Documentation on the GBNP package

Version 0.9.5

5 June 2007

Abstract

We provide algorithms, written in the GAP 4 programming language, for computing Gröbner bases of non-commutative 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 non-commutative 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 non-commutative Gröbner basis algorithm to terminate, as it would imply solvability of the word problem for finitely presented (semi)groups.

This documentation gives a short description of the mathematical content, explains the functions of the package, and provides more than twenty worked out examples.

Acknowledgements

1. Introduction
   1.1 Installation
   1.2 Using the package
   1.3 Further documentation
2. Description
   2.1 Non-commutative Polynomials (NPs)
   2.2 Non-commutative 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 Introduction
      3.7-2 EvalTrace
      3.7-3 PrintTraceList
      3.7-4 PrintTracePol
      3.7-5 PrintNPListTrace
      3.7-6 SGrobnerTrace
      3.7-7 StrongNormalFormTraceDiff
   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 The order of the Weyl group of type E_6
   A.6 The gcd of some univariate polynomials
   A.7 From the Tapas book
   A.8 The Birman-Murakami-Wenzl algebra of type A_3
   A.9 The Birman-Murakami-Wenzl algebra of type A_2
   A.10 A commutative example by Mora
   A.11 Tracing an example by Mora
   A.12 Finiteness of the Weyl group of type E_6
   A.13 Preprocessing for Weyl group computations
   A.14 A quotient algebra with exponential growth
   A.15 A commutative quotient algebra of polynomial growth
   A.16 An algebra over a finite field
   A.17 The dihedral group of order 8
   A.18 The dihedral group of order 8 on another module
   A.19 The dihedral group on a non-cyclic module
   A.20 The icosahedral group
   A.21 The symmetric inverse monoid for a set of size four
   A.22 A module of the Hecke algebra of type A_3 over GF(3)
   A.23 Generalized Temperley-Lieb algebras
   A.24 The universal enveloping algebra of a Lie algebra
   A.25 Serre's exercise
   A.26 Baur and Draisma's transformations

Documentation on the GBNP package

Abstract

Acknowledgements

Contents