restart:with(linalg): with(combinat): with(LinearAlgebra):with(ListTools):libname:= `/homes/vector4/k/kapovich/CONVEX`, libname: with(convex):The goal of the program irred.mw is to show that the system of stability and chamber inequalities is irredundant and that it determines a cone D=D(SO(8)) which has 306 facets and 81 extremal rays. StackMatrix:= proc(A,B) local T:
T:=convert(stackmatrix(A,B), Matrix): end:TranSpose:= proc(M) local T:
T:=convert(transpose(M),Matrix): end: Triality action on the given n-by-12 Matrix M.
We start with the empty 0-by-12 matrix S. We convert it to 6*n-by-12 matrix S obtained by applying triality permutations to M. Namely, each row r of M is written in the form [w1, w2, w3]. Triality acts by simultaneously permuting 1, 3, 4 coordinates of the vectors wi. We will be primarily using it for n=162.
Below is a program for the triality permutation:
triality:=proc(M) local n, i, r, w, V, j, k, v, L, LL, s, ll, u, S; S:=matrix(0,12): n:=RowDimension(M):
for i from 1 to n do
r:=Row(M, i): w[1]:=convert(SubVector(r, [1..4]), vector): w[2]:=convert(SubVector(r, [5..8]), vector): w[3]:=convert(SubVector(r, [9..12]), vector):
V:=convert(stackmatrix(w[1], w[2], w[3]), Matrix):
for j from 1 to 4 do v[j]:= Column(V, j): od: L:=[v[1], v[3], v[4]]: #list of columns to permute
LL:=permute(L): #permutation of the columns
s:=matrix(0,12):
for k from 1 to 6 do #begin smallcycle
ll[k]:= augment(LL[k,1], v[2], LL[k,2], LL[k,3]):
u[k]:= convert(ll[k], vector):
s:=stackmatrix(s, u[k]):
od: S:= stackmatrix(S,s): od:
S:=convert(S,Matrix):
#print(S):
end: #we are NOT printing the output matrix
PermutationBelow is a program for permutations of the rows of an n-by-12 matrix X. Each row r is written in the form r=[w1, w2, w3] and then the subvectors wi are permuted. The output is the matrix T which is 6*n-by-12.permutation:=proc(X) local n, i, W, w, Lis, lis, j, l, T; T:=Matrix(0,12): n:=RowDimension(X):
for i from 1 to n do
W:=Row(X,i): w[1]:=SubVector(W, [1..4]): w[2]:=SubVector(W, [5..8]): w[3]:=SubVector(W, [9..12]):
Lis:=[w[1], w[2], w[3]]: lis:=permute(Lis):
for j from 1 to 6 do
l[j]:=convert(lis[j],Vector[row]): T:=StackMatrix(T, l[j]):
od:
od:
#print(T); #We are NOT printing the output matrix
end:
Procedure remdup removes duplicate members of a listremdup:=proc(L) local LL, n, k, x:
n:=nops(L); # number of entries in L
if n<=1 then
return L; # if L has 0 or 1 entries we are done
fi;
LL:=[L[1]]; # add the first entry of L to LL
for k from 2 to n do
if member(L[k],L[1..k-1]) then
; # do not add any entries which come earlier
else # (so in effect remove later duplicates)
LL:=[op(LL),L[k]];
fi;
od;
LL:
end:Procedure Cull removes duplicate rows of a matrix: Cull:=proc(M) local n, m, r, i, L, l, N :
n:=RowDimension(M): m:=ColumnDimension(M):
L:=[]:
for i from 1 to n do r:=[seq(M[i,j], j=1..m)]:
L:= [op(L), r]: od:
l:=remdup(L):
N:=convert(l,Matrix);
