Version 0.9.4

30 May 2007

**Arjeh M. Cohen
**

e-mail: A.M.Cohen@tue.nl

**Address:**

TU/e,

POB 513, 5600 MB Eindhoven, the Netherlands

We provide algorithms, written in the **GAP** 4 programming language, for computing Gröbner bases of noncommutative polynomials, and some variations, such as a weighted and truncated version and a tracing facility. In addition, there are algorithms for analyzing the quotient of a noncommutative polynomial algebra by a 2-sided ideal generated by a set of polynomials whose Gröbner basis has been determined and for computing quotient modules of free modules over quotient algebras.

The notion of algorithm is interpreted loosely: in general one cannot expect a noncommutative Gröbner basis algorithm to terminate, as it would imply solvability of the word problem for finitely presented (semi)groups.

The package is based on an earlier version by Rosane Ushirobira.

The bulk of the package is written by Arjeh M. Cohen and Dié A.H. Gijsbers.

The theory is mainly taken from literature by Teo Mora [M] and Edward L. Green [G99].

From 0.8.3 on the package has three additional files (

`fincheck.g`

,`tree.g`

`graphs.g`

) with routines for finding the Hilbert function and testing finite dimensionality when given a Gröbner basis by Chris Krook [K03], based on work by Victor Ufnarovski [U89].From 0.9 on the package is enriched with support for fields implemented in GAP and additional prefix rules for quotient modules, as well as some speed improvements by Jan Willem Knopper. The documentation system from which this text is derived is also due to him.

1.1 Installation

1.2 Introduction to the package

1.3 Other parts of the documentation

2. Description

2.1 Noncommutative Polynomials (NPs)

2.2 Noncommutative Polynomials for Modules (NPMs)

2.3 Core functions

2.4 About the implementation

2.5 Tracing variant

2.6 Truncation variant

2.7 Prefix and Two-Sided (PTS) variant

2.8 Gröbner basis records

2.9 Dimensionality of a quotient algebra

3. Functions

3.1 Converting Polynomials into different formats

3.1-1 GP2NP

3.1-2 GP2NPList

3.1-3 NP2GP

3.1-4 NP2GPList

3.2 Printing polynomials in NP format

3.2-1 GBNP.ConfigPrint

3.2-2 PrintNP

3.2-3 PrintNPList

3.3 Calculating with polynomials in NP format

3.3-1 AddNP

3.3-2 BimulNP

3.3-3 CleanNP

3.3-4 GtNP

3.3-5 LtNP

3.3-6 LTermsNP

3.3-7 MkMonicNP

3.3-8 MulNP

3.4 Gröbner functions, standard variant

3.4-1 Grobner

3.4-2 SGrobner

3.5 Finite-dimensional quotient algebras

3.5-1 BaseQA

3.5-2 DimQA

3.5-3 MatricesQA

3.5-4 MatricesQAC

3.5-5 MatrixQA

3.5-6 MatrixQAC

3.5-7 MulQA

3.5-8 StrongNormalFormNP

3.6 Finiteness and Hilbert Series

3.6-1 Introduction

3.6-2 DetermineGrowth

3.6-3 DetermineGrowthObs

3.6-4 FinCheck

3.6-5 HilbertSeries

3.6-6 Preprocess

3.7 Functions of the trace variant

3.7-1 PrintTraceList

3.7-2 PrintTracePol

3.7-3 PrintNPListTrace

3.7-4 SGrobnerTrace

3.7-5 StrongNormalFormTraceNP

3.8 Functions of the truncated variant

3.8-1 Examples

3.8-2 SGrobnerTrunc

3.8-3 MakeArgumentList

3.9 Functions with prefix and two-sided rules

3.9-1 BaseQAPTS

3.9-2 DimQAPTS

3.9-3 MulQAPTS

3.9-4 SGrobnerPTS

3.9-5 StrongNormalFormPTSNP

4. Info Level

4.1 Introduction

4.2 InfoGBNP

4.2-1 InfoGBNP

4.2-2 What will be printed at level 0

4.2-3 What will be printed at level 1

4.2-4 What will be printed at level 2

4.3 InfoGBNPTime

4.3-1 InfoGBNPTime

4.3-2 What will be printed at level 0

4.3-3 What will be printed at level 1

4.3-4 What will be printed at level 2

A. Examples

A.1 Introduction

A.2 A simple commutative Gröbner Basis computation

A.3 A truncated Gröbner basis for Leonard pairs

A.4 The truncated variant on two weighted homogeneous polynomials

A.5 More details on a truncated version for Leonard pairs

A.6 The order of the Weyl group of type E_6

A.7 The gcd of some univariate polynomials

A.8 From the Tapas book

A.9 The Birman-Murakami-Wenzl algebra of type A_3

A.10 The Birman-Murakami-Wenzl algebra of type A2

A.11 A commutative example by Mora

A.12 Tracing an example by Mora

A.13 Finiteness of the Weyl group of type E_6

A.14 Preprocessing for Weyl group computations

A.15 A quotient algebra with exponential growth

A.16 A commutative quotient algebra of polynomial growth

A.17 An algebra over a finite field

A.18 The dihedral group of order 8

A.19 The dihedral group of order 8 on another module

A.20 The dihedral group on a non-cyclic module

A.21 The icosahedral group

A.22 The symmetric inverse monoid for a set of size four

A.23 A module of the Hecke Algebra of type A_3 over GF(3)

A.24 Generalized Temperley-Lieb algebras

B. Files

B.1 Files: Introduction

B.2 Main files

B.3 Auxiliary files

generated by GAPDoc2HTML