One part of the manual is devoted to the link with the mathematical theory and another chapter is devoted to examples. If you want more information beyond what the manual offers, have interesting algorithms you may want us to know, have any other helpful comments, please contact Arjeh M. Cohen , email amc@win.tue.nl or Marc A.A. van Leeuwen , email maavl@cwi.nl.

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 height partial ordering: lambda is higher than mu iff lambda-mu is a sum of positive roots. 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

cox_mat(grp g)= { loc m=id(Lie_rank(g)) # start with ones on the diagonal ; for i=1 to n_rows(m)-1 do for j=i+1 to n_rows(m) # for off-diagonal entries use ord do m[i,j]=ord(W_action([i,j],g)); m[j,i]=m[i,j] od od ; m } ord(mat m) = # the multiplicative order of a matrix { loc p=m; loc i=1; loc idmat=id(n_rows(m)) ; while p != idmat do p=p*m; i+=1 od ; i }

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Last updated: December 19, 1996
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