Math.Algebra.Group.PermutationGroup

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.

Documentation

rotateL :: [a] -> [a]

newtype Permutation a

A type for permutations, considered as functions or actions which can be performed on an underlying set.

Constructors

P (Map a a)

Instances

(Eq k, Num k) => HopfAlgebra k (Permutation Int)

(Eq k, Num k) => Bialgebra k (Permutation Int)

(Eq k, Num k) => Coalgebra k (Permutation Int)

(Eq k, Num k) => Algebra k (Permutation Int)

(Eq k, Num k) => Module k (Permutation Int) Int

(Eq k, Num k) => Module k (Permutation Int) [Int]

Eq a => Eq (Permutation a)

Ord a => Num (Permutation a)

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.

Ord a => Ord (Permutation a)

(Ord a, Show a) => Show (Permutation a)

Ord a => HasInverses (Permutation a)

The HasInverses instance is what enables us to write g^-1 for the inverse of a group element.

HasInverses (GroupAlgebra Q)

Note that the inverse of a group algebra element can only be efficiently calculated if the group generated by the non-zero terms is very small (eg <100 elements).

fmapP :: Ord a => (t -> a) -> Permutation t -> Permutation a

p :: Ord a => [[a]] -> Permutation a

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.

fromPairs :: Ord a => [(a, a)] -> Permutation a

fromPairs' :: Ord a => [(a, a)] -> Permutation a

toPairs :: Permutation a -> [(a, a)]

fromList :: Ord a => [a] -> Permutation a

supp :: Permutation a -> [a]

(.^) :: Ord a => a -> Permutation a -> a

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]

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.

fromCycles :: Ord a => [[a]] -> Permutation a

toCycles :: Ord a => Permutation a -> [[a]]

cycleOf :: Ord a => Permutation a -> a -> [a]

parity :: Ord a => Permutation a -> Int

sign :: (Ord a1, Num a) => Permutation a1 -> a

orderElt :: Ord a => Permutation a -> Int

(~^) :: Ord a => Permutation a -> Permutation a -> Permutation a infix 8

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.

comm :: (HasInverses a, Num a) => a -> a -> a

closureS :: Ord a => [a] -> [a -> a] -> Set a

closure :: Ord a => [a] -> [a -> a] -> [a]

orbit :: Ord a => (a -> t -> a) -> a -> [t] -> [a]

(.^^) :: Ord a => a -> [Permutation a] -> [a]

x .^^ gs returns the orbit of the point or vertex x under the action of the gs

orbitP :: Ord a => [Permutation a] -> a -> [a]

orbitV :: Ord a => [Permutation a] -> a -> [a]

(-^^) :: Ord a => [a] -> [Permutation a] -> [[a]]

b -^^ gs returns the orbit of the block or edge b under the action of the gs

orbitB :: Ord a => [Permutation a] -> [a] -> [[a]]

orbitE :: Ord a => [Permutation a] -> [a] -> [[a]]

action :: Ord a => [a] -> (a -> a) -> Permutation a

orbits :: Ord a => [Permutation a] -> [[a]]

_C :: Integral a => a -> [Permutation a]

_C n returns generators for Cn, the cyclic group of order n

_D :: Integral a => a -> [Permutation a]

_D2 :: Integral a => a -> [Permutation a]

_S :: Integral a => a -> [Permutation a]

_S n returns generators for Sn, the symmetric group on [1..n]

_A :: Integral a => a -> [Permutation a]

_A n returns generators for An, the alternating group on [1..n]

dp :: (Ord a, Ord b) => [Permutation a] -> [Permutation b] -> [Permutation (Either a b)]

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)

wr :: (Ord t1, Ord t) => [Permutation t] -> [Permutation t1] -> [Permutation (t, t1)]

toSn :: (Ord a, Ord k, Num a, Enum a) => [Permutation k] -> [Permutation a]

fromDigits :: (Ord a, Num a) => Permutation [a] -> Permutation a

fromDigits' :: Num a => [a] -> a

fromBinary :: (Ord a, Num a) => Permutation [a] -> Permutation a

fromBinary' :: Num a => [a] -> a

elts :: (Num a, Ord a) => [a] -> [a]

