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:

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
or ft95 at

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

Platforms currently supported:

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:

Among the special features of Schur

What has Schur done?

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

Applications of Schur

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

Comments from the literature

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.