-- Copyright (c) 2012, David Amos. All rights reserved. {-# LANGUAGE MultiParamTypeClasses, FlexibleInstances, NoMonomorphismRestriction, ScopedTypeVariables, DeriveFunctor #-} -- |A module defining the following Combinatorial Hopf Algebras, together with coalgebra or Hopf algebra morphisms between them: -- -- * Sh, the Shuffle Hopf algebra -- -- * SSym, the Malvenuto-Reutnenauer Hopf algebra of permutations -- -- * YSym, the (dual of the) Loday-Ronco Hopf algebra of binary trees -- -- * QSym, the Hopf algebra of quasi-symmetric functions (having a basis indexed by compositions) -- -- * Sym, the Hopf algebra of symmetric functions (having a basis indexed by integer partitions) -- -- * NSym, the Hopf algebra of non-commutative symmetric functions module Math.Combinatorics.CombinatorialHopfAlgebra where -- Sources: -- Structure of the Malvenuto-Reutenauer Hopf algebra of permutations -- Marcelo Aguiar and Frank Sottile -- http://www.math.tamu.edu/~sottile/research/pdf/SSym.pdf -- Structure of the Loday-Ronco Hopf algebra of trees -- Marcelo Aguiar and Frank Sottile -- http://www.math.tamu.edu/~sottile/research/pdf/Loday.pdf -- Hopf Structures on the Multiplihedra -- Stefan Forcey, Aaron Lauve and Frank Sottile -- http://www.math.tamu.edu/~sottile/research/pdf/MSym.pdf -- Lie Algebras and Hopf Algebras -- Michiel Hazewinkel, Nadiya Gubareni, V.V.Kirichenko import Data.List as L import Data.Maybe (fromJust) import qualified Data.Set as S import Math.Core.Field import Math.Core.Utils import Math.Algebras.VectorSpace hiding (E) import Math.Algebras.TensorProduct import Math.Algebras.Structures import Math.Combinatorics.Poset -- import Math.Algebra.Group.PermutationGroup import Math.CommutativeAlgebra.Polynomial -- SHUFFLE ALGEBRA -- This is just the tensor algebra, but with shuffle product (and deconcatenation coproduct) -- |A basis for the shuffle algebra. As a vector space, the shuffle algebra is identical to the tensor algebra. -- However, we consider a different algebra structure, based on the shuffle product. Together with the -- deconcatenation coproduct, this leads to a Hopf algebra structure. newtype Shuffle a = Sh [a] deriving (Eq,Ord,Show) -- |Construct a basis element of the shuffle algebra sh :: [a] -> Vect Q (Shuffle a) sh = return . Sh shuffles (x:xs) (y:ys) = map (x:) (shuffles xs (y:ys)) ++ map (y:) (shuffles (x:xs) ys) shuffles xs [] = [xs] shuffles [] ys = [ys] instance (Eq k, Num k, Ord a) => Algebra k (Shuffle a) where unit x = x *> return (Sh []) mult = linear mult' where mult' (Sh xs, Sh ys) = sumv [return (Sh zs) | zs <- shuffles xs ys] deconcatenations xs = zip (inits xs) (tails xs) instance (Eq k, Num k, Ord a) => Coalgebra k (Shuffle a) where counit = unwrap . linear counit' where counit' (Sh xs) = if null xs then 1 else 0 comult = linear comult' where comult' (Sh xs) = sumv [return (Sh us, Sh vs) | (us, vs) <- deconcatenations xs] instance (Eq k, Num k, Ord a) => Bialgebra k (Shuffle a) where {} instance (Eq k, Num k, Ord a) => HopfAlgebra k (Shuffle a) where antipode = linear (\(Sh xs) -> (-1)^length xs *> return (Sh (reverse xs))) -- SSYM: PERMUTATIONS -- (This is permutations considered as combinatorial objects rather than as algebraic objects) -- Permutations with shifted shuffle product and flattened deconcatenation coproduct -- This is the Malvenuto-Reutenauer Hopf algebra of permutations, SSym. -- It is neither commutative nor co-commutative -- ssymF xs is the fundamental basis F_xs (Aguiar and Sottile) -- |The fundamental basis for the Malvenuto-Reutenauer Hopf algebra of permutations, SSym. newtype SSymF = SSymF [Int] deriving (Eq) instance Ord SSymF where compare (SSymF xs) (SSymF ys) = compare (length xs, xs) (length ys, ys) instance Show SSymF where show (SSymF xs) = "F " ++ show xs -- |Construct a fundamental basis element in SSym. -- The list of ints must be a permutation of [1..n], eg [1,2], [3,4,2,1]. ssymF :: [Int] -> Vect Q SSymF ssymF xs | L.sort xs == [1..n] = return (SSymF xs) | otherwise = error "Not a permutation of [1..n]" where n = length xs -- so this is a candidate mult. It is associative and SSymF [] is obviously a left and right identity -- (need quickcheck properties to prove that) shiftedConcat (SSymF xs) (SSymF ys) = let k = length xs in SSymF (xs ++ map (+k) ys) prop_Associative f (x,y,z) = f x (f y z) == f (f x y) z -- > quickCheck (prop_Associative shiftedConcat) -- +++ OK, passed 100 tests. instance (Eq k, Num k) => Algebra k SSymF where unit x = x *> return (SSymF []) mult = linear mult' where mult' (SSymF xs, SSymF ys) = let k = length xs in sumv [return (SSymF zs) | zs <- shuffles xs (map (+k) ys)] -- standard permutation, also called flattening, eg [6,2,5] -> [3,1,2] flatten xs = let mapping = zip (L.sort