Module

# Data.SymmetricGroup

Package
purescript-symmetric-groups
Repository
hdgarrood/purescript-symmetric-groups

One important example of a group which arises very often in group theory and its applications is the symmetric group on some set X, which is the group of bijective functions from X to itself. The group operation here is function composition and the identity element is the identity function. (If I've lost you by this point, the first few of chapters of the PureScript numeric hierarchy guide should help.)

We often restrict our attention to just the symmetric groups on finite sets, since they are a little easier to deal with. Since we are dealing with a finite set, we might as well label the elements of the set from 1 up to n (where n is the size of the set); the group structure will be the same no matter how we have labelled the elements of the underlying set. The symmetric group on a set X can be written S(X); if X is the set {1,2,...,n} we often abuse notation and write the group S(X) as just S(n).

Some vocabulary first: if f is a function from the set X to itself, we say that x is a fixed point of f if and only if f(x) = x. We sometimes say that "f fixes x" instead, since this is less of a mouthful than "x is a fixed point of f".

When attempting to represent the group S(n) in PureScript, one approach might be to define a type constructor of kind `Type -> Type`, where the argument is a type-level natural number representing n. This approach quickly runs into a problem, though, which is that there is no (ergonomic) type for natural numbers less than or equal to a certain number. For example, Idris has `Fin n`, which is the type of integers between 0 and n-1. Of course, it is possible to define a similar type in PureScript, but without dependent types it will not be nearly as comfortable to use as Idris' `Fin`. However, we want to have some way of converting elements of this set to standard PureScript functions*, but without an ergonomic type `Fin` we will need to use `Partial` constraints (yuck).

We also would like to be able to do things like embed S(n) into S(n+1) without too much effort (that is, without having to convert between two different types): note that every permutation f on a set of n elements can be extended to a permutation f' on a set of n+1 elements just by saying that f' fixes n+1 and does the same as f for everything else, i.e. f'(k) = f(k) for all k ≤ n.

There is a simple trick we can use to address this. We say that a permutation is finitary if and only if it fixes all but finitely many points. For example, the permutation on the set of natural numbers which swaps 1 and 2 and fixes everything else is finitary; the permutation which swaps every even number with the odd number preceding it is not. It turns out that the product of two finitary permutations is itself finitary, and also that the inverse of a finitary permutation is finitary (exercise!). Therefore the set of finitary permutations on a set X is a group (in fact a subgroup of S(X)), which we will refer to as FS(X).

The design adopted by this library is to define a type representing the group FS(ℕ) of finitary permutations on the natural numbers. Then, for any natural number n, there is a natural embedding of S(n) into FS(ℕ) by just fixing everything greater than or equal to n; in the same way there is a natural embedding of S(k) into S(n) (within FS(ℕ)) whenever k < n.

Perhaps surprisingly, Cayley's Theorem tells us that any finite group is isomorphic to a subgroup of S(n), so we can in fact represent any finite group at all using this library (although this fact might mostly just be a curiosity).

*note: we don't use standard PureScript functions directly, as it is not an efficient representation for many of the operations we would like to be able to do, and also it is harder to guarantee that the function in question is bijective.

### #SymSource

``newtype Sym``

The type `Sym` represents the group FS(ℕ) of finitary permutations of the set of natural numbers. Values of this type cannot be constructed from functions of type `Int -> Int`, because we cannot easily verify that these are bijective. Instead, use `fromCycle` or `fromCycles`.

The runtime representation of a value of this type is an array with k elements, where k is the largest number which is not a fixed point of f; if k is very large this may be a problem.

#### Instances

• `Eq Sym`
• `Ord Sym`
• `Show Sym`
• `Semigroup Sym`
• `Monoid Sym`
• `Group Sym`

### #fromCycleSource

``fromCycle :: List Int -> Sym``

Generate a permutation given a single cycle. If the given list includes nonpositive or duplicate elements they will be ignored.

``````fromCycle (1:2:Nil) == fromCycle (1:2:1:Nil)
fromCycle (1:2:Nil) == fromCycle (1:2:0:Nil)
fromCycle (1:2:Nil) == fromCycle (1:2:(-1):Nil)
``````

### #fromCyclesSource

``fromCycles :: List (List Int) -> Sym``

Construct a permutation from a list of cycles.

``````f = asFunction (fromCycles ((1 : 3 : Nil) : (2 : 4 : Nil) : Nil))

f 1 == 3
f 2 == 4
f 3 == 1
f 4 == 2
``````

### #asCyclesSource

``asCycles :: Sym -> List (List Int)``

Represent a permutation as a list of cycles. Note that `asCycles <<< fromCycles` is not equal to `id`, because in general there are lots of different ways to write any given permutation as a product of cycles. However, `fromCycles <<< asCycles` is equal to `id`.

``````asCycles (fromCycles ((1 : 2 : 3 : Nil) : Nil))
== (1 : 2 : 3 : Nil) : Nil
``````

