-- Copyright (c) David Amos, 2008-2012. All rights reserved. {-# LANGUAGE NoMonomorphismRestriction #-} -- |A module for doing arithmetic in permutation groups. -- -- Group elements are represented as permutations of underlying sets, and are entered and displayed -- using a Haskell-friendly version of cycle notation. For example, the permutation (1 2 3)(4 5) -- would be entered as @p [[1,2,3],[4,5]]@, and displayed as [[1,2,3],[4,5]]. Permutations can be defined -- over arbitrary underlying sets (types), not just the integers. -- -- If @g@ and @h@ are group elements, then the expressions @g*h@ and @g^-1@ calculate product and inverse respectively. module Math.Algebra.Group.PermutationGroup where import qualified Data.List as L import qualified Data.Map as M import qualified Data.Set as S import Math.Common.ListSet (toListSet, union, (\\) ) -- a version of union which assumes the arguments are ascending sets (no repeated elements) import Math.Core.Utils hiding (elts) import Math.Algebra.LinearAlgebra hiding (inverse) -- only needed for use in ghci infix 8 ~^ rotateL (x:xs) = xs ++ [x] -- PERMUTATIONS -- |A type for permutations, considered as functions or actions which can be performed on an underlying set. newtype Permutation a = P (M.Map a a) deriving (Eq,Ord) -- |Construct a permutation from a list of cycles. -- For example, @p [[1,2,3],[4,5]]@ returns the permutation that sends 1 to 2, 2 to 3, 3 to 1, 4 to 5, 5 to 4. p :: (Ord a) => [[a]] -> Permutation a p = fromCycles fromPairs xys | isValid = fromPairs' xys | otherwise = error "Not a permutation" where (xs,ys) = unzip xys (xs',ys') = (L.sort xs, L.sort ys) isValid = xs' == ys' && all ((==1) . length) (L.group xs') -- ie the domain and range are the same, and are *sets* fromPairs' xys = P $ M.fromList $ filter (uncurry (/=)) xys -- we remove fixed points, so that the derived Eq instance works as expected toPairs (P g) = M.toList g fromList xs = fromPairs $ zip xs (L.sort xs) -- for example, fromList [2,3,1] is [[1,3,2]] - because the 1 moved to the 3 position -- the support of a permutation is the points it moves (returned in ascending order) supp (P g) = M.keys g -- (This is guaranteed not to contain fixed points provided the permutations have been constructed using the supplied constructors) -- |x .^ g returns the image of a vertex or point x under the action of the permutation g. -- For example, @1 .^ p [[1,2,3]]@ returns 2. -- The dot is meant to be a mnemonic for point or vertex. (.^) :: (Ord a) => a -> Permutation a -> a x .^ P g = case M.lookup x g of Just y -> y Nothing -> x -- if x `notElem` supp (P g), then x is not moved -- |b -^ g returns the image of an edge or block b under the action of the permutation g. -- For example, @[1,2] -^ p [[1,4],[2,3]]@ returns [3,4]. -- The dash is meant to be a mnemonic for edge or line or block. (-^) :: (Ord a) => [a] -> Permutation a -> [a] xs -^ g = L.sort [x .^ g | x <- xs] -- construct a permutation from cycles fromCycles cs = fromPairs $ concatMap fromCycle cs where fromCycle xs = zip xs (rotateL xs) -- convert a permutation to cycles toCycles g = toCycles' $ supp g where toCycles' ys@(y:_) = let c = cycleOf g y in c : toCycles' (ys L.\\ c) toCycles' [] = [] cycleOf g x = cycleOf' x [] where cycleOf' y ys = let y' = y .