The Littlewood-Richardson Rule is a fundamental formula in combinatorics and representation theory. On the one-hand it can be thought of as describing induction in symmetric groups:
S_n x S_m --> S_(n+m)
On the other hand, it can also be thought of as describing the decomposition of the tensor products of irreducible representations of unitary groups, whose characters are Schur functions. References for the Littlewood-Richardson rule are Macdonald, Symmetric Functions and Hall Polynomials, and Fulton and Harris, Representation Theory: A First Course.
Here from Bayreuth is a web site providing access to the Littlewood-Richardson Rule. This is an on-line implementation of a function from the public-domain Symmetrica program, a collection of routines performing useful functions related to representation theory, particularly representation theory of the symmetric group.
Here is an on-line implementation of LiE, a program to do Lie group calculations, including the Littlewood-Richardson rule. The formerly commercial program LiE has been freely available since July 1996. It is the work of Arjeh Cohen, Marc van Leeuwen and Bert Lisser. The web page of Marc van Leeuwen contains other material on the Littlewood-Richardson rule and combinatorial algorithms.
ACE is a Maple package of combinatorial algorithms including the Littlewood-Richardson rule. There's an on-line version available at their web-site.
Schur is a commercial package for calculations involving Lie group representations and symmetric functions. It too contains an implementation of the Littlewood-Richardson rule.
Here is a (comparatively slow) Mathematica Program which implements the Littlewood-Richardson Rule. For our purposes a partition is a list of positive integers in descending order. If p and q are partitions, the function LRRule[p,q] returns a list of partitions in reverse lexicographic order which arise from p and q by the Littlewood-Richardson rule. Partitions can occur multiple times in the list -- they will appear with the correct multiplicity. For example
LRRule[{2,1},{2,1}]
returns
{{4,2}, {4,1,1}, {3,3}, {3,2,1}, {3,2,1}, {3,1,1,1},
{2,2,2},{2,2,1,1}}
In this list of partitions, {3,2,1} appears twice, as it should, since its coefficient is 2.
An alternative form for the Littlewood-Richardson rule is given by the function schur, which implements the arithmetic of Schur functions. Thus
schur[2,1]*schur[2,1]
returns:
schur[3,3]+schur[4,2]+schur[2,2,2]+2*schur[3,2,1]+schur[4,1,1]+
schur[2,2,1,1]+schur[3,1,1,1] .
Illustrating the Jacobi-Trudi identities, define
e[n_]:=If[n<0,0,schur @@ Table[1,{n}]]
so e[3]=schur[1,1,1] etc. Then
Det[{{e[5],e[6],e[7]},{e[2],e[3],e[4]},{0,e[0],e[1]}}]
returns schur[3,2,2,1,1] .
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Last modified: Sat Sep 6 14:11:33 PDT 1997