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)

eltsS :: (Ord a, Num a) => [a] -> Set a

order :: (Num a, Ord a) => [a] -> Int

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)

isMember :: (Ord a, Num a) => [a] -> a -> Bool

minsupp :: Permutation c -> c

orderTGS :: (Ord a, Num a1) => [Permutation a] -> a1

eltsTGS :: Ord a => [Permutation a] -> [Permutation a]

tgsFromSgs :: Ord a => [Permutation a] -> [Permutation a]

orderSGS :: Ord a => [Permutation a] -> Integer

Given a strong generating set, return the order of the group it generates. Note that the SGS is assumed to be relative to the natural order of the points on which the group acts.

orderBSGS :: Ord a => ([a], [Permutation a]) -> Integer

Given a base and strong generating set, return the order of the group it generates.

gens :: (Ord a, Num a) => [a] -> [a]

(~^^) :: Ord a => Permutation a -> [Permutation a] -> [Permutation a]

conjClass :: Ord a => [Permutation a] -> Permutation a -> [Permutation a]

conjClassReps :: (Ord a, Show a) => [Permutation a] -> [(Permutation a, Int)]

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).

reduceGens :: (Ord a, Num a) => [a] -> [a]

isSubgp :: (Ord a, Num a) => [a] -> [a] -> Bool

subgps :: (Ord a, Show a) => [Permutation a] -> [[Permutation a]]

Return the subgroups of a group. Only suitable for use on small groups (eg < 100 elts)

isMinimal :: Ord a => Permutation a -> Bool

centralizer :: (Ord t, Num t) => [t] -> [t] -> [t]

centre :: (Ord t, Num t) => [t] -> [t]

normalizer :: Ord a => [Permutation a] -> [Permutation a] -> [Permutation a]

stabilizer :: Ord a => [Permutation a] -> a -> [Permutation a]

ptStab :: Ord a => [Permutation a] -> [a] -> [Permutation a]

setStab :: Ord a => [Permutation a] -> [a] -> [Permutation a]

normalClosure :: Ord a => [Permutation a] -> [Permutation a] -> [Permutation a]

commutatorGp :: Ord a => [Permutation a] -> [Permutation a] -> [Permutation a]

derivedSubgp :: Ord a => [Permutation a] -> [Permutation a]

(-*-) :: (Ord b, Num b) => [b] -> [b] -> [b]

(-*) :: (Ord a, Num a) => [a] -> a -> [a]

(*-) :: (Ord a, Num a) => a -> [a] -> [a]

isNormal :: (Ord a, Show a) => [Permutation a] -> [Permutation a] -> Bool

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>).

normalSubgps :: (Ord a, Show a) => [Permutation a] -> [[Permutation a]]

Return the normal subgroups of a group. Only suitable for use on small groups (eg < 100 elts)

isSimple :: (Show a, Ord a) => [Permutation a] -> Bool

cosets :: (Ord t, Num t) => [t] -> [t] -> [[t]]

quotientGp :: (Ord a, Show a) => [Permutation a] -> [Permutation a] -> [Permutation Int]

quotientGp gs ks returns <gs> / <ks>

(//) :: (Ord a, Show a) => [Permutation a] -> [Permutation a] -> [Permutation Int]

Synonym for quotientGp

(~~^) :: Ord a => [Permutation a] -> Permutation a -> [Permutation a]

conjugateSubgps :: Ord a => [Permutation a] -> [Permutation a] -> [[Permutation a]]

subgpAction :: (Ord a1, Ord a, Num a1, Enum a1) => [Permutation a] -> [Permutation a] -> [Permutation a1]

rrpr :: (Ord a, Num a) => [a] -> a -> Permutation a

rrpr' :: (Ord a, Num a) => [a] -> a -> Permutation a

permutationMatrix :: (Ord a, Num t) => [a] -> Permutation a -> [[t]]