^ g in if y' == x then reverse (y:ys) else cycleOf' y' (y:ys) instance (Ord a, Show a) => Show (Permutation a) where show g | g == 1 = "1" | otherwise = show (toCycles g) parity g = let cs = toCycles g in (length (concat cs) - length cs) `mod` 2 -- parity' g = length (filter (even . length) $ toCycles g) `mod` 2 sign g = (-1)^(parity g) orderElt g = foldl lcm 1 $ map length $ toCycles g -- == order [g] -- |The Num instance is what enables us to write @g*h@ for the product of group elements and @1@ for the group identity. -- Unfortunately we can't of course give sensible definitions for the other functions declared in the Num typeclass. instance (Ord a, Show a) => Num (Permutation a) where g * h = fromPairs' [(x, x .^ g .^ h) | x <- supp g `union` supp h] -- signum = sign -- doesn't work, complains about no (+) instance fromInteger 1 = P $ M.empty _ + _ = error "(Permutation a).+: not applicable" negate _ = error "(Permutation a).negate: not applicable" abs _ = error "(Permutation a).abs: not applicable" signum _ = error "(Permutation a).signum: not applicable" -- |The HasInverses instance is what enables us to write @g^-1@ for the inverse of a group element. instance (Ord a, Show a) => HasInverses (Permutation a) where inverse (P g) = P $ M.fromList $ map (\(x,y)->(y,x)) $ M.toList g -- |g ~^ h returns the conjugate of g by h, that is, h^-1*g*h. -- The tilde is meant to a mnemonic, because conjugacy is an equivalence relation. (~^) :: (Ord a, Show a) => Permutation a -> Permutation a -> Permutation a g ~^ h = h^-1 * g * h -- commutator comm g h = g^-1 * h^-1 * g * h -- ORBITS closureS xs fs = closure' S.empty (S.fromList xs) where closure' interior boundary | S.null boundary = interior | otherwise = let interior' = S.union interior boundary boundary' = S.fromList [f x | x <- S.toList boundary, f <- fs] S.\\ interior' in closure' interior' boundary' closure xs fs = S.toList $ closureS xs fs orbit action x gs = closure [x] [ (`action` g) | g <- gs] -- |x .^^ gs returns the orbit of the point or vertex x under the action of the gs (.^^) :: (Ord a) => a -> [Permutation a] -> [a] x .^^ gs = orbit (.^) x gs orbitP gs x = orbit (.^) x gs orbitV gs x = orbit (.^) x gs -- |b -^^ gs returns the orbit of the block or edge b under the action of the gs (-^^) :: (Ord a) => [a] -> [Permutation a] -> [[a]] b -^^ gs = orbit (-^) b gs orbitB gs b = orbit (-^) b gs orbitE gs b = orbit (-^) b gs action xs f = fromPairs [(x, f x) | x <- xs] -- find all the orbits of a group -- (as we typically work with transitive groups, this is more useful for studying induced actions) -- (Note that of course this won't find orbits of points which are fixed by all elts of G) orbits gs = let xs = foldl union [] $ map supp gs in orbits' xs where orbits' [] = [] orbits' (x:xs) = let o = x .^^ gs in o : orbits' (xs L.\\ o) -- GROUPS -- Some standard sequences of groups, and constructions of new groups from old -- |_C n returns generators for Cn, the cyclic group of order n _C :: (Integral a) => a -> [Permutation a] _C n | n >= 2 = [p [[1..n]]] -- D2n, dihedral group of order 2n, symmetry group of n-gon -- For example, _D 8 == _D2 4 == symmetry group of square _D n | r == 0 = _D2 q where (q,r) = n `quotRem` 2 _D2 n | n >= 3 = [a,b] where a = p [[1..n]] -- rotation b = p [[i,n+1-i] | i <- [1..n `div` 2]] -- reflection -- b = fromPairs $ [(i,n+1-i) | i <- [1..n]] -- reflection -- |_S n returns generators for Sn, the symmetric group on [1..n] _S :: (Integral a) => a -> [Permutation a] _S n | n >= 3 = [s,t] | n == 2 = [t] | n == 1 = [] where s = p [[1..n]] t = p [[1,2]] -- |_A n returns generators for An, the alternating group on [1..n] _A :: (Integral a) => a -> [Permutation a] _A n | n > 3 = [s,t] | n == 3 = [t] | n == 2 = [] where s | odd n = p [[3..n]] | even n = p [[1,2], [3..n]] t = p [[1,2,3]] -- |Given generators for groups H and K, acting on sets A and