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# GBNP

## computing Gröbner bases of noncommutative polynomials

1.0.5

9 March 2022

A.M. Cohen
Email: A.M.Cohen@tue.nl

J.W. Knopper
Email: J.W.Knopper@tue.nl

Address:
TU/e,
POB 513, 5600 MB Eindhoven, the Netherlands

### 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.

### Copyright

© 2001-2010 by Arjeh M. Cohen, Dié A.H. Gijsbers, Jan Willem Knopper, Chris Krook. Address: Discrete Algebra and Geometry (DAM) group at the Department of Mathematics and Computer Science of Eindhoven University of Technology.

### Acknowledgements

- 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 [Mor94] and Edward L. Green [Gre99].

- From Version 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 [Kro03], based on work by Victor Ufnarovski [Ufn89].

- From Version 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. Knopper has also formatted the documentation in GAPDoc [LN06].

- From Version 1.0 on the package is extended with NMO (for Noncommutative Monomial Orderings) by Randall Cone. This enables the GBNP user to choose a wider selection of monomial orderings than the standard one built into GBNP itself. Documentation on NMO can be found in the NMO manual [Con10].

### Contents

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 Module variant
2.8 Gröbner basis records
2.9 Quotient algebras
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 PrintNP
3.2-2 GBNP.ConfigPrint
3.2-3 PrintNPList
3.3 Calculating with polynomials in NP format
3.3-1 NumAlgGensNP
3.3-2 NumAlgGensNPList
3.3-3 NumModGensNP
3.3-4 NumModGensNPList
3.3-5 AddNP
3.3-6 BimulNP
3.3-7 CleanNP
3.3-8 GtNP
3.3-9 LtNP
3.3-10 LMonsNP
3.3-11 MkMonicNP
3.3-12 MulNP
3.4 Gröbner functions, standard variant
3.4-1 Grobner
3.4-2 SGrobner
3.4-3 IsGrobnerBasis
3.4-4 IsStrongGrobnerBasis
3.4-5 IsGrobnerPair
3.4-6 MakeGrobnerPair
3.5 Finite-dimensional quotient algebras
3.5-1 BaseQA
3.5-2 DimQA
3.5-3 MatrixQA
3.5-4 MatricesQA
3.5-5 MulQA
3.5-6 StrongNormalFormNP
3.6 Finiteness and Hilbert series
3.6-1 DetermineGrowthQA
3.6-2 FinCheckQA
3.6-3 HilbertSeriesQA
3.6-4 PreprocessAnalysisQA
3.7 Functions of the trace variant
3.7-1 EvalTrace
3.7-2 PrintTraceList
3.7-3 PrintTracePol
3.7-4 PrintNPListTrace
3.7-5 SGrobnerTrace
3.7-6 StrongNormalFormTraceDiff
3.8 Functions of the truncated variant
3.8-1 Examples
3.8-2 SGrobnerTrunc
3.8-3 CheckHomogeneousNPs
3.8-4 BaseQATrunc
3.8-5 DimsQATrunc
3.8-6 FreqsQATrunc
3.9 Functions of the module variant
3.9-1 SGrobnerModule
3.9-2 BaseQM
3.9-3 DimQM
3.9-4 MulQM
3.9-5 StrongNormalFormNPM
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
A.27 The cola gene puzzle
References
Index
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