Safe Haskell | None |
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Words in free groups (and free powers of cyclic groups)

- data Generator a
- type Word a = [Generator a]
- inverseGen :: Generator a -> Generator a
- inverseWord :: Word a -> Word a
- allWords :: Int -> Int -> [Word Int]
- allWordsNoInv :: Int -> Int -> [Word Int]
- multiplyFree :: Eq a => Word a -> Word a -> Word a
- reduceWordFree :: Eq a => Word a -> Word a
- countIdentityWordsFree :: Int -> Int -> Integer
- countWordReductionsFree :: Int -> Int -> Int -> Integer
- multiplyZ2 :: Eq a => Word a -> Word a -> Word a
- multiplyZ3 :: Eq a => Word a -> Word a -> Word a
- multiplyZm :: Eq a => Int -> Word a -> Word a -> Word a
- reduceWordZ2 :: Eq a => Word a -> Word a
- reduceWordZ3 :: Eq a => Word a -> Word a
- reduceWordZm :: Eq a => Int -> Word a -> Word a
- countIdentityWordsZ2 :: Int -> Int -> Integer
- countWordReductionsZ2 :: Int -> Int -> Int -> Integer
- countIdentityWordsZ3NoInv :: Int -> Int -> Integer

# Documentation

A generator of a (free) group

type Word a = [Generator a]Source

A *word*, describing (non-uniquely) an element of a group.
The identity element is represented (among others) by the empty word.

inverseGen :: Generator a -> Generator aSource

The inverse of a generator

inverseWord :: Word a -> Word aSource

The inverse of a word

Lists all words of the given length (total number will be `(2g)^n`

).
The numbering of the generators is `[1..g]`

.

Lists all words of the given length which do not contain inverse generators
(total number will be `g^n`

).
The numbering of the generators is `[1..g]`

.

# The free group on `g`

generators

multiplyFree :: Eq a => Word a -> Word a -> Word aSource

Multiplication of the free group (returns the reduced result). It is true for any two words w1 and w2 that

multiplyFree (reduceWordFree w1) (reduceWord w2) = multiplyFree w1 w2

reduceWordFree :: Eq a => Word a -> Word aSource

Reduces a word in a free group by repeatedly removing `x*x^(-1)`

and
`x^(-1)*x`

pairs. The set of *reduced words* forms the free group; the
multiplication is obtained by concatenation followed by reduction.

Counts the number of words of length `n`

which reduce to the identity element.

Generating function is `Gf_g(u) = \frac {2g-1} { g-1 + g \sqrt{ 1 - (8g-4)u^2 } }`

:: Int | g = number of generators in the free group |

-> Int | n = length of the unreduced word |

-> Int | k = length of the reduced word |

-> Integer |

Counts the number of words of length `n`

whose reduced form has length `k`

(clearly `n`

and `k`

must have the same parity for this to be nonzero):

countWordReductionsFree g n k == sum [ 1 | w <- allWords g n, k == length (reduceWordFree w) ]

# Free powers of cyclic groups

multiplyZm :: Eq a => Int -> Word a -> Word a -> Word aSource

Multiplication in free products of Zm's

reduceWordZ2 :: Eq a => Word a -> Word aSource

Reduces a word, where each generator `x`

satisfies the additional relation `x^2=1`

(that is, free products of Z2's)

reduceWordZ3 :: Eq a => Word a -> Word aSource

Reduces a word, where each generator `x`

satisfies the additional relation `x^3=1`

(that is, free products of Z3's)

reduceWordZm :: Eq a => Int -> Word a -> Word aSource

Reduces a word, where each generator `x`

satisfies the additional relation `x^m=1`

(that is, free products of Zm's)

Counts the number of words (without inverse generators) of length `n`

which reduce to the identity element, using the relations `x^2=1`

.

Generating function is `Gf_g(u) = \frac {2g-2} { g-2 + g \sqrt{ 1 - (4g-4)u^2 } }`

The first few `g`

cases:

A000984 = [ countIdentityWordsZ2 2 (2*n) | n<-[0..] ] = [1,2,6,20,70,252,924,3432,12870,48620,184756...] A089022 = [ countIdentityWordsZ2 3 (2*n) | n<-[0..] ] = [1,3,15,87,543,3543,23823,163719,1143999,8099511,57959535...] A035610 = [ countIdentityWordsZ2 4 (2*n) | n<-[0..] ] = [1,4,28,232,2092,19864,195352,1970896,20275660,211823800,2240795848...] A130976 = [ countIdentityWordsZ2 5 (2*n) | n<-[0..] ] = [1,5,45,485,5725,71445,925965,12335685,167817405,2321105525,32536755565...]

:: Int | g = number of generators in the free group |

-> Int | n = length of the unreduced word |

-> Int | k = length of the reduced word |

-> Integer |

Counts the number of words (without inverse generators) of length `n`

whose
reduced form in the product of Z2-s (that is, for each generator `x`

we have `x^2=1`

)
has length `k`

(clearly `n`

and `k`

must have the same parity for this to be nonzero):

countWordReductionsZ2 g n k == sum [ 1 | w <- allWordsNoInv g n, k == length (reduceWordZ2 w) ]

countIdentityWordsZ3NoInvSource

Counts the number of words (without inverse generators) of length `n`

which reduce to the identity element, using the relations `x^3=1`

.

countIdentityWordsZ3NoInv g n == sum [ 1 | w <- allWordsNoInv g n, 0 == length (reduceWordZ2 w) ]

In mathematica, the formula is: `Sum[ g^k * (g-1)^(n-k) * k/n * Binomial[3*n-k-1, n-k] , {k, 1,n} ]`