B respectively, -- return generators for the direct product H*K, acting on the disjoint union A+B (= Either A B) dp :: (Ord a, Ord b) => [Permutation a] -> [Permutation b] -> [Permutation (Either a b)] dp hs ks = [P $ M.fromList $ map (\(x,x') -> (Left x,Left x')) $ M.toList h' | P h' <- hs] ++ [P $ M.fromList $ map (\(y,y') -> (Right y,Right y')) $ M.toList k' | P k' <- ks] -- Wreath product of groups -- Given generators for H and K, acting on sets X and Y respectively, -- return generators for H wr K, acting on X*Y (== (X,Y)) -- (Cameron, Combinatorics, p229-230; Cameron, Permutation Groups, p11-12) wr hs ks = let _X = S.toList $ foldl S.union S.empty [M.keysSet h' | P h' <- hs] -- set on which H acts _Y = S.toList $ foldl S.union S.empty [M.keysSet k' | P k' <- ks] -- set on which K acts -- Then the wreath product acts on cartesian product X * Y, -- regarded as a fibre bundle over Y of isomorphic copies of X _B = [P $ M.fromList $ map (\(x,x') -> ((x,y),(x',y))) $ M.toList h' | P h' <- hs, y <- _Y] -- bottom group B applies the action of H within each fibre _T = [P $ M.fromList [((x,y),(x,y')) | x <- _X, (y,y') <- M.toList k'] | P k' <- ks] -- top group T uses the action of K to permute the fibres in _B ++ _T -- semi-direct product of B and T -- !! Why using M.keysSet rather than supp? -- embed group elts into Sn - ie, convert so that the set acted on is [1..n] toSn gs = [toSn' g | g <- gs] where _X = L.sort $ foldl union [] $ map supp gs -- the set on which G acts mapping = M.fromList $ zip _X [1..] -- the mapping from _X to [1..n] toSn' g = fromPairs' $ map (\(x,x') -> (mapping M.! x, mapping M.! x')) $ toPairs g -- Given a permutation over lists of small positive integers, such as [1,2,3], -- return a permutation over the integers obtained by interpreting the lists as digits. -- For example, [1,2,3] -> 123. fromDigits g = fromPairs [(fromDigits' x, fromDigits' y) | (x,y) <- toPairs g] fromDigits' xs = f (reverse xs) where f (x:xs) = x + 10 * f xs f [] = 0 -- Given a permutation over lists of 0s and 1s, -- return the permutation obtained by interpreting these as binary digits. -- For example, [1,1,0] -> 6. fromBinary g = fromPairs [(fromBinary' x, fromBinary' y) | (x,y) <- toPairs g] fromBinary' xs = f (reverse xs) where f (x:xs) = x + 2 * f xs f [] = 0 -- INVESTIGATING GROUPS -- Functions to investigate groups in various ways -- Most of these functions will only be efficient for small groups (say |G| < 10000) -- For larger groups we will need to use Schreier-Sims and associated algorithms -- |Given generators for a group, return a (sorted) list of all elements of the group. -- Implemented using a naive closure algorithm, so only suitable for small groups (|G| < 10000) elts :: (Num a, Ord a) => [a] -> [a] elts gs = closure [1] [ (*g) | g <- gs] eltsS gs = closureS [1] [ (*g) | g <- gs] -- |Given generators for a group, return the order of the group (the number of elements). -- Implemented using a naive closure algorithm, so only suitable for small groups (|G| < 10000) order :: (Num a, Ord a) => [a] -> Int order gs = S.size $ eltsS gs -- length $ elts gs isMember gs h = h `S.member` eltsS gs -- h `elem` elts gs -- TRANSVERSAL GENERATING SETS -- The functions graphAuts2 and graphAuts3 return generating sets consisting of successive transversals -- In this case, we don't need to run Schreier-Sims to list elements or calculate order minsupp = head . supp -- calculate the order of the group, given a "transversal generating set" orderTGS tgs = let transversals = map (1:) $ L.groupBy (\g h -> minsupp g == minsupp h) tgs in