Littlewood-Richardson Calculator version 2.1
Copyright (C) 1999- Anders S. Buch (asbuch at math rutgers edu)


INSTALLATION
============

This file explains how to install and use the programs.  See the file
ChangeLog for a list of bug-fixes, enhancements, and contributors.

This project uses the standard installation procedure:

    ./configure
    make
    sudo make install

The last step is optional and installs the programs

    lrcalc  and  schubmult

in the directory /usr/local/bin, as well as the maple interface
lrcalc.maple in /usr/local/share/lrcalc.  It also installs the lrcalc
library and header files.  (The prefix /usr/local can be changed, see
the file INSTALL for details.)

On some Linux distributions you need to set the environment variable
LD_LIBRARY_PATH to load shared libraries installed in /usr/local/lib.
If you use Bash, then this can be done by adding the following line to
the file $HOME/.bashrc and starting a new shell:

export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:/usr/local/lib

Notice also that the Littlewood-Richardson Calculator is now
integrated in the SAGE open-source mathematics system.  You may prefer
to use lrcalc from SAGE to take advantage of the additional functions
in this system.


USAGE
=====

We give a very brief description of the programs.

The program lrcalc can perform various operations on symmetric
functions, and its first argument must specify the operation to be
used.  There are five of these operations:


lrcalc coef:  Compute a single Littlewood-Richardson coefficient.
-----------

The command "lrcalc coef z - x - y" computes the Littlewood-Richardson
coefficient c^z_{x,y} which is the coefficient of the Schur function
s_z in the product s_x * s_y.

Example:

% lrcalc coef 3 2 1 - 2 1 - 2 1
2


lrcalc skew:  Expand a skew Schur function in the basis of Schur
-----------   functions.

The command "lrcalc skew z / x" lists all partitions y for which the
Littlewood-Richardson coefficient c^z_{x,y} is non-zero, with that
coefficient in front.

Example:

% lrcalc skew 3 2 1 / 2 1
2  (2,1)
1  (1,1,1)
1  (3)


lrcalc mult:  Expand a product of two Schur functions in the basis of
-----------   Schur functions.

The command "lrcalc mult x - y" lists all partitions z for which the
Littlewood-Richardson coefficient c^z_{x,y} is non-zero, with that
coefficient in front.

Example:

% lrcalc mult 2 1 - 2 1
2  (3,2,1)
1  (4,2)
1  (3,1,1,1)
1  (3,3)
1  (2,2,1,1)
1  (2,2,2)
1  (4,1,1)


The option -r <rows> can be used to limit the partitions in the output
to a given number of rows.


The command Lrcalc mult can also compute products in the small quantum
cohomology ring of a Grassmann variety.  The option "-q m,k" will use
the quantum ring of the Grassmannian Gr(m,m+k).  If -q is replaced
with -f, then the same product is computed, but the output is given in
fusion ring notation.

Example:

% lrcalc mult -q 3,2 3 2 1 - 3 2 1
1  (2)
1  (1,1)

% lrcalc mult -f 3,2 3 2 1 - 3 2 1
1  (4,4,4)
1  (5,4,3)


lrcalc coprod:  Calculate coproducts in the ring of symmetric
-------------   functions.

The command "lrcalc coprod z" lists all pairs of partitions x and y
for which the Littlewood-Richardson coefficient c^z_{x,y} is non-zero,
with that coefficient in front.

Example:

% lrcalc coprod 3 2 1
1  (2,1,1)  (1,1)
1  (3,2,1)  ()
1  (3,1,1)  (1)
1  (2,2)  (1,1)
2  (2,1)  (2,1)
1  (3,2)  (1)
1  (3,1)  (2)
1  (1,1,1)  (2,1)
1  (2,2,1)  (1)
1  (2,1,1)  (2)
1  (2,2)  (2)
1  (2,1)  (3)
1  (3,1)  (1,1)


schubmult:  Expand a product of two Schubert polynomials in the basis
---------   of Schubert polynomials.

Example:

% schubmult 1 3 2 - 1 3 2
1  (2,3,1)
1  (1,4,2,3)


The option -s tells schubmult to use "string" notation:

% schubmult -s 0 1 0 2 1 - 1 0 1 0 2
1  (1,1,0,2,0)
1  (1,2,0,0,1)
1  (1,0,2,1,0)


MAPLE INTERFACE
===============

The file lrcalc.maple in the directory /usr/local/share/lrcalc
provides an interface between the above C programs and Maple.  It
allows you to combine the notational convenience of Maple with the
speed of C programs.

To use the interface, simply type

    read "/usr/local/share/lrcalc/lrcalc.maple";

after starting Maple.

You can copy the file lrcalc.maple to a more convenient directory to
avoid typing the full path every time you wish to load it.

The notation used by the Maple interface is the same as in John
Stembridge's "SF" Maple package for computing with symmetric
functions.  I hope this will make it easier to use SF with the Maple
interface.

The Maple interface defines three Maple functions:

    tos, skew, lrcoef

The following Maple session gives an example of how to use them:


% maple
    |\^/|     Maple V Release 4 (WMI Campus Wide License)
._|\|   |/|_. Copyright (c) 1981-1996 by Waterloo Maple Inc. All rights
 \  MAPLE  /  reserved. Maple and Maple V are registered trademarks of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.
> read "/usr/local/share/lrcalc/lrcalc.maple";
> lrcoef([3,2,1], [2,1], [2,1]);
                                       2

> skew(s[3,2,1], s[2,1]);
                         s[3] + 2 s[2, 1] + s[1, 1, 1]

> tos(s[2,1] * s[2,1]);
s[3, 3] + s[4, 2] + s[2, 2, 1, 1] + s[2, 2, 2] + 2 s[3, 2, 1] + s[4, 1, 1]

     + s[3, 1, 1, 1]

> lrcoef( s[12,11,10,9,8,7,6,5,4,3,2,1],\
  s[8,7,6,5,4,3,2,1], s[8,7,6,6,5,4,3,2,1]);
                                    7869992


Enjoy!

Anders S. Buch  (January 27, 2021)
