A Maple package for the decomposition of certain tensor products of representations using crystal graphs

by David Joyner, Roland Martin, Micheal Fourte

This document is an introduction to the Maple package crystal, version 2.5.

For downloading information, see the entry under crystal.

section 1: Introduction

This document is an introduction to crystal, a Maple package which helps decompose the tensor product of two irreducible finite dimensional representations of a simple Lie algebra into irreducible constituents using crystal graphs. Basically this amounts to implementing a theorem of Kashiwara : the irreducible constituents of such a tensor product are in a natural 1-1 correspondence with the connected components of the crystal graph of the tensor product. The crystal package contains programs to

  1. compute the crystal graph of an irreducible representation, associated to a fundamental weight, provided it is multiplicity free when restricted to the Cartan subalgebra,
  2. compute the crystal graph of the tensor product of two irreducible representations, each of which is obtained from #1,
  3. display the crystal graph of the tensor product obtained in #2 (using our own routines, which are better suited for our purposes than the networks package draw command),
  4. display the crystal graph of a representation obtained in #1,
  5. computing all the weights occurring in a given irreducible representation,
  6. compute the restriction of an irreducible representation obtained in #1 to a simple subalgebra obtained by omitting one of the simple roots.
This package makes much use of John Stembridge's packages coxeter and weyl [S]. In fact, a slightly modified form of the coxeter example program cartan_matrix has been included in crystal, with Professor Stembridge's kind permission. This package also makes use of the standard maple packages linalg, networks, and plots.

We hope in a future version to extend the algorithms to cover decompositions of a larger class of tensor products.

section 2: Definitions

This is not the place for an exposition of representation theory, Lie algebras, or crystal graphs. However, we do need to make some of our conventions clear, especially for crystal graphs. We refer to Humphreys [H] and Kashiwara [K] for a more thorough discussion.

A Lie algebra is a finite dimensional vector space of square matrices L over the reals R, having a bracket

[ , ] : L x L ----> L
defined by [X,Y] = XY-YX (for X,Y in L) satisfying certain properties. We shall always assume that L is simple in the sense that [L,L] is not equal to 0 and L contains no non-trivial ideals. The simple Lie algebras are denoted by the strings A1, A2, ..., B1, B2, ..., C1, C2, ..., D1, D2, ..., E6, E7, E8, F4, and G2. We may assume that L is a subset of the set of all n x n matrices, for some n. Let H be a subalgebra of L: 
          /  [ h1   0   ...  0  ]           \  
         |   [  0   h2       .  ]           |
     H= <    [  .        .    .  ]   in  L   >
         |   [  .          .  0  ]           |    
          \  [  0   ...  0   hn ]           /
and define lambdai by 
                      [ h1   0   ...  0  ]           
                      [  0   h2       .  ]           
       lambdai   :    [ .        .    .  ]   |---->  hi,
                      [  .          .  0 ]           
                      [  0   ...  0   hn ]
for 1 <= i <= n.

Written additively, the Z-linear combinations of these linear maps or weights lambdai : H ---> R generate a weight lattice P subset Rr, where the rank r of this lattice is the rank of L. Here Z denotes the sets of all integers.

After a possible relabelling, let lambda1, ..., lambdar denote an integral basis of P and call them the fundamental weights. The finite dimensional representations pi of L are in 1-1 correspondence with finite subsets Spi of P satisfying certain properties. One of these properties is that Spi has two distinguished elements: the highest dominant weight lambdapi+ and the lowest weight lambdapi-. In fact, either lambdapi+ or lambdapi- uniquely determine the finite set Spi (we refer to [H] for the explanation of why this is true).. In particular, we can and do use the highest dominant weight lambda to determine the representation pi, lambda = lambdapi+. Note that crystal[weight_system](v,R) uses a weight vector v, represented as a linear combination c1e1+...+cr*er, where -c1*e1-...-cr*er occurs in weyl[weights](R).

A root system of L is a finite subset R of Qr, whose elements are roots, satisfying certain axioms (see [H] ). Here Q denotes the set of rational numbers. These roots generate a sublattice Q of P called the root lattice. There is a subset S = {alpha1,...,alphar} of R which forms a Z-basis for Q and with the property that for all alpha in R,

alpha = c1alpha1 + ... + cralphar,
with either all ci >= 0 (such a root is called positive) or all ci <= 0 (such a root is called negative). Such a set S is called a set of simple roots. Weights may be written as a rational linear combination of the roots. Those weights whose coefficients in this linear combination are all non-negative are called dominant. We often denote the Lie algebra and the root system by the same notation. For example, in the case of R=A2, i.e., L= sl3, lambda1= e1, lambda2=e1+e2 and S={alpha1,alpha2}, where alpha1=e1-e2, alpha2=e2-e3.

A crystal graph of a representation pi is a weighted graph whose vertices are indexed by the elements of Spi. The edge [lambda, lambda'] exists and is labeled by the index i of a simple root alphai if lambda' - lambda =alphai . A crystal graph product of two crystal graphs is a crystal graph obtained from the two given graphs by rules given in Kashiwara's paper [K]. The idea is that you can get infomation about an arbitrary irreducible representations pi of R with highest weight lambdapi+ by building its crystal graph out of the crystal graphs associated to the fundamental weights occurring in

lambdapi+ = c1lambda1 +...+ crlambdar,
ci being non-negative integers.

section 2: Global variables created by crystal

Since we use coxeter and weyl, some of the notation we use has already been explained in Stembridge's introduction to those packages [S]. We refer the reader there for more details on the notation used in coxeter and weyl.

We have already mentioned that the simple Lie algebra is to be denoted in the computer by its root system A1,..,G2. The A1,...,G2 are global variables and should not be used for anything else. Roots and weights which are not fundamental are to be written as rational linear combinations of the ei's (square roots and decimals are not permitted), 1 <= i<= r=rank(R). The e1,e2, ... are also globally defined.

An irreducible representation is denoted by its highest dominant weight vector, written as a linear combination of the ei's. For example, the identity representation of A3 is denoted e1 and its contragredient is denoted by e1+e2+e3. The fundamantal weights of A3 are e1,e1+e2,e1+e2+e3. To compute the list of all weight vectors occuring in the identity representation of A3, load crystal and type 

crystal[weight_system](e1,A3);
The output will be a list of lists:
[[1,0,0],[-1,1,0],[0,-1,1],[0,0,-1]]
These are the coordinates of the weights occuring in the identity representation in the basis e1,e1+e2,e1+e2+e3. For example, [1,0,0] represents e1, the highest dominant and [0,0,-1] represents e4=-e1-e2-e3, the lowest weight.

In order to be able to look at relatively large graphs, say a 20 x 20, with their vertices labeled and edges weighted, on the computer screen, we wrote plotting procedures which plot only a portion of the graph. These may be printed out and pieced together by the user if necessary. Our pictures use the same orientation as Mathieu [Ma], which is slightly different from that of Kashiwara [K] .

Permission is granted to anyone to to use, modify, or redistribute this software freely, subject to the following restrictions:

  1. The authors accepts no responsibility for any consequences of this software and makes no guarantee that the software is free of defects.
  2. The origin of this software must not be misrepresented, either by explicit claim or by omission.
  3. This notice and the copyleft must be included in all copies or altered versions of this software.
  4. This software may not be included or redistributed as part of any package to be sold for profit without the explicit written permission of the authors.
Another crystal page.

wdj@usna.edu

Last updated 6-6-2004.