product $ map L.genericLength transversals -- list the elts of the group, given a "transversal generating set" eltsTGS tgs = let transversals = map (1:) $ L.groupBy (\g h -> minsupp g == minsupp h) tgs in map product $ sequence transversals -- recover a transversal generating set from a strong generating set -- A strong generating set is a generating set gs such that <gs intersect si> = si -- ie, its intersection with each successive stabiliser in the chain generates the stabiliser tgsFromSgs sgs = concatMap transversal bs where bs = toListSet $ map minsupp sgs transversal b = closure b $ filter ( (b <=) . minsupp ) sgs closure b gs = closure' M.empty (M.fromList [(b, 1)]) where closure' interior boundary | M.null boundary = filter (/=1) $ M.elems interior | otherwise = let interior' = M.union interior boundary boundary' = M.fromList [(x .^ g, h*g) | (x,h) <- M.toList boundary, g <- gs] M.\\ interior' in closure' interior' boundary' -- For example, sgs (_A 5) == [[[1,2,3]],[[2,4,5]],[[3,4,5]]] -- So we need all three to generate the first transversal, then the last two to generate the second transversal, etc -- |Given a strong generating set, return the order of the group it generates orderSGS :: (Ord a) => [Permutation a] -> Integer orderSGS sgs = product $ map (L.genericLength . fundamentalOrbit) bs where bs = toListSet $ map minsupp sgs fundamentalOrbit b = b .^^ filter ( (b <=) . minsupp ) sgs -- MORE INVESTIGATIONS -- given the elts of a group, find generators gens hs = gens' [] (S.singleton 1) hs where gens' gs eltsG (h:hs) = if h `S.member` eltsG then gens' gs eltsG hs else gens' (h:gs) (eltsS $ h:gs) hs gens' gs _ [] = reverse gs -- conjClass gs h = orbit (~^) gs h -- Conjugacy class - should only be used for small groups h ~^^ gs = conjClass gs h conjClass gs h = closure [h] [ (~^ g) | g <- gs] -- conjClass gs h = h ~^^ gs -- |conjClassReps gs returns conjugacy class representatives and sizes for the group generated by gs. -- This implementation is only suitable for use with small groups (|G| < 10000). conjClassReps :: (Ord a, Show a) => [Permutation a] -> [(Permutation a, Int)] conjClassReps gs = conjClassReps' (elts gs) where conjClassReps' (h:hs) = let cc = conjClass gs h in (h, length cc) : conjClassReps' (hs \\ cc) conjClassReps' [] = [] -- using the ListSet implementation of \\, since we know both lists are sorted {- -- This is just the orbits under conjugation. Can we generalise "orbits" to help us here? conjClasses gs = conjClasses' (elts gs) where conjClasses' [] = [] conjClasses' (h:hs) = let c = conjClass gs h in c : conjClasses' (hs L.\\ c) -} -- given list of generators, try to find a shorter list reduceGens (1:gs) = reduceGens gs reduceGens (g:gs) = reduceGens' ([g], eltsS [g]) gs where reduceGens' (gs,eltsgs) (h:hs) = if h `S.member` eltsgs then reduceGens' (gs,eltsgs) hs else reduceGens' (h:gs, eltsS $ h:gs) hs reduceGens' (gs,_) [] = reverse gs -- SUBGROUPS isSubgp hs gs = all (`S.member` gs') hs where gs' = eltsS gs -- The following is similar to the "cyclic extension" method - Holt p385 -- However, Holt only looks at normal cyclic extensions (ie, by an elt of prime order), and so only finds solvable subgps -- |Return the subgroups of a group. Only suitable for use on small groups (eg < 100 elts) subgps :: (Ord a, Show a) => [Permutation a] -> [[Permutation a]] subgps gs = [] : subgps' S.empty [] (map (:[]) hs) where hs = filter isMinimal $ elts gs subgps' found ls (r:rs) = let ks = elts r in if ks `S.member` found then subgps' found ls rs else r : subgps' (S.insert ks found) (r:ls) rs subgps' found [] [] = [] subgps' found ls [] = subgps' found [] [l ++ [h] | l <- reverse ls, h <- hs, last l < h] -- g is the minimal elt in the cyclic subgp it generates isMinimal 1 = False isMinimal g = all (g <=) primitives -- g == minimum primitives where powers = takeWhile (/=1) $ tail $ iterate (*g) 1 n = orderElt g -- == length powers + 1 primitives = filter (\h -> orderElt h == n) powers -- centralizer of a subgroup or a set of elts -- the centralizer of H in G is the set of elts of G which commute with all elts of H centralizer gs hs = [k | k <- elts gs, all (\h -> h*k == k*h) hs] -- the centre of G is the set of elts of G which commute with all other elts centre gs = centralizer gs gs -- normaliser of a subgroup -- the normaliser of H in G is {g <- G | g^-1Hg == H} -- it is a subgroup of G, and H is a normal subgroup of it: H <|= N_G(H) <= G normalizer gs hs = [g | g <- elts gs, all (\h -> h~^g `elem` elts hs) hs] -- stabilizer of a point stabilizer gs x = [g | g <- elts gs, x .^ g == x] -- pointwise stabiliser of a set ptStab gs xs = [g | g <- elts gs, and [x .^ g == x | x <- xs] ] -- setwise stabiliser of a set setStab gs xs = [g | g <- elts gs, xs -^ g == xs] -- normal closure of H in G normalClosure gs hs = reduceGens $ hs ++ [h ~^ g | h <- hs, g <- gs ++ map inverse gs] -- commutator gp of H and K commutatorGp hs ks = normalClosure (hsks) [h^-1 * k^-1 * h * k | h <- hs', k <- ks'] where hs' = reduceGens hs ks' = reduceGens ks hsks = reduceGens (hs' ++ ks') -- no point processing more potential generators than we have to -- derived subgroup derivedSubgp gs = commutatorGp gs gs -- ACTION ON COSETS, QUOTIENT GROUPS xs -*- ys = toListSet [x*y | x <- xs, y <- ys] xs -* y = L.sort [x*y | x <- xs] -- == xs -*- [y] x *- ys = L.sort [x*y | y <- ys] -- == [x] -*- ys -- |isNormal gs ks returns True if \<ks\> is normal in \<gs\>. -- Note, it is caller's responsibility to ensure that \<ks\> is a subgroup of \<gs\> (ie that each k is in \<gs\>). isNormal :: (Ord a, Show a) => [Permutation a] -> [Permutation a] -> Bool isNormal gs ks = all (== ks') [ (g^-1) *- ks' -* g | g <- gs] where ks' = elts ks -- |Return the normal subgroups of a group. Only suitable for use on small groups (eg < 100 elts) normalSubgps :: (Ord a, Show a) => [Permutation a] -> [[Permutation a]] normalSubgps gs = filter (isNormal gs) (subgps gs) isSimple gs = length (normalSubgps gs) == 2 -- Note: caller must ensure that hs is a subgp of gs cosets gs hs = orbit (-*) hs' gs where hs' = elts hs -- |quotientGp gs ks returns \<gs\> / \<ks\> quotientGp :: (Ord a, Show a) => [Permutation a] -> [Permutation a] -> [Permutation Int] quotientGp gs ks | ks `isNormal` gs = gens $ toSn [action cosetsK (-* g) | g <- gs] | otherwise = error "quotientGp: not well defined unless ks normal in gs" where cosetsK = cosets gs ks -- |Synonym for quotientGp (//) :: (Ord a, Show a) => [Permutation a] -> [Permutation a] -> [Permutation Int] gs // ks = quotientGp gs ks -- action of group element on a subset by conjugation xs ~~^ g = L.sort [x ~^ g | x <- xs] conjugateSubgps gs hs = orbit (~~^) hs' gs where hs' = elts hs -- not necessarily transitive on isomorphic subgps - eg a gp with an outer aut subgpAction gs hs = let conjugatesH = conjugateSubgps gs hs in toSn [action conjugatesH (~~^ g) | g <- gs] -- in cube gp, the subgps all appear to correspond to stabilisers of subsets, or of blocks -- right regular permutation representation rrpr gs h = rrpr' (elts gs) h rrpr' gs h = fromPairs [(g, g*h) | g <- gs] permutationMatrix xs g = [ [if x .^ g == y then 1 else 0 | y <- xs] | x <- xs ]