restart:

with(linalg): with(combinat): with(LinearAlgebra):with(ListTools):with(weyl):with(coxeter):

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the assigned name fibonacci now has a global binding

Warning, the protected name Chi has been redefined and unprotected

Warning, the assigned name GramSchmidt now has a global binding

Warning, the names DotProduct and Transpose have been rebound

Warning, the assigned name Group now has a global binding

Warning, the protected name tensor has been redefined and unprotected

Warning, the name rank has been redefined

StackMatrix:= proc(A,B) local T: T:=convert(stackmatrix(A,B), Matrix): end:

TranSpose:= proc(M) local T: T:=convert(transpose(M),Matrix): end:

We first download matrix K of the inequalities in varpi-coordinates:

K:=ImportMatrix("K.txt"):

Transition matrix from {al_i, zeta_j}- to varpi-coordinates:

Lambda:=Matrix([[2, -1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0], [-1, 2, -1, -1, 0, 0, -1, -1, 0, 0, 0, 0], [0, -1, 2, 0, 0, 0, 0, 0, -1, -1, 0, 0], [0, -1, 0, 2, 0, 0, 0, 0, 0, 0, -1, -1], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]]);

NiM+SSdMYW1iZGFHNiItSSdSVEFCTEVHRiU2KyIqIXlaXThJKWFueXRoaW5nR0kqcHJvdGVjdGVkR0YrSSdNYXRyaXhHNiRGK0koX3N5c2xpYkdGJUkscmVjdGFuZ3VsYXJHRiVJLkZvcnRyYW5fb3JkZXJHRiU3IiIiIzsiIiIiIzdGMw==

evalm(Lambda);

NiMtSSdtYXRyaXhHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjNy43LiIiIyEiIiIiIUYuRi1GLUYuRi5GLkYuRi5GLjcuRi1GLEYtRi1GLkYuRi1GLUYuRi5GLkYuNy5GLkYtRixGLkYuRi5GLkYuRi1GLUYuRi43LkYuRi1GLkYsRi5GLkYuRi5GLkYuRi1GLTcuRi5GLkYuRi4iIiJGLkYuRi5GLkYuRi5GLjcuRi5GLkYuRi5GLkYuRjNGLkYuRi5GLkYuNy5GLkYuRi5GLkYuRi5GLkYuRjNGLkYuRi43LkYuRi5GLkYuRi5GLkYuRi5GLkYuRjNGLjcuRi5GLkYuRi5GLkYzRi5GLkYuRi5GLkYuNy5GLkYuRi5GLkYuRi5GLkYzRi5GLkYuRi43LkYuRi5GLkYuRi5GLkYuRi5GLkYzRi5GLjcuRi5GLkYuRi5GLkYuRi5GLkYuRi5GLkYz

U:=Multiply(K, Lambda):

tU:=TranSpose(U);#transpose of U

NiM+SSN0VUc2Ii1JJ1JUQUJMRUdGJTYrIipfZmRRIkkpYW55dGhpbmdHSSpwcm90ZWN0ZWRHRitJJ01hdHJpeEc2JEYrSShfc3lzbGliR0YlSSxyZWN0YW5ndWxhckdGJUkuRm9ydHJhbl9vcmRlckdGJTciIiIjOyIiIiIjNztGNCIkMSQ=

We save this matrix in the file tU.txt :

ExportMatrix("tU.txt", tU):

Now copy the file tU.txt to the file the file tU.lat (you might need tU.txt later on!) and edit the latter by inserting an extra row: 12 306

which are the dimensions of the matrix tU. We then run the program 4ti2 on the file tU.lat :

>./hilbert --quiet tU

Remarks:

1. Option --quiet ensures that there is a minimal output on the terminal while the program is working.

2. It would be a mistake to run ./hilbert tU.lat : We are grateful to Raymond Hemmecke for this explanation

After running for about 10 minutes the program produced the output file tU.hil

We then convert this output file into MAPLE format:

>./output --maple tU.hil

which yields the file tU.hil.maple

This file contains 82-by-306 matrix whose rows are elements of the Hilbert basis of the semigroup generated by the rows of the matrix tU.