xs) [1..] in [y | x <- xs, let Just y = lookup x mapping] instance (Eq k, Num k) => Coalgebra k SSymF where counit = unwrap . linear counit' where counit' (SSymF xs) = if null xs then 1 else 0 comult = linear comult' where comult' (SSymF xs) = sumv [return (SSymF (st us), SSymF (st vs)) | (us, vs) <- deconcatenations xs] st = flatten instance (Eq k, Num k) => Bialgebra k SSymF where {} instance (Eq k, Num k) => HopfAlgebra k SSymF where antipode = linear antipode' where antipode' (SSymF []) = return (SSymF []) antipode' x@(SSymF xs) = (negatev . mult . (id `tf` antipode) . removeTerm (SSymF [],x) . comult . return) x -- This expression for antipode is derived from mult . (id `tf` antipode) . comult == unit . counit -- It's possible because this is a graded, connected Hopf algebra. (connected means the counit is projection onto the grade 0 part) -- It would be nicer to have an explicit expression for antipode. {- instance (Eq k, Num k) => HopfAlgebra k SSymF where antipode = linear antipode' where antipode' (SSymF v) = sumv [lambda v w *> return (SSymF w) | w <- L.permutations v] lambda v w = length [s | s <- powerset [1..n-1], odd (length s), descentSet (w^-1 * v_s) `isSubset` s] - length [s | s <- powerset [1..n-1], even (length s), descentSet (w^-1 * v_s) `isSubset` s] -} instance HasInverses SSymF where inverse (SSymF xs) = SSymF $ map snd $ L.sort $ map (\(s,t)->(t,s)) $ zip [1..] xs -- Hazewinkel p267 -- |A pairing showing that SSym is self-adjoint instance (Eq k, Num k) => HasPairing k SSymF SSymF where pairing = linear pairing' where pairing' (x,y) = delta x (inverse y) -- Not entirely clear to me why this works -- The pairing is *not* positive definite (Hazewinkel p267) -- eg (\x -> pairing' x x >= 0) (ssymF [1,3,2] + ssymF [2,3,1] - ssymF [3,1,2]) == False -- |An alternative \"monomial\" basis for the Malvenuto-Reutenauer Hopf algebra of permutations, SSym. -- This basis is related to the fundamental basis by Mobius inversion in the poset of permutations with the weak order. newtype SSymM = SSymM [Int] deriving (Eq) instance Ord SSymM where compare (SSymM xs) (SSymM ys) = compare (length xs, xs) (length ys, ys) instance Show SSymM where show (SSymM xs) = "M " ++ show xs -- |Construct a monomial basis element in SSym. -- The list of ints must be a permutation of [1..n], eg [1,2], [3,4,2,1]. ssymM :: [Int] -> Vect Q SSymM ssymM xs | L.sort xs == [1..n] = return (SSymM xs) | otherwise = error "Not a permutation of [1..n]" where n = length xs inversions xs = let ixs = zip [1..] xs in [(i,j) | ((i,xi),(j,xj)) <- pairs ixs, xi > xj] -- should really check that xs and ys have the same length, and perhaps insist also on same type weakOrder xs ys = inversions xs `isSubsetAsc` inversions ys mu (set,po) x y = mu' x y where mu' x y | x == y = 1 | po x y = negate $ sum [mu' x z | z <- set, po x z, po z y, z /= y] | otherwise = 0 -- |Convert an element of SSym represented in the monomial basis to the fundamental basis ssymMtoF :: (Eq k, Num k) => Vect k SSymM -> Vect k SSymF ssymMtoF = linear ssymMtoF' where ssymMtoF' (SSymM u) = sumv [mu (set,po) u v *> return (SSymF v) | v <- set, po u v] where set = L.permutations u po = weakOrder -- |Convert an element of SSym represented in the fundamental basis to the monomial basis ssymFtoM :: (Eq k, Num k) => Vect k SSymF -> Vect k SSymM ssymFtoM = linear ssymFtoM' where ssymFtoM' (SSymF u) = sumv [return (SSymM v) | v <- set, po u v] where set = L.permutations u po = weakOrder -- (p,q)-shuffles: permutations of [1..p+q] having at most one descent, at position p -- denoted S^{(p,q)} in Aguiar&Sottile -- (Grassmannian permutations?) -- pqShuffles p q = [u++v | u <- combinationsOf p [1..n], let v = [1..n] `diffAsc` u] where n = p+q -- The inverse of a (p,q)-shuffle. -- The special form of (p,q)-shuffles makes an O(n) algorithm possible -- pqInverse :: Int -> Int -> [Int] -> [Int] {- -- incorrect pqInverse p q xs = pqInverse' [1..p] [p+1..p+q] xs where pqInverse' (l:ls) (r:rs) (x:xs) = if x <= p then l : pqInverse' ls (r:rs) xs else r : pqInverse' (l:ls) rs xs pqInverse' ls rs _ = ls ++ rs -- one of them is null -} -- pqInverseShuffles p q = shuffles [1..p] [p+1..p+q] instance (Eq k, Num k) => Algebra k SSymM where unit x = x *> return (SSymM []) mult = ssymFtoM . mult . (ssymMtoF `tf` ssymMtoF) {- mult2 = linear mult' where mult' (SSymM u, SSymM v) = sumv [alpha u v w *> return (SSymM w) | w <- L.permutations [1..p+q] ] where p = length u; q = length v alpha u v w = length [z | z <- pqInverseShuffles p q, let uv = shiftedConcat u v, uv * z `weakOrder` w, u and v are maximal, ie no transposition of adjacents in either also works] where p = length u q = length v -- so we need to define (*) for permutations in row form -} instance (Eq k, Num k) => Coalgebra k SSymM where counit = unwrap . linear counit' where counit' (SSymM xs) = if null xs then 1 else 0 -- comult = (ssymFtoM `tf` ssymFtoM) . comult . ssymMtoF comult = linear comult' where comult' (SSymM xs) = sumv [return (SSymM (flatten ys), SSymM (flatten zs)) | (ys,zs) <- deconcatenations xs, minimum (infinity:ys) > maximum (0:zs)] -- ie deconcatenations at a global descent infinity = maxBound :: Int instance (Eq k, Num k) => Bialgebra k SSymM where {} instance (Eq k, Num k) => HopfAlgebra k SSymM where antipode = ssymFtoM . antipode . ssymMtoF -- Hazewinkel p265 instance (Eq k, Num k) => Algebra k (Dual SSymF) where unit x = x *> return (Dual (SSymF [])) mult = linear mult' where mult' (Dual (SSymF xs), Dual (SSymF ys)) = sumv [(return . Dual . SSymF) (xs'' ++ ys'') | xs' <- combinationsOf r [1..r+s], let ys' = diffAsc [1..r+s] xs', xs'' <- L.permutations xs', flatten xs'' == xs, ys'' <- L.permutations ys', flatten ys'' == ys ] where r = length xs; s = length ys -- In other words, mult x y is the sum of those z whose comult (in SSymF) has an (x,y) term -- So the matrix for mult is the transpose of the matrix for comult in SSymF instance (Eq k, Num k) => Coalgebra k (Dual SSymF) where counit = unwrap . linear counit' where counit' (Dual (SSymF xs)) = if null xs then 1 else 0 comult = linear comult' where comult' (Dual (SSymF xs)) = sumv [return (Dual (SSymF ys), Dual (SSymF (flatten zs))) | i <- [0..n], let (ys,zs) = L.partition (<=i) xs ] where n = length xs -- In other words, comult x is the sum of those (y,z) whose mult (in SSymF) has a z term -- So the matrix for comult is the transpose of the matrix for mult in SSymF instance (Eq k, Num k) => Bialgebra k (Dual SSymF) where {} instance (Eq k, Num k) => HopfAlgebra k (Dual SSymF) where antipode = linear antipode' where antipode' (Dual (SSymF [])) = return (Dual (SSymF [])) antipode' x@(Dual (SSymF xs)) = (negatev . mult . (id `tf` antipode) . removeTerm (Dual (SSymF []),x) . comult . return) x -- This pairing is positive definite (Hazewinkel p267) instance (Eq k, Num k) => HasPairing k SSymF (Dual SSymF) where pairing = linear pairing' where pairing' (x, Dual y) = delta x y -- |The isomorphism from SSym to its dual that takes a permutation in the fundamental basis to its inverse in the dual basis ssymFtoDual :: (Eq k, Num k) => Vect k SSymF -> Vect k (Dual SSymF) ssymFtoDual = nf . fmap (Dual . inverse) -- This is theta on Hazewinkel p266 (though later he also uses theta for the inverse of this map) -- YSYM: PLANAR BINARY TREES -- These are really rooted planar binary trees. -- It's because they're planar that we can distinguish left and right child branches. -- (Non-planar would be if we considered trees where left and right children are swapped relative to one another as the same tree) -- It is neither commutative nor co-commutative -- |A type for (rooted) planar binary trees. The basis elements of the Loday-Ronco Hopf algebra are indexed by these. -- -- Although the trees are labelled, we're really only interested in the shapes of the trees, and hence in the type PBT (). -- The Algebra, Coalgebra and HopfAlgebra instances all ignore the labels. -- However, it is convenient to allow labels, as they can be useful for seeing what is going on, and they also make it possible -- to define various ways to create trees from lists of labels. data PBT a = T (PBT a) a (PBT a) | E deriving (Eq, Show, Functor) instance Ord a => Ord (PBT a) where compare u v = compare (shapeSignature u, prefix u) (shapeSignature v, prefix v) -- |The fundamental basis for (the dual of) the Loday-Ronco Hopf algebra of binary trees, YSym. newtype YSymF a = YSymF (PBT a) deriving (Eq, Ord, Functor) instance Show a => Show (YSymF a) where show (YSymF t) = "F(" ++ show t ++ ")" -- |Construct the element of YSym in the fundamental basis indexed by the given tree ysymF :: PBT a -> Vect Q (YSymF a) ysymF t = return (YSymF t) {- depth (T l x r) = 1 + max (depth l) (depth r) depth E = 0 -} nodecount (T l x r) = 1 + nodecount l + nodecount r nodecount E = 0 -- in fact leafcount t = 1 + nodecount t (easiest to see with a picture) leafcount (T l x r) = leafcount l + leafcount r leafcount E = 1 prefix E = [] prefix (T l x r) = x : prefix l ++ prefix r -- The shape signature uniquely identifies the shape of a tree. -- Trees with distinct shapes have distinct signatures. -- In addition, if sorting on shapeSignature, smaller trees sort before larger trees, -- and leftward leaning trees sort before rightward leaning trees shapeSignature t = shapeSignature' (nodeCountTree t) where shapeSignature' E = [0] -- not [], otherwise we can't distinguish T (T E () E) () E from T E () (T E () E) shapeSignature' (T l x r) = x : shapeSignature' r ++ shapeSignature' l nodeCountTree E = E nodeCountTree (T l _ r) = T l' n r' where l' = nodeCountTree l r' = nodeCountTree r n = 1 + (case l' of E -> 0; T _ lc _ -> lc) + (case r' of E -> 0; T _ rc _ -> rc) leafCountTree E = E leafCountTree (T l _ r) = T l' n r' where l' = leafCountTree l r' = leafCountTree r n = (case l' of E -> 1; T _ lc _ -> lc) + (case r' of E -> 1; T _ rc _ -> rc) -- A tree that counts nodes in left and right subtrees lrCountTree E = E lrCountTree (T l _ r) = T l' (lc,rc) r' where l' = lrCountTree l r' = lrCountTree r lc = case l' of E -> 0; T _ (llc,lrc) _ -> 1 + llc + lrc rc = case r' of E -> 0; T _ (rlc,rrc) _ -> 1 + rlc + rrc shape :: PBT a -> PBT () shape t = fmap (\_ -> ()) t -- label the nodes of a tree in infix order while preserving its shape numbered t = numbered' 1 t where numbered' _ E = E numbered' i (T l x r) = let k = nodecount l in T (numbered' i l) (i+k) (numbered' (i+k+1) r) -- could also pair the numbers with the input labels splits E = [(E,E)] splits (T l x r) = [(u, T v x r) | (u,v) <- splits l] ++ [(T l x u, v) | (u,v) <- splits r] instance (Eq k, Num k, Ord a) => Coalgebra k (YSymF a) where counit = unwrap . linear counit' where counit' (YSymF E) = 1; counit' (YSymF (T _ _ _)) = 0 comult = linear comult' where comult' (YSymF t) = sumv [return (YSymF u, YSymF v) | (u,v) <- splits t] -- using sumv rather than sum to avoid requiring Show a -- so again this is a kind of deconcatenation coproduct multisplits 1 t = [ [t] ] multisplits 2 t = [ [u,v] | (u,v) <- splits t ] multisplits n t = [ u:ws | (u,v) <- splits t, ws <- multisplits (n-1) v ] graft [t] E = t graft ts (T l x r) = let (ls,rs) = splitAt (leafcount l) ts in T (graft ls l) x (graft rs r) instance (Eq k, Num k, Ord a) => Algebra k (YSymF a) where unit x = x *> return (YSymF E) mult = linear mult' where mult' (YSymF t, YSymF u) = sumv [return (YSymF (graft ts u)) | ts <- multisplits (leafcount u) t] -- using sumv rather than sum to avoid requiring Show a instance (Eq k, Num k, Ord a) => Bialgebra k (YSymF a) where {} instance (Eq k, Num k, Ord a) => HopfAlgebra k (YSymF a) where antipode = linear antipode' where antipode' (YSymF E) = return (YSymF E) antipode' x = (negatev . mult . (id `tf` antipode) . removeTerm (YSymF E,x) . comult . return) x -- |An alternative \"monomial\" basis for (the dual of) the Loday-Ronco Hopf algebra of binary trees, YSym. newtype YSymM = YSymM (PBT ()) deriving (Eq, Ord) instance Show YSymM where show (YSymM t) = "M(" ++ show t ++ ")" -- |Construct the element of YSym in the monomial basis indexed by the given tree ysymM :: PBT () -> Vect Q YSymM ysymM t = return (YSymM t) -- |List all trees with the given number of nodes trees :: Int -> [PBT ()] trees 0 = [E] trees n = [T l () r | i <- [0..n-1], l <- trees (n-1-i), r <- trees i] -- |The covering relation for the Tamari partial order on binary trees tamariCovers :: PBT a -> [PBT a] tamariCovers E = [] tamariCovers (T t@(T u x v) y w) = [T t' y w | t' <- tamariCovers t] ++ [T t y w' | w' <- tamariCovers w] ++ [T u y (T v x w)] -- Note that this preserves the descending property, and hence the bijection with permutations -- If we were to swap x and y, we would preserve the binary search tree property instead (if our trees had it) tamariCovers (T E x u) = [T E x u' | u' <- tamariCovers u] -- |The up-set of a binary tree in the Tamari partial order tamariUpSet :: Ord a => PBT a -> [PBT a] tamariUpSet t = upSet' [] [t] where upSet' interior boundary = if null boundary then interior else let interior' = setUnionAsc interior boundary boundary' = toSet $ concatMap tamariCovers boundary in upSet' interior' boundary' -- tamariOrder1 u v = v `elem` upSet u -- |The Tamari partial order on binary trees. -- This is only defined between trees of the same size (number of nodes). -- The result between trees of different sizes is undefined (we don't check). tamariOrder :: PBT a -> PBT a -> Bool tamariOrder u v = weakOrder (minPerm u) (minPerm v) -- It should be possible to unpack this to be a statement purely about trees, but probably not worth it -- |Convert an element of YSym represented in the monomial basis to the fundamental basis ysymMtoF :: (Eq k, Num k) => Vect k YSymM -> Vect k (YSymF ()) ysymMtoF = linear ysymMtoF' where ysymMtoF' (YSymM t) = sumv [mu (set,po) t s *> return (YSymF s) | s <- set] where po = tamariOrder set = tamariUpSet t -- [s | s <- trees (nodecount t), t `tamariOrder` s] -- |Convert an element of YSym represented in the fundamental basis to the monomial basis ysymFtoM :: (Eq k, Num k) => Vect k (YSymF ()) -> Vect k YSymM ysymFtoM = linear ysymFtoM' where ysymFtoM' (YSymF t) = sumv [return (YSymM s) | s <- tamariUpSet t] -- sumv [return (YSymM s) | s <- trees (nodecount t), t `tamariOrder` s] instance (Eq k, Num k) => Algebra k YSymM where unit x = x *> return (YSymM E) mult = ysymFtoM . mult . (ysymMtoF `tf` ysymMtoF) instance (Eq k, Num k) => Coalgebra k YSymM where counit = unwrap . linear counit' where counit' (YSymM E) = 1; counit' (YSymM (T _ _ _)) = 0 -- comult = (ysymFtoM `tf` ysymFtoM) . comult . ysymMtoF comult = linear comult' where comult' (YSymM t) = sumv [return (YSymM r, YSymM s) | (rs,ss) <- deconcatenations (underDecomposition t), let r = foldl under E rs, let s = foldl under E ss] instance (Eq k, Num k) => Bialgebra k YSymM where {} instance (Eq k, Num k) => HopfAlgebra k YSymM where antipode = ysymFtoM . antipode . ysymMtoF -- QSYM: QUASI-SYMMETRIC FUNCTIONS -- The following is the Hopf algebra QSym of quasi-symmetric functions -- using the monomial and fundamental bases (indexed by compositions) -- compositions in ascending order -- might be better to use bfs to get length order -- |List the compositions of an integer n. For example, the compositions of 4 are [[1,1,1,1],[1,1,2],[1,2,1],[1,3],[2,1,1],[2,2],[3,1],[4]] compositions :: Int -> [[Int]] compositions 0 = [[]] compositions n = [i:is | i <- [1..n], is <- compositions (n-i)] -- can retrieve subsets of [1..n-1] from compositions n as follows -- > map (tail . scanl (+) 0) (map init $ compositions 4) -- [[],[3],[2],[2,3],[1],[1,3],[1,2],[1,2,3]] -- quasi shuffles of two compositions quasiShuffles :: [Int] -> [Int] -> [[Int]] quasiShuffles (x:xs) (y:ys) = map (x:) (quasiShuffles xs (y:ys)) ++ map ((x+y):) (quasiShuffles xs ys) ++ map (y:) (quasiShuffles (x:xs) ys) quasiShuffles xs [] = [xs] quasiShuffles [] ys = [ys] -- |A type for the monomial basis for the quasi-symmetric functions, indexed by compositions. newtype QSymM = QSymM [Int] deriving (Eq) instance Ord QSymM where compare (QSymM xs) (QSymM ys) = compare (sum xs, xs) (sum ys, ys) instance Show QSymM where show (QSymM xs) = "M " ++ show xs -- |Construct the element of QSym in the monomial basis indexed by the given composition qsymM :: [Int] -> Vect Q QSymM qsymM xs | all (>0) xs = return (QSymM xs) | otherwise = error "qsymM: not a composition" instance (Eq k, Num k) => Algebra k QSymM where unit x = x *> return (QSymM []) mult = linear mult' where mult' (QSymM alpha, QSymM beta) = sumv [return (QSymM gamma) | gamma <- quasiShuffles alpha beta] instance (Eq k, Num k) => Coalgebra k QSymM where counit = unwrap . linear counit' where counit' (QSymM alpha) = if null alpha then 1 else 0 comult = linear comult' where comult' (QSymM gamma) = sumv [return (QSymM alpha, QSymM beta) | (alpha,beta) <- deconcatenations gamma] instance (Eq k, Num k) => Bialgebra k QSymM where {} instance (Eq k, Num k) => HopfAlgebra k QSymM where antipode = linear antipode' where antipode' (QSymM alpha) = (-1)^length alpha * sumv [return (QSymM beta) | beta <- coarsenings (reverse alpha)] -- antipode' (QSymM alpha) = (-1)^length alpha * sumv [return (QSymM (reverse beta)) | beta <- coarsenings alpha] coarsenings (x1:x2:xs) = map (x1:) (coarsenings (x2:xs)) ++ coarsenings ((x1+x2):xs) coarsenings xs = [xs] -- for xs a singleton or null refinements (x:xs) = [y++ys | y <- compositions x, ys <- refinements xs] refinements [] = [[]] -- |A type for the fundamental basis for the quasi-symmetric functions, indexed by compositions. newtype QSymF = QSymF [Int] deriving (Eq) instance Ord QSymF where compare (QSymF xs) (QSymF ys) = compare (sum xs, xs) (sum ys, ys) instance Show QSymF where show (QSymF xs) = "F " ++ show xs -- |Construct the element of QSym in the fundamental basis indexed by the given composition qsymF :: [Int] -> Vect Q QSymF qsymF xs | all (>0) xs = return (QSymF xs) | otherwise = error "qsymF: not a composition" -- |Convert an element of QSym represented in the monomial basis to the fundamental basis qsymMtoF :: (Eq k, Num k) => Vect k QSymM -> Vect k QSymF qsymMtoF = linear qsymMtoF' where qsymMtoF' (QSymM alpha) = sumv [(-1) ^ (length beta - length alpha) *> return (QSymF beta) | beta <- refinements alpha] -- |Convert an element of QSym represented in the fundamental basis to the monomial basis qsymFtoM :: (Eq k, Num k) => Vect k QSymF -> Vect k QSymM qsymFtoM = linear qsymFtoM' where qsymFtoM' (QSymF alpha) = sumv [return (QSymM beta) | beta <- refinements alpha] -- ie beta <- up-set of alpha instance (Eq k, Num k) => Algebra k QSymF where unit x = x *> return (QSymF []) mult = qsymMtoF . mult . (qsymFtoM `tf` qsymFtoM) instance (Eq k, Num k) => Coalgebra k QSymF where counit = unwrap . linear counit' where counit' (QSymF xs) = if null xs then 1 else 0 comult = (qsymMtoF `tf` qsymMtoF) . comult . qsymFtoM instance (Eq k, Num k) => Bialgebra k QSymF where {} instance (Eq k, Num k) => HopfAlgebra k QSymF where antipode = qsymMtoF . antipode . qsymFtoM -- QUASI-SYMMETRIC POLYNOMIALS -- the above induces Hopf algebra structure on quasi-symmetric functions via -- m_alpha -> sum [product (zipWith (^) (map x_ is) alpha | is <- combinationsOf k [] ] where k = length alpha xvars n = [glexvar ("x" ++ show i) | i <- [1..n] ] -- |@qsymPoly n is@ is the quasi-symmetric polynomial in n variables for the indices is. (This corresponds to the -- monomial basis for QSym.) For example, qsymPoly 3 [2,1] == x1^2*x2+x1^2*x3+x2^2*x3. qsymPoly :: Int -> [Int] -> GlexPoly Q String qsymPoly n is = sum [product (zipWith (^) xs' is) | xs' <- combinationsOf r xs] where xs = xvars n r = length is -- SYM, THE HOPF ALGEBRA OF SYMMETRIC FUNCTIONS -- |A type for the monomial basis for Sym, the Hopf algebra of symmetric functions, indexed by integer partitions newtype SymM = SymM [Int] deriving (Eq,Show) instance Ord SymM where compare (SymM xs) (SymM ys) = compare (sum xs, ys) (sum ys, xs) -- note the order reversal in snd -- |Construct the element of Sym in the monomial basis indexed by the given integer partition symM :: [Int] -> Vect Q SymM symM xs | all (>0) xs = return (SymM $ sortDesc xs) | otherwise = error "symM: not a partition" instance (Eq k, Num k) => Algebra k SymM where unit x = x *> return (SymM []) mult = linear mult' where mult' (SymM lambda, SymM mu) = sumv [return (SymM nu) | nu <- symMult lambda mu] -- multisetPermutations = toSet . L.permutations -- compositionsFromPartition2 = foldl (\xss ys -> concatMap (shuffles ys) xss) [[]] . L.group -- compositionsFromPartition2 = foldl (\ls r -> concat [shuffles l r | l <- ls]) [[]] . L.group -- The partition must be in either ascending or descending order (so that L.group does as expected) compositionsFromPartition = foldr (\l rs -> concatMap (shuffles l) rs) [[]] . L.group -- In effect, we multiply in Sym by converting to QSym, multiplying there, and converting back. -- It would be nice to find a more direct method. symMult xs ys = filter isWeaklyDecreasing $ concat [quasiShuffles xs' ys' | xs' <- compositionsFromPartition xs, ys' <- compositionsFromPartition ys] instance (Eq k, Num k) => Coalgebra k SymM where counit = unwrap . linear counit' where counit' (SymM lambda) = if null lambda then 1 else 0 comult = linear comult' where comult' (SymM lambda) = sumv [return (SymM mu, SymM nu) | mu <- toSet (powersetdfs lambda), let nu = diffDesc lambda mu] instance (Eq k, Num k) => Bialgebra k SymM where {} instance (Eq k, Num k) => HopfAlgebra k SymM where antipode = linear antipode' where antipode' (SymM []) = return (SymM []) antipode' x = (negatev . mult . (id `tf` antipode) . removeTerm (SymM [],x) . comult . return) x -- |The elementary basis for Sym, the Hopf algebra of symmetric functions. Defined informally as -- > symE [n] = symM (replicate n 1) -- > symE lambda = product [symE [p] | p <- lambda] newtype SymE = SymE [Int] deriving (Eq,Ord,Show) symE :: [Int] -> Vect Q SymE symE xs | all (>0) xs = return (SymE $ sortDesc xs) | otherwise = error "symE: not a partition" instance (Eq k, Num k) => Algebra k SymE where unit x = x *> return (SymE []) mult = linear (\(SymE lambda, SymE mu) -> return $ SymE $ multisetSumDesc lambda mu) instance (Eq k, Num k) => Coalgebra k SymE where counit = unwrap . linear counit' where counit' (SymE lambda) = if null lambda then 1 else 0 comult = linear comult' where comult' (SymE [n]) = sumv [return (e i, e (n-i)) | i <- [0..n] ] comult' (SymE lambda) = product [comult' (SymE [n]) | n <- lambda] e 0 = SymE [] e i = SymE [i] instance (Eq k, Num k) => Bialgebra k SymE where {} -- |Convert from the elementary to the monomial basis of Sym symEtoM :: (Eq k, Num k) => Vect k SymE -> Vect k SymM symEtoM = linear symEtoM' where symEtoM' (SymE [n]) = return (SymM (replicate n 1)) symEtoM' (SymE lambda) = product [symEtoM' (SymE [p]) | p <- lambda] -- |The complete basis for Sym, the Hopf algebra of symmetric functions. Defined informally as -- > symH [n] = sum [symM lambda | lambda <- integerPartitions n] -- == all monomials of weight n -- > symH lambda = product [symH [p] | p <- lambda] newtype SymH = SymH [Int] deriving (Eq,Ord,Show) symH :: [Int] -> Vect Q SymH symH xs | all (>0) xs = return (SymH $ sortDesc xs) | otherwise = error "symH: not a partition" instance (Eq k, Num k) => Algebra k SymH where unit x = x *> return (SymH []) mult = linear (\(SymH lambda, SymH mu) -> return $ SymH $ multisetSumDesc lambda mu) instance (Eq k, Num k) => Coalgebra k SymH where counit = unwrap . linear counit' where counit' (SymH lambda) = if null lambda then 1 else 0 comult = linear comult' where comult' (SymH [n]) = sumv [return (h i, h (n-i)) | i <- [0..n] ] comult' (SymH lambda) = product [comult' (SymH [n]) | n <- lambda] h 0 = SymH [] h i = SymH [i] instance (Eq k, Num k) => Bialgebra k SymH where {} -- |Convert from the complete to the monomial basis of Sym symHtoM :: (Eq k, Num k) => Vect k SymH -> Vect k SymM symHtoM = linear symHtoM' where symHtoM' (SymH [n]) = sumv [return (SymM mu) | mu <- integerPartitions n] symHtoM' (SymH lambda) = product [symHtoM' (SymH [p]) | p <- lambda] -- NSYM, THE HOPF ALGEBRA OF NON-COMMUTATIVE SYMMETRIC FUNCTIONS -- |A basis for NSym, the Hopf algebra of non-commutative symmetric functions, indexed by compositions newtype NSym = NSym [Int] deriving (Eq,Ord,Show) nsym :: [Int] -> Vect Q NSym nsym xs = return (NSym xs) nsym xs | all (>0) xs = return (NSym xs) | otherwise = error "nsym: not a composition" instance (Eq k, Num k) => Algebra k NSym where unit x = x *> return (NSym []) mult = linear mult' where mult' (NSym xs, NSym ys) = return $ NSym $ xs ++ ys instance (Eq k, Num k) => Coalgebra k NSym where counit = unwrap . linear counit' where counit' (NSym zs) = if null zs then 1 else 0 comult = linear comult' where comult' (NSym [n]) = sumv [return (z i, z (n-i)) | i <- [0..n] ] comult' (NSym zs) = product [comult' (NSym [n]) | n <- zs] z 0 = NSym [] z i = NSym [i] instance (Eq k, Num k) => Bialgebra k NSym where {} -- Hazewinkel et al p233 instance (Eq k, Num k) => HopfAlgebra k NSym where antipode = linear antipode' where antipode' (NSym alpha) = sumv [(-1)^length beta *> return (NSym beta) | beta <- refinements (reverse alpha)] -- MAPS BETWEEN (POSETS AND) HOPF ALGEBRAS -- A descending tree is one in which a child is always less than a parent. descendingTree [] = E descendingTree [x] = T E x E descendingTree xs = T l x r where x = maximum xs (ls,_:rs) = L.break (== x) xs l = descendingTree ls r = descendingTree rs -- This is a bijection from permutations to "ordered trees". -- It is order-preserving on trees with the same nodecount. -- We can recover the permutation by reading the node labels in infix order. -- This is the map called lambda in Loday.pdf -- |Given a permutation p of [1..n], we can construct a tree (the descending tree of p) as follows: -- -- * Split the permutation as p = ls ++ [n] ++ rs -- -- * Place n at the root of the tree, and recursively place the descending trees of ls and rs as the left and right children of the root -- -- * To bottom out the recursion, the descending tree of the empty permutation is of course the empty tree -- -- This map between bases SSymF -> YSymF turns out to induce a morphism of Hopf algebras. descendingTreeMap :: (Eq k, Num k) => Vect k SSymF -> Vect k (YSymF ()) descendingTreeMap = nf . fmap (YSymF . shape . descendingTree') where descendingTree' (SSymF xs) = descendingTree xs -- This is the map called Lambda in Loday.pdf, or tau in MSym.pdf -- It is an algebra morphism. -- One of the ideas in the MSym paper is to look at the intermediate result (fmap descendingTree' x), -- which is an "ordered tree", and consider the map as factored through this -- The map is surjective but not injective. The fibers tau^-1(t) are intervals in the weak order on permutations -- "inverse" for descendingTree -- These are the maps called gamma in Loday.pdf minPerm t = minPerm' (lrCountTree t) where minPerm' E = [] minPerm' (T l (lc,rc) r) = minPerm' l ++ [lc+rc+1] ++ map (+lc) (minPerm' r) maxPerm t = maxPerm' (lrCountTree t) where maxPerm' E = [] maxPerm' (T l (lc,rc) r) = map (+rc) (maxPerm' l) ++ [lc+rc+1] ++ maxPerm' r -- The composition of [1..n] obtained by treating each left-facing leaf as a cut -- Specifically, we visit the nodes in infix order, cutting after a node if it does not have an E as its right child -- This is the map called L in Loday.pdf leftLeafComposition E = [] leftLeafComposition t = cuts $ tail $ leftLeafs t where leftLeafs (T l x E) = leftLeafs l ++ [False] leftLeafs (T l x r) = leftLeafs l ++ leftLeafs r leftLeafs E = [True] cuts bs = case break id bs of (ls,r:rs) -> (length ls + 1) : cuts rs (ls,[]) -> [length ls] leftLeafComposition' (YSymF t) = QSymF (leftLeafComposition t) -- |A Hopf algebra morphism from YSymF to QSymF leftLeafCompositionMap :: (Eq k, Num k) => Vect k (YSymF a) -> Vect k QSymF leftLeafCompositionMap = nf . fmap leftLeafComposition' -- The descent set of a permutation is [i | x_i > x_i+1], where we start the indexing from 1 descents [] = [] descents xs = map (+1) $ L.elemIndices True $ zipWith (>) xs (tail xs) -- The composition of [1..n] obtained by treating each descent as a cut descentComposition [] = [] descentComposition xs = dc $ zipWith (>) xs (tail xs) ++ [False] where dc bs = case break id bs of (ls,r:rs) -> (length ls + 1) : dc rs (ls,[]) -> [length ls] -- |Given a permutation of [1..n], its descents are those positions where the next number is less than the previous