### #asFunctionSource

``asFunction :: Sym -> Int -> Int``

Convert a finitary permutation into a regular function. The resulting function fixes all negative numbers.

``````f = asFunction (fromCycle (1 : 2 : 3 : Nil))

f 1 == 2
f 2 == 3
f 3 == 1
f 4 == 4
f (-1) == (-1)
``````

### #cycleOfSource

``cycleOf :: Sym -> Int -> List Int``

Compute the cycle containing a given point.

``````f = fromCycles ((1 : 3 : Nil) : (2 : 4 : Nil) : Nil)

cycleOf f 1 == 1 : 3 : Nil
cycleOf f 2 == 2 : 4 : Nil
cycleOf f 5 == Nil
``````

### #minNSource

``minN :: Sym -> Int``

The smallest natural number N for which the given permutation fixes all numbers greater than or equal to N.

``````minN (fromCycle (1:2:Nil)) == 3
``````

### #inversionsSource

``inversions :: Sym -> Array (Tuple Int Int)``

The inversions of a permutation, i.e. for a permutation f, this function returns all pairs of points x, y such that `x < y` and `f x > f y`.

The parity of the number of inversions of a permutation is equal to the parity of the permutation itself.

### #paritySource

``parity :: Sym -> Parity``

The parity of a permutation. All permutations can be expressed as products of 2-cycles: for example (1 2 3) can be written as (2 3)(1 3). The parity of a permutation is defined as the parity of the number of 2-cycles when it is written as a product of 2-cycles, so e.g. (1 2 3) is even.

This function is a group homomorphism from the group Sym to the additive group of the field of two elements (here represented by the `Parity` type); that is,

``````parity f + parity g = parity (f <> g)
``````

holds for all permutations `f`, `g`.

The parity of a permutation is sometimes also called the "sign" or "signature".

### #orderSource

``order :: Sym -> Int``

The order of a permutation; the smallest positive integer n such that s^n is the identity. Restricting `Sym` to finitary permutations ensures that this is always finite.

### #permutationsSource

``permutations :: Int -> Array (Array Int)``

Returns all permutations of the array with elements from 1 up to n.

### #symmetricSource

``symmetric :: Int -> Array Sym``

`symmetric n` gives you every element of the group S(n) in an array.

### #alternatingSource

``alternating :: Int -> Array Sym``

`alternating n` gives you every element of the group A(n) -- that is, the subgroup of S(n) given by the even permutations -- in an array.

### #trivialSubgroupSource

``trivialSubgroup :: forall a. Ord a => Group a => Set a``

The set containing just the identity element of a group (i.e. `mempty`).

### #subgroupSource

``subgroup :: Set Sym -> Set Sym``

Given a set of permutations, form the subgroup generated by that set. This function is idempotent, that is, `subgroup (subgroup a) = subgroup a`.

### #actLeftSource

``actLeft :: forall a. Ord a => Group a => a -> Set a -> Set a``

If `h` is a subgroup, then `actLeft s h` gives the coset formed by applying `s` to each element of `h` on the left.

### #cosetsSource

``cosets :: forall a. Ord a => Group a => Set a -> Set a -> Set (Set a)``

If `h` is a subgroup of `g`, then `cosets h g` gives the set of cosets of `h` in `g`. Otherwise, the behaviour of this function is undefined.

## Re-exports from Data.Int

### #ParitySource

``data Parity``

A type for describing whether an integer is even or odd.

The `Ord` instance considers `Even` to be less than `Odd`.

The `Semiring` instance allows you to ask about the parity of the results of arithmetical operations, given only the parities of the inputs. For example, the sum of an odd number and an even number is odd, so `Odd + Even == Odd`. This also works for multiplication, eg. the product of two odd numbers is odd, and therefore `Odd * Odd == Odd`.

More generally, we have that

``````parity x + parity y == parity (x + y)
parity x * parity y == parity (x * y)
``````

for any integers `x`, `y`. (A mathematician would say that `parity` is a ring homomorphism.)

After defining addition and multiplication on `Parity` in this way, the `Semiring` laws now force us to choose `zero = Even` and `one = Odd`. This `Semiring` instance actually turns out to be a `Field`.

#### Constructors

• `Even`
• `Odd`

#### Instances

• `Eq Parity`
• `Ord Parity`
• `Show Parity`
• `Bounded Parity`
• `Semiring Parity`
• `Ring Parity`
• `CommutativeRing Parity`
• `EuclideanRing Parity`
• `DivisionRing Parity`
• `Field Parity`