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 a => Eq (Permutation a)

(Ord a, Show 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)

Mon (Permutation Int)

(Ord a, Show 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).

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

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

(~^) :: (Ord a, Show a) => Permutation a -> Permutation a -> Permutation a

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

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

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

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

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

minsupp :: Permutation c -> c

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

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

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

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

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

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

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

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

isSubgp :: (Num a, Ord 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, Show a) => Permutation a -> Bool

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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