number. -- For example, the permutation [2,3,5,1,6,4] has descents from 5 to 1 and from 6 to 4. The descents can be regarded as cutting -- the permutation sequence into segments - 235-16-4 - and by counting the lengths of the segments, we get a composition 3+2+1. -- This map between bases SSymF -> QSymF turns out to induce a morphism of Hopf algebras. descentMap :: (Eq k, Num k) => Vect k SSymF -> Vect k QSymF descentMap = nf . fmap (\(SSymF xs) -> QSymF (descentComposition xs)) -- descentMap == leftLeafCompositionMap . descendingTreeMap underComposition (QSymF ps) = foldr under (SSymF []) [SSymF [1..p] | p <- ps] where under (SSymF xs) (SSymF ys) = let q = length ys zs = map (+q) xs ++ ys -- so it has a global descent at the split in SSymF zs -- This is a poset morphism (indeed, it forms a Galois connection with descentComposition) -- but it does not extend to a Hopf algebra morphism. -- (It does extend to a coalgebra morphism.) -- (It is picking the maximum permutation having a given descent composition, -- so there's an element of arbitrariness to it.) -- This is the map called Z (Zeta?) in Loday.pdf {- -- This is O(n^2), whereas an O(n) implementation should be possible -- Also, we would really like the associated composition (obtained by treating each global descent as a cut)? globalDescents xs = globalDescents' 0 [] xs where globalDescents' i ls (r:rs) = (if minimum (infinity:ls) > maximum (0:r:rs) then [i] else []) ++ globalDescents' (i+1) (r:ls) rs globalDescents' n _ [] = [n] infinity = maxBound :: Int -- The idea is that this leads to a map from SSymM to QSymM globalDescentComposition [] = [] globalDescentComposition (x:xs) = globalDescents' 1 x xs where globalDescents' i minl (r:rs) = if minl > maximum (r:rs) then i : globalDescents' 1 r rs else globalDescents' (i+1) r rs globalDescents' i _ [] = [i] globalDescentMap :: (Eq k, Num k) => Vect k SSymM -> Vect k QSymM globalDescentMap = nf . fmap (\(SSymM xs) -> QSymM (globalDescentComposition xs)) -} -- A multiplication operation on trees -- (Connected with their being cofree) -- (intended to be used as infix) under E t = t under (T l x r) t = T l x (under r t) isUnderIrreducible (T l x E) = True isUnderIrreducible _ = False underDecomposition (T l x r) = T l x E : underDecomposition r underDecomposition E = [] -- GHC7.4.1 doesn't like the following type signature - a bug. -- ysymmToSh :: (Eq k, Num k) => Vect k (YSymM) => Vect k (Shuffle (PBT ())) ysymmToSh = fmap ysymmToSh' where ysymmToSh' (YSymM t) = Sh (underDecomposition t) -- This is a coalgebra morphism (but not an algebra morphism) -- It shows that YSym is co-free {- -- This one not working yet - perhaps it needs an nf, or to go via S/YSymF, or ... ssymmToSh = nf . fmap ssymmToSh' where ssymmToSh' (SSymM xs) = (Sh . underDecomposition . shape . descendingTree) xs -} -- |The injection of Sym into QSym (defined over the monomial basis) symToQSymM :: (Eq k, Num k) => Vect k SymM -> Vect k QSymM symToQSymM = linear symToQSymM' where symToQSymM' (SymM ps) = sumv [return (QSymM c) | c <- compositionsFromPartition ps] -- We could equally well send NSym -> SymE, since the algebra and coalgebra definitions for SymE and SymH are exactly analogous. -- However, NSym -> SymH is more natural, since it is consistent with the duality pairings below. -- eg Hazewinkel 238ff -- (Why do SymE and SymH have the same definitions? They're not dual bases. It's because of the Wronski relations.) -- |A surjection of NSym onto Sym (defined over the complete basis) nsymToSymH :: (Eq k, Num k) => Vect k NSym -> Vect k SymH nsymToSymH = linear nsymToSym' where nsymToSym' (NSym zs) = return (SymH $ sortDesc zs) -- The Hopf algebra morphism NSym -> Sym factors through NSym -> SSym -> YSym -> Sym (contained in QSym) -- (?? This map NSym -> SSym is the dual of the descent map SSym -> QSym ??) -- (Loday.pdf, p30) -- (See also Hazewinkel p267-9) nsymToSSym = linear nsymToSSym' where nsymToSSym' (NSym xs) = product [return (SSymF [1..n]) | n <- xs] -- |A duality pairing between the complete and monomial bases of Sym, showing that Sym is self-dual. instance (Eq k, Num k) => HasPairing k SymH SymM where pairing = linear pairing' where pairing' (SymH alpha, SymM beta) = delta alpha beta -- Kronecker delta -- Hazewinkel p178 -- Actually to show duality you would need to show that the map SymH -> SymM*, v -> <v,.> is onto -- |A duality pairing between NSym and QSymM (monomial basis), showing that NSym and QSym are dual. instance (Eq k, Num k) => HasPairing k NSym QSymM where pairing = linear pairing' where pairing' (NSym alpha, QSymM beta) = delta alpha beta -- Kronecker delta -- Hazewinkel p236-7 -- Actually to show duality you would need to show that the map NSym -> QSymM*, v -> <v,.> is onto