GAP (HAP commands) - Chapter 20: Words in free \(ZG\)-modules

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20 Words in free \(ZG\)-modules
20.1
20.1-1 AddFreeWords
20.1-2 AddFreeWordsModP
20.1-3 AlgebraicReduction
20.1-4 MultiplyWord
20.1-5 Negate
20.1-6 NegateWord
20.1-7 PrintZGword
20.1-8 TietzeReduction
20.1-9 ResolutionBoundaryOfWord

### 20 Words in free \(ZG\)-modules

#### 20.1

##### 20.1-1 AddFreeWords

`‣ AddFreeWords`( v, w ) | ( function )

Inputs two words \(v,w\) in a free \(ZG\)-module and returns their sum \(v+w\). If the characteristic of \(Z\) is greater than \(0\) then the next function might be more efficient.

Examples:

##### 20.1-2 AddFreeWordsModP

`‣ AddFreeWordsModP`( v, w, p ) | ( function )

Inputs two words \(v,w\) in a free \(ZG\)-module and the characteristic \(p\) of \(Z\). It returns the sum \(v+w\). If \(p=0\) the previous function might be fractionally quicker.

Examples:

##### 20.1-3 AlgebraicReduction

`‣ AlgebraicReduction`( w ) | ( function )

`‣ AlgebraicReduction`( w, p ) | ( function )

Inputs a word \(w\) in a free \(ZG\)-module and returns a reduced version of the word in which all pairs of mutually inverse letters have been cancelled. The reduction is performed in a free abelian group unless the characteristic \(p\) of \(Z\) is entered.

Examples:

##### 20.1-4 MultiplyWord

`‣ MultiplyWord`( n, w ) | ( function )

Inputs a word \(w\) and integer \(n\). It returns the scalar multiple \(n\cdot w\).

Examples:

##### 20.1-5 Negate

`‣ Negate`( [i, j] ) | ( function )

Inputs a pair \([i,j]\) of integers and returns \([-i,j]\).

Examples:

##### 20.1-6 NegateWord

`‣ NegateWord`( w ) | ( function )

Inputs a word \(w\) in a free \(ZG\)-module and returns the negated word \(-w\).

Examples:

##### 20.1-7 PrintZGword

`‣ PrintZGword`( w, elts ) | ( function )

Inputs a word \(w\) in a free \(ZG\)-module and a (possibly partial but sufficient) listing elts of the elements of \(G\). The function prints the word \(w\) to the screen in the form

\(r_1E_1 + \ldots + r_nE_n\)

where \(r_i\) are elements in the group ring \(ZG\), and \(E_i\) denotes the \(i\)-th free generator of the module.

Examples: 1

##### 20.1-8 TietzeReduction

`‣ TietzeReduction`( S, w ) | ( function )

Inputs a set \(S\) of words in a free \(ZG\)-module, and a word \(w\) in the module. The function returns a word \(w'\) such that {\(S,w'\)} generates the same abelian group as {\(S,w\)}. The word \(w'\) is possibly shorter (and certainly no longer) than \(w\). This function needs to be improved!

Examples:

##### 20.1-9 ResolutionBoundaryOfWord

`‣ ResolutionBoundaryOfWord`( R, n, w ) | ( function )

Inputs a resolution \(R\), a positive integer \(n\) and a list \(w\) representing a word in the free module \(R_n\). It returns the image of \(w\) under the \(n\)-th boundary homomorphism.

Examples: 1

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