			LiE


LiE is the name of a software package that enables mathematicians 
and physicists to perform computations of a Lie group theoretic 
nature. It focuses on the representation theory of complex 
semisimple (reductive) Lie groups and algebras, and on the 
structure of their Weyl groups and root systems.

LiE does not compute directly with elements of the Lie groups and 
algebras themselves; it rather computes with weights, roots, 
characters and similar objects. Some specialities of LiE are: 
tensor product decompositions, branching to subgroups, Weyl group 
orbits, reduced elements in Weyl groups, distinguished coset 
representatives and much more. These operations have been 
compiled into the program which results in fast execution: 
typically one or two orders of magnitude faster than similar 
programs written in a general purpose program.

The LiE programming language makes it possible to customise and 
extend the package with more mathematical functions. A user manual 
is provided containing many examples. 

LiE establishes an interactive environment from which commands 
can be given that involve basic programming primitives and 
powerful built-in functions. These commands are read by an 
interpreter built into the package and passed to the core of the 
system. This core consists of programs representing some 100 
mathematical functions. The interpreter offers on-line facilities 
which explain operations and functions, and which give background 
information about Lie group theoretical concepts and about 
currently valid definitions and values. 


Availability

LiE is written in C, and can be made available on any system 
running UNIX or comparable operating systems, and (with a little 
more effort) on many other machines with a C compiler. The 
present version is available for the following computers: Sun 3, 
Sun 4 and SparcStation, DEC VAX, DEC Station 3100, Silicon Graphics 
workstations, IBM RS 6000 series, NeXT, PC-compatible computers 
(MS-DOS), Atari and Apple Macintosh. On Unix type computers the
input preprocessor Pre makes editing and repeating commands an 
easy job.


Support and information 

A manual, which treats the mathematical functions, the language 
and the interpreter, comes with the package. It can be ordered 
separately as well. One part of the manual is devoted to the 
link with the mathematical theory and another chapter is devoted 
to examples. The CAN Expertise Centre maintains the software, 
conducts user support, and deals with bug reports, and so on. 
If you want more information beyond what the manual offers, 
have interesting algorithms you may want us to know, have any 
other helpful comments, or if you want to order the package, 
please contact 

CAN Expertise Centre 
Kruislaan 413, 1098 SJ Amsterdam
The Netherlands 
Phone:	+31 20 5926050 
Fax:	+31 20 5924199 
E-mail:	lie@can.nl



Some facts about LiE

Size

300 KB object code and 875 KB is needed for the whole LiE 
directory on a Sun 4. LiE claims memory dynamically. The source 
code consists of about 17,000 lines of C code.

LiE is a very open system: all source files come with the package!


Authors

LiE was developed by the computer algebra group at the Centre for 
Mathematics and Computer Science in Amsterdam. Arjeh M. Cohen, 
who is the leader of this group, started and coordinated the 
project. Marc van Leeuwen was the main author of the LiE manual. 
Bert Lisser has been the scientific programmer.


LiE's role in mathematics

LiE has been used to solve Kostant's conjecture dating from 1983 
regarding subgroups of exceptional Lie groups.


Examples of built-in mathematical functions 


LiE provides data of Lie theoretical nature:

> diagram( F4 )

O---O=>=O---O
1   2   3   4 
F4


On-line help is available, e.g. about the function to compute 
Kazhdan-Lusztig polynomials:

> ?KL_poly

KL_poly(vec,vec,grp)-> pol
KL_poly(x,y,g) [x,y: Weyl word, 
result: polynomial]. 
Returns the Kazhdan-Lusztig 
polynomial P_{x,y}.


An example: 

> KL_poly( [1,2], [1,2,3,4,2,1,2,3,2,4], F4 )

In the response below you see LiE's notation for the polynomial 
1+2x+x^2: 

1X[0] + 2X[1] + 1X[2]


You can learn Lie theoretical concepts with LiE:

> learn highest root

Highest root This is the maximum of  the set of roots with 
respect to the partial ordering <<. It is the highest weight of 
the adjoint representation.


The polynomial representing the dominant part of the character of 
the B3 -module with highest weight [1,1,1] is computed by:

> dom_char( all_one(3), B3 )

14X[0,0,1] + 2X[0,0,3] + 4X[0,1,1]
+ 8X[1,0,1] + 1X[1,1,1] + 2X[2,0,1]


The decomposition polynomial of the third symmetric tensor power 
of the A3-module with highest weight [1,1,1] is computed by:

> plethysm( [3], [1,1,1], A3 )

 4X[0,0,2] + 1X[0,0,6] +  3X[0,1,0] + 3X[0,1,4] + 8X[0,2,2] + 
 6X[0,3,0] + 1X[0,3,4] +  2X[0,4,2] + 1X[0,5,0] + 7X[1,0,3] +
11X[1,1,1] + 1X[1,1,5] +  6X[1,2,3] + 8X[1,3,1] + 1X[1,5,1] +
 4X[2,0,0] + 4X[2,0,4] + 12X[2,1,2] + 8X[2,2,0] + 1X[2,2,4] + 
 3X[2,3,2] + 2X[2,4,0] +  7X[3,0,1] + 1X[3,0,5] + 4X[3,1,3] + 
 6X[3,2,1] + 1X[3,3,3] +  4X[4,0,2] + 3X[4,1,0] + 1X[4,2,2] + 
 1X[4,3,0] + 1X[5,0,3] +  1X[5,1,1] + 1X[6,0,0]


The types of the maximal proper subgroups of E8 are:

> max_sub( E8 )

G2F4,C2,A1A2,A1,A1,A1,D8,A8,A7A1,A5
A2A1,A4A4,D5A3,E6A2,E7A1


Programming example

The Coxeter matrix of a Weyl group is the matrix with entries m_ij 
equal to the order of the product r_i r_j of the fundamental 
reflections ri and rj. Below is a way to compute the matrix 
based on its definition. It illustrates the programming capacity 
of LiE:

cox_mat(grp g)= 
{loc m=id(Lie_rank(g)); 
for i=1 to n_rows(m)-1 do 
  for j=i+1 to n_rows(m) do 
    m[i,j]=ord(W_action([i,j],g)); 
    m[j,i]=m[i,j] 
od od;
m } 

ord(mat m) = 
{loc p=m; loc i=1; 
loc idmat=id(n_rows(m));
while p != idmat do p=p*m;i+=1 od;
i }



"Our primary goal in realising the present version has been to cover
(on-line) the mathematical content of the following three books:
J.Tits, Tabellen zu den einfachen LieGrfuppen und ihren Darstellungen,
        Lecture Notes in Math. 40, Springer, Berlin, 1967.
M.R. Bremner, R.V. Moody, J. Patera, Tables of dominant weight
multiplicities for representations of simple Lie algebras, Monographs and
        Textbooks in Pure and Appl. Math. 90, Dekker, New York, 1985.
W.G. McKay & J. Patera, Tables of dimensions, indices and vranching rules 
        for representations of simple Lie algebras, Lecture Notes in Pure 
        and Appl. Math 69, Dekker, New York, 1981."

