Schur Group Theory Software

by Brian G. Wybourne

Professor Brian Wybourne died in Torun, Poland, on 26 November 2003. He is greatly missed by family, friends and colleagues.

Following Brian Wybourne's death we (Franck Butelle, Ronald King and Fréric Toumazet) felt that his programme SCHUR should be maintained, and if possible enhanced, with a view to making it freely available to the mathematical and physics research community.

To this end we have produced a new version of SCHUR, currently SCHUR 6.02. While we have carried out a number of tests of the new version on different platforms with different operating systems, it comes with no guarantee. It may be downloaded from:
http://sourceforge.net/projects/schur
We are extremely grateful to Brian Wybourne's family for allowing us to make this programme freely available. It was constructed over many years by Brian and a succession of his students and colleagues, with the version SCHUR 5.0 first marketed through the good offices of Steven Christensen.

We hope that SCHUR will be of use in various research activities, and would appreciate feedback about its effectiveness and any bugs that you may find. It is anticipated that updates will be made available from time to time.

Please send feedback to: fb95 at users.sourceforge.net
or ft95 at users.sourceforge.net

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

Functions to treat non-compact groups.
Now over 200 functions.
Updated manual - now over 220 pages.

Platforms currently supported:

Intel-compatible PC's (DOS or DOS Window under Windows 98, XP (with the help of cygwin) )
Intel-compatible PC's (Linux)
SPARC/Solaris 2.5-10 and x86/Solaris 8-10 (from version 6.02c) see http://www.sunfreeware.com for binary packages
We expect more platform releases soon.

What is Schur?

Schur is a stand alone C program for interactively calculating properties of Lie groups and symmetric functions. Schur has been designed to answer questions of relevance to a wide range of problems of special interest to chemists, mathematicians and physicists - particularly for persons who need specific knowledge relating to some aspect of Lie groups or symmetric functions and yet do not wish to be encumbered with complex algorithms. The objective of Schur is to supply results with the complexity of the algorithms hidden from view so that the user can effectively use Schur as a scratch pad, obtaining a result and then using that result to derive new results in a fully interactive manner. Schur can be used as a tool for calculating branching rules, Kronecker products, Casimir invariants, dimensions, plethysms, S-function operations, Young diagrams and their hook lengths etc.

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?

Among the many tasks amenable to Schur are the following:

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?

Schur has found important applications in diverse research topics such as:

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

Schur has already been involved in many applications. Among these are:

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

The manual that comes with the Schur software provides not only extensive dicussion of how to use Schur, but also considerable mathematical discussion of Lie groups and symmetric functions and general group theory.