ g | i <- [1..n]]?
Well, suppose i .^ g = j. This says that g moves i to the j position. But we want to know what ends up in the i position. Suppose that j .^ g = i, for some j. Applying g^-1 to both sides, we see that j = i .^ (g^-1).

Okay, so given a permutation, in either form, we can reconstruct it as a reduced word in the generators.

We saw last time that the length of a reduced word is also the length of the shortest path from 1 to the element in the Cayley graph. Distance in the Cayley graph is a metric on the group, so the length of a reduced word tells us "how far" the element is from being the identity.

If it's only this distance that we're interested in, then there is a more direct way to work it out. Given a permutation g of [1..n], then an inversion is a pair (i,j) with i < j but i .^ g > j .^ g. In Haskell:

inversions g = [(i,j) | i <- [1..n], j <- [i+1..n], i < j, i .^ g > j .^ g]
    where n = maximum $ supp g

For example:

> inversions $ fromList [1,4,3,2]

With a little thought, you should be able to convince yourself that the number of inversions is equal to the length of the reduced word for g - because each swap that we perform during bubblesort corrects exactly one inversion.

Okay, so this is all very nice, but what use is it? Well, of course, maths doesn't have to be useful, any more than any other aesthetic pursuit. However, as it happens, in this case it is.

In statistics, the Kendall tau test gives an indicator of the correlation between two measured quantities (for example, the height and weight of the test subjects). Suppose that we are given a list of pairs (eg (height,weight) pairs), and we want to know how strongly correlated the first and second quantities are.

Ok, so what we do is, we rank the first quantities from lowest to highest, and replace each quantity by its rank (a number from 1 to n). We do the same for the second quantities. So we end up with a list of pairs of numbers from 1 to n. Now, we sort the list on the first element, and then count the number of inversions in the second element.

For example, suppose our original list was [(1.55m, 60kg), (1.8m, 80kg), (1.5m, 70kg), (1.6m, 72kg)]. Converting to ranks, we get [(2nd,1st),(4th,4th),(1st,2nd),(3rd,3rd)]. Sorting on fst, we get [(1,2),(2,1),(3,3),(4,4)]. Looking at snd, we see that we have just one inversion. The idea is that the fewer inversions we have, the better correlated the two quantities. (Of course in reality there's a bit more to it than that - to convert the number of inversions into a probability, we need to know the distribution of word lengths for Sn, where n is the number of pairs of test data that we have.)

So you can think of the Kendall tau test as saying: What permutation has been applied in moving from the first quantities (the heights) to the second quantities (the weights)? How far is that permutation from the identity (on the Cayley graph)? What proportion of all permutations (in Sn) lie at that distance or less from the identity? (Even more concretely, we can imagine colouring in successive shells of the Cayley graph, working out from the identity, until we hit the given permutation, and then asking what proportion of the "surface" of the graph has been coloured.)

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