Schur Group Theory Software

by Brian G. Wybourne

An Interactive Program For Calculating Properties Of
Lie Groups and Symmetric Functions

Wybourne's Home Page with lots more about Group Theory and Schur.

Version Schur 5.3.1

• Functions to treat non-compact groups.
• Now over 160 functions.
• Updated manual - now over 220 pages.

Platforms currently supported:

• Intel-compatible PC's (DOS or DOS Window under Windows 3.1, 95, 98, NT, 2000 )
• Sun SPARC (Solaris 2.5-8)
• Intel-compatible PC's (Solaris 2.6-8)
• Intel-compatible PC's (Red Hat Linux)
• We expect to have a Max OS X release soon.

Pricing and Ordering

What is Schur?

As well as being a research tool Schur forms an excellent tool for helping students to independently explore the properties of Lie groups and symmetric functions and to test their understanding by creating simple examples and moving on to more complex examples. The user has at his or her disposal over 160 commands which may be nested to give a vast variety of potential operations. Every command, with examples, is described in a 200 page manual. Attention has been given to input/output issues to simplify input and to give a well organized output. The output may be obtained in TeX form if desired. Log files may be created for subsequent editing. On line help files may be brought to screen at any time.

Place Schur in your workstation, PC or portable notebook and you have available a host of information on Lie groups and symmetric functions. A tool both for teaching and research.

What can Schur do?

• The calculation of Kronecker products for all the compact Lie groups and for the ordinary and spin representations of the symmetric group. Not only for individual irreducible representations but also lists of irreducible representations. List handling is a general feature of Schur.
• The calculation of branching rules with the ability to successively branch through a chain of nested groups.
• The calculation of the properties of irreducible representations such as dimensions, second-order Casimir and Dynkin invariants, the trace of the n-th order Casimir invariants and the conversion between partition and Dynkin labelling of irreducible representations.
• The handling of direct products of several groups.
• The computation of a wide range of properties related to Schur function operations such as the Littlewood-Richardson rule, inner products, skew products, and plethysms as well as the inclusion of commands for generating the terms in infinite series of Schur functions up to a user defined cutoff.
• The computation of the properties of the symmetric Q-functions with respect to operations such as the analogous Littlewood-Richardson rule, skew and inner products.
• The standardisation of non-standard representations of groups by the use of modification procedures.
• Calculation of properties of the classical symmetric functions.

Among the special features of Schur

• All operations can be made on lists of irreducible representations and not just single irreducible representations.
• Sequences of instructions may be set as functions (which may be saved on disk) allowing easy extension of Schur to implement user defined rules.
• Results of a session with Schur may be saved as a logfile for future record or editing.
• Over 160 commands allow a wide variety of applications of Schur.
• Schur can be a valuable tool in the teaching of the properties of groups as students and teachers can readily create examples. Taken with this manual it can be used as a self-paced learning tool.
• Schur can be used as a research tool in many studies.

What has Schur done?

• Constructing character tables for the Hecke algebras H_{n}(q) of type A_{n-1}.
• Symmetry properties of the Riemann tensor.
• Group properties of the Interacting Boson Model of nuclei.
• Non-compact group properties such as branching rules and Kronecker products.
• Problems in supersymmetry.
• Evaluation of the properties of one- and two-photon processes in rare earth ions.
• Symplectic models of nuclei.
• Studies of the mathematical properties of the exceptional Lie groups.
• Studies of the symmetric functions such as Schur functions, Q-functions and Hall-Littlewood polynomials.

Applications of Schur

• Application to the analysis and classification of the normal forms for tensor polynomials involving the Riemann tensor making extensive use of the commands plethysm, o_sfnproduct, sk_sfn, std, branch, dimension. See Fulling et al, Class. Quantum Grav. 9, 1151 (1992).
• Application to the interacting boson model of nuclei making use of the commands branch, series, dimension, Casimir. See Morrison et al, J. Math. Phys. 32, 356 (1992).
• Application to the calculation of the characters of Hecke algebras H_n(q) of type A_(n-1) using the commands o_sfnproduct, product, sb_tex, p_to_s. See King and Wybourne, J. Math. Phys. 33, 4 (1992).
• Application to non-compact groups to the nuclear symplectic Sp(6,R) shell model using the commands rule, i_plethysmrd, std, branch, series, weight. See Wybourne, J. Phys. A: Math. Gen. 25, 4389 (1992).
• Application to the electronic f-shell using the automorphisms of SO(8) using the commands auto, product, branch, dimension, rule, fn, series. See Wybourne, J. Phys. B: At. Mol. Opt. Phys. 25, 1683 (1992).
• Application to the analysis of the S-function content of generating functions using the commands o_sfnproduct, sk_sfn, plethysm, series. See King et al, J. Phys. A: Math. Gen. 22 , 4519 (1989).
• Application to Q-functions using the commands o_qfnproduct, std_qfn, branch, dimension, spin, rule, fn. See Salam and Wybourne, J. Math. Phys. 31, 1310 (1989); J. Phys. A: Math. Gen. 22, 3771 (1989).