It is very likely that you would have to restart MAPLE at this point. Rather then recomputing the matrix U it is better to use the one

which was saved in the end of the previous session:

tu:=ImportMatrix("tU.txt");

NiM+SSN0dUc2Ii1JJ1JUQUJMRUdGJTYrIipXUUlOIkkpYW55dGhpbmdHSSpwcm90ZWN0ZWRHRitJJ01hdHJpeEc2JEYrSShfc3lzbGliR0YlSSxyZWN0YW5ndWxhckdGJUkuRm9ydHJhbl9vcmRlckdGJTciIiIjOyIiIiIjNztGNCIkMSQ=

hbmatrix:=ImportMatrix("tU.hil.maple"):

Dimensions(hbmatrix);

NiQiIiMiIiI=

HB:=Matrix(hbmatrix[1,1]);#The rows of this matrix are the above elements of the Hilbert basis

NiM+SSNIQkc2Ii1JJ1JUQUJMRUdGJTYrIipXXCUpUiJJKWFueXRoaW5nR0kqcHJvdGVjdGVkR0YrSSdNYXRyaXhHNiRGK0koX3N5c2xpYkdGJUkscmVjdGFuZ3VsYXJHRiVJLkZvcnRyYW5fb3JkZXJHRiU3IiIiIzsiIiIiIyMpO0Y0IiQxJA==

T:=TranSpose(tu);

NiM+SSJURzYiLUknUlRBQkxFR0YlNisiKnMtKFw4SSlhbnl0aGluZ0dJKnByb3RlY3RlZEdGK0knTWF0cml4RzYkRitJKF9zeXNsaWJHRiVJLHJlY3Rhbmd1bGFyR0YlSS5Gb3J0cmFuX29yZGVyR0YlNyIiIiM7IiIiIiQxJDtGNCIjNw==

#evalm(T);

for i from 1 to 82 do v[i]:=linsolve(T, Row(HB,i)): od:

We now define a matrix V whose rows are the vectors v[i] (i=1,...,82):

We next convert from {alpha_i, zeta_j}-coordinates to the varpi-coordinates.

It is done by multiplying V on the right by transpose of the matrix Lambda. The result is written in the matrix Z:

V:=Matrix(1..82, 1..12):

for i from 1 to 82 do for j from 1 to 12 do V[i,j]:=v[i][j]: od: od:

Z:=Multiply(V, TranSpose(Lambda)):

evalm(Z);

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

Rows of Z are elements of the Hilbert basis in varpi-coordinates.

We now do tensor product computations. We download weyl and coxeter packages written by John Stembridge (consult his homepage

for the installation instructions). Unfortunately, these packages appear to be incompatible with CONVEX.

with(weyl):with(coxeter):

The following procedure ten(lambda, mu, nu) determines if the tensor product of

V[lambda] and V[mu] contains V[nu] among its irreducible summamnds. If it does, the the output of the procedure is TRUE,

otherwise it is FALSE. Here lambda, mu, nu are dominant weights written in the varpi-coordinates.

Note that in this procedure we reverse the order of the fundamental coweights varpi_i from its Bourbaki

order to the opposite one, used by Stembridge's weyl package.

w:=weights(D4):

ten:= proc(lambda, mu, nu) local l, m, t, tau: l:=lambda[1]*w[4]+lambda[2]*w[3]+lambda[3]*w[2]+lambda[4]*w[1]: m:=mu[1]*w[4]+mu[2]*w[3]+mu[3]*w[2]+mu[4]*w[1]: t:=tensor(l, m, D4): if coeff(t, X[nu[4], nu[3], nu[2], nu[1]])=1 then tau:=true else tau:=false: fi: print(tau): end:

The next procedure check takes an n-by-12 matrix R as its input. For each row R[i] of R, it writes R[i] as the

triple of vectors (lambda, mu, nu) and then applies

ten(lambda, mu, nu)

to determine if the tensor product of V[lambda] and V[mu] contains V[nu] among its irreducible summamnds.

check:=proc(R, n) local i, r, lambda, mu, nu: for i from 1 to n do lambda:= convert(R[i,1..4], Vector): mu:= convert(R[i,5..8], Vector): nu:= convert(R[i,9..12], Vector): r:=ten(lambda, mu, nu): print(r): od: end:

check(Z, 82):