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MathGroup Archive 1993

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Schur 5.1 Released

  • To: mathgroup at
  • Subject: Schur 5.1 Released
  • From: Steven M. Christensen <stevec>
  • Date: Tue, 7 Sep 93 01:12:53 EDT


As moderator, I am taking some liberties here to post about a new
product Brian Wybourne and I have been working on for about
a year.  Brian has written a program called Schur, which I
have helped him to put into product form and now to distribute.
It is not written in Mathematica, but we hope to do so in
the future and also to create a MathLink to Mathematica.
Please pass this along to anyone who might be interested and also
send comments on problems that might be of interest to us in
future Schur development.


Steve Christensen



           *                     Schur 5.1                     * 
           *                 by Brian G. Wybourne              * 
           *                                                   *  
           *              An Interactive Program For           * 
           *              Calculating Properties Of            *
           *         Lie Groups and Symmetric Functions        *

September 6, 1993 -

Brian Wybourne and Steven Christensen are pleased to announce the
release of Schur(tm) Version 5.1.   Schur previously ran on IBM PC
compatibles and was written in Turbo Pascal.  Schur 5.1 is completely
rewritten in C giving the program significantly better performance and
improved use of memory for large computations.  Many of the 179 functions
are upgraded and the 200 page user manual has been significantly enlarged
and updated.

Schur 5.1 now runs on:

o 386 or 486 PC's (DOS) with numerical coprocessor

o Sun Sparc (SunOS 4.1.X and Solaris 2.X)
o Sun-3 with 68881 (SunOS 4.1.X) 
o NeXT (NeXTStep)
o IBM RS/6000 (AIX 3.2)
o HP 9000 series (HP/UX)
o Silicon Graphics Iris (IRIX System V.3)
o DEC Alpha (OSF/1)

Versions for Macintosh, PowerPC, Pentium, PC Windows, and others are
under development.  Prices and availability are subject to change
without notice.

For information on availability and pricing, contact: 

Dr. Steven M. Christensen
P.O. Box 16175
Chapel Hill, NC 27516
Phone/Fax:  919-967-9853
Email: stevec at

Technical questions about Schur can be mailed or emailed to:

Dr. Brian G. Wybourne
Instytut Fisyki
Uniwersytet Mikolaja Kopernika
ul. Gradsiadaka 5/7
87-100 Torun
Email: BGW at

o 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.

o What can Schur do? 

Among the many tasks amenable to Schur are the following:

1. Calculation of Kronecker products for all compact Lie groups and
for ordinary and spin representations of the symmetric group. Not only
for individual representations but also for lists of representations.
List handling and sorting is a general feature of Schur.

2. Calculation of branching rules with the ability to successively
branch through a chain of nested groups.

3. Calculation of properties of representations such as dimensions,
Casimir invariants, conversion between Dynkin and partition labels.

4. Calculation of a wide range of properties related to Schur
function operations such as the Littlewood-Richardson rule, inner
products, skew products, plethysms, Young diagrams and generating the
terms in infinite series of Schur functions up to a user defined

5. Calculation of properties of the symmetric Q-functions with respect
to operations such as the analogous Littlewood-Richardson rule, skew
and inner products.

6. Calculation of transformations between the various classical
symmetric functions.

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

1. 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).
2. 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).
3. 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).
4. 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).
5. 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).
6. 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).
7. 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).

o Comments from the literature 

1. "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).

2. "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).

3. "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).

o The Author:

The author Professor B. G. Wybourne obtained his Ph.D. in Physics from
the University of Canterbury, Christchurch, New Zealand and is
currently a Professor of Physics at Uniwersytet Mikolaja Kopernika, in
Torun, Poland.  He has published over 120 papers, largely concerning
symmetry, combinatorics and group theory in physics. Among his
publications are the three books Spectroscopic Properties of Rare
Earths, Symmetry Principles and Atomic Spectroscopy, and Classical
Groups for Physicists. While Professor of Physics at the University of
Canterbury he, and his students, developed algorithms for calculating
diverse properties of Lie groups and symmetric functions which have
been built into the program Schur. Since coming to Poland he has
continued his research activities and the development of Schur.

o The Distributor:

Steven M. Christensen is active in the promotion of the use of
symbolic computing in scientific research.  He is the co-developer
of the MathTensor, Mathematica-based tensor analysis software,
distributed by MathSolutions, Inc.  He is founder of the MathGroup,
Mathematica user group and has been a consultant to Wolfram Research,
IBM, Sun Microsystems and other computer hardware and software 
manufacturers.  He was the originator of the SunSITE software and
information archive and is an Adjunct Professor of Physics at the
University of North Carolina at Chapel Hill.  He has published
in the areas of quantum gravity and quantum field theory, black holes
and cosmology.   He obtained his Ph.D. in Physics from the University
of Texas at Austin.

o Schur is a Trademark of Brian G. Wybourne.

o All information is Copyright (c) 1993 by Brian G. Wybourne and
  Steven M. Christensen


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