Comments from the literature

• "In particular, his package Schur must be regarded as necessary to both mathematicians and physicists whose work is dependent on calculations involving compact Lie groups and Schur functions" Mathematical Reviews 93f: 05101 (1993).
• "Finally, we should mention that Wybourne and his colleagues at the University of Canterbury in Christchurch, New Zealand have developed a nice package called Schur which run's on PC's and which computes all the above products of Schur functions plus a great deal more branching rules, etc for Lie groups." Acta Appled Mathematics 21, 105 (1990).
• "Over two decades, Wybourne and his students have developed a computer program, Schur, which performs many of the required calculations." Classical and Quantum Gravity 9, 1151 (1992).

The Schur Manual

Introducing Schur

• What is Schur?
• What can Schur do ?
• What has Schur done ?
• Working through the Manual

Basics of Schur: Input and Output

• Input of Lists to Schur
• Output of Lists from Schur
• The Schur Modes
• Sample Input and Output Lists
• Commands and Expressions
• Accessing Help Files

Combinatorics and Schur functions

• Partitions
• Young Diagrams
• Skew Frames
• Frobenius notation for partitions
• Young Tableaux
• Hook Lengths and Dimensions for S_n
• Unitary Numbering of Young Tableaux
• Young Tableaux and Monomials
• Monomial Symmetric Functions
• The Classical Symmetric Functions
• The Schur Functions
• Calculation of the Elements of the Kostka Matrix
• Classical Definition of the S-function
• Non-standard S-functions
• Skew S-functions
• The Littlewood-Richardson Rule
• Relationship to the Unitary Group
• Inner Products of S-functions
• Reduced Inner Products
• Plethysm of S-functions
• Inner Plethysm
• S-function Series
• Symbolic Manipulation
• The U_n -> U_{n-1} Branching Rule
• Schur's Q-functions
• Non-standard Q-functions
• Young's Raising Operators

Notation for Lie Groups

• Unitary Group Labels
• Orthogonal and Symplectic Group Labels
• Associate Irreducible Representations
• Irreducible Representations of O_n and SO_n
• Irreducible Representations of Exceptional Groups
• The Super Lie Groups
• Notation for the Symmetric and Alternating Groups
• Standard Labels for Lie Groups
• Standard Labels and Dynkin Labels
• Modification Rules
• Fusion Modification Rules
• Dimensions of Irreducible Representations
• Casimir and Dynkin Invariants
• Kronecker Products
• Plethysms in Lie Groups
• Automorphisms and Isomorphisms in Lie Groups
• Branching Rules
• Odds and Ends

Tutorials in using Schur

• Introduction to Tutorials
• Tutorial 1 : Getting Started in the SFNmode
• Tutorial 2 : Exploring the REPmode
• Tutorial 3 : The Branching Rule Mode
• Tutorial 4 : Introduction to the DPMode
• Exercises

Advanced Tutorials in using Schur

• Advanced Tutorial 1 : Writing User Defined Functions
• Advanced Tutorial 2 : Using the Rule command
• The U_1 trick in Schur
• The Final Test

Examples of Schur in Physics, Chemistry and Mathematics

• The Simple SU_3 Quark Model of Baryons and Mesons
• Unification Models and QCD
• Electronic States of the N_2 Molecule
• Plethysm and Asymptopia

Further Reading for Users of Schur

• Introduction
• References

Every Command in Schur Described

• Introduction

The Schur Help Files

• The Schur Help Files
• The Function Files

Practical Details

• Introduction
• Setting up directories
• Limitations and set dimensions
• Error messages and runtime errors

Producing TeX tables

• Introduction
• Making a TeX Table

The Non-compact Groups Sp(2n,R) and Mp(2n)

• Introduction
• Labelling the Irreps of Non-compact Lie Groups
• Branching Rules for subgroups of Mp(2n) and Sp(2n,R)
• Kronecker Products for Sp(2n,R)

Index to Schur

• Index

Tables

• Brackets used in the output of lists by Schur
• Standard labels for irreducible representations of the Lie groups of rank k
• Relationship between standard Schur labels and the corresponding Dynkin labels for the classic Lie groups
• Relationship between standard Schur labels and the corresponding Dynkin labels for the exceptional Lie groups
• The modification rules appropriate to the classical Lie groups
• Spectroscopic terms of the d^5 electron configuration
• The numbers c[lambda][mu][2^2] for irreducible representations of SO_5
• All the commands in Schur
• The branching rule table
• Formats for entry of groups in Schur
• Groups and classes of representations available for calculating Kronecker products in Schur
• The Schur help files

Send comments or questions to steve@smc.vnet.net

© Copyright 2002 Steven M. Christensen and Associates, Inc.

This page was last updated on March 1, 2002.