end: Below we are entering generalized triangle inequalities for SO(8). We first enter unpermuted inequalities; we then apply S_3 x S_3 action to permute them. We start with the Grassmanian G/P_2.b0:=matrix(1,4,[1,1,0,0]): b1:=matrix(1,4,[1,0,1,0]): b21:=matrix(1,4,[0,1,1,0]): b22:=matrix(1,4,[1,0,0,1]): b23:=matrix(1,4,[1,0,0,-1]): b31:=matrix(1,4,[0,1,0,1]): b32:=matrix(1,4,[0,1,0,-1]): b33:=matrix(1,4,[1,0,-1,0]): b41:=matrix(1,4,[0,0,1,1]): b42:=matrix(1,4,[0,1,-1,0]): b43:=matrix(1,4,[0,0,1,-1]):b44:=matrix(1,4,[1,-1,0,0]):ERI (essential reduced inequalities) for G/P_2:BB1:=augment(b1, b1, -b21):BB2:=augment(b1, b21, -b31):BB12:=augment(b22, b23, -b42):BB11:=augment(b22, b22, -b41):BB10:=augment(b21, b23, -b42):BB9:=augment(b21, b22, -b42):BB8:=augment(b21, b21, -b41):BB7:=augment(b1, b33, -b42):BB6:=augment(b1, b32, -b42):BB5:=augment(b1, b31, -b42):BB4:=augment(b1, b31, -b41):BB3:=augment(b1, b22, -b31):ERI2:=stackmatrix(BB1,BB2,BB3,BB4,BB5,BB6,BB7,BB8,BB9,BB10,BB11,BB12):#These are the ERI for G/P_2Weak Triange Inequalities (WTI) for G/P_2:BB13:=augment(b0,b0,-b0): BB14:=augment(b1,b0,-b1): BB15:=augment(b21,b0,-b21): BB16:=augment(b31,b0,-b31): BB17:=augment(b41,b0,-b41): BB18:=augment(b42,b0,-b42):WTI2:=stackmatrix(BB13,BB14,BB15,BB16,BB17,BB18):#WTE for G/P_2We now enter inequalities for the Grassmanian G/P_1Generators: d0:=matrix(1,4,[1,0,0,0]):d1:=matrix(1,4,[0,1,0,0]):d2:=matrix(1,4,[0,0,1,0]): d32:=matrix(1,4,[0,0,0,-1]): d31:=matrix(1,4,[0,0,0,1]):d4:=matrix(1,4,[0,0,-1,0]): d5:=-d1: d6:=-d0:ERI1:=stackmatrix(augment(d1,d1,d4), augment(d1,d2,d31), augment(d2,d32,-d5)):WTI1:=stackmatrix(augment(d0,d0,d6), augment(d1,d0,d5), augment(d31,d0,d32), augment(d2,d0,d4)): Below is the full matrix of unpermuted inequalities for SO(8):inequalities:=stackmatrix(ERI1, WTI1, ERI2, WTI2):M:=convert(inequalities, Matrix):Dimensions(M);The matrix Inequalities contains all the unpermuted inequalities written in Bourbaki coordinates. We next convert to the fundamental weight coordinates.We then apply triality and permutation to the above 25 inequalities to produce the full system of inequalities. We first write the transition matrix. o:=Matrix(12,12);#zero matrixomega := Matrix([[1, 1, 1/2, 1/2], [0, 1, 1/2, 1/2], [0, 0, 1/2, 1/2], [0, 0, (-1)/2, 1/2]]);#matrix of fundamental weightso[1..4,1..4]:=omega: o[5..8,5..8]:=omega: o[9..12,9..12]:=omega: Omega:=o;#transition matrixid:=IdentityMatrix(12):m:=Multiply(M,Omega);We now permute matrix m. We first apply permutation and then triality: am:=permutation(m,25):Dimensions(am);pm:=triality(am,150):Dimensions(pm);k:=Cull(pm):RowDimension(k);#This is the number of permuted inequalities leftK:=StackMatrix(k,id);#This is the final inequality matrix in varpi-coordinates, including the chamber inequalitiesExportMatrix("K.txt", K):
#Saving matrix K for the future use, e.g. for computing Hilbert basis. The matrix k had only 294 distinct rows. The matrix k is the full matrix of stability inequalities in varpi-coordinates. What is missing are the chamber inequalities which amount to requiring that all vectors in question belong to the positive orthant C. Below we use the package CONVEX to compute the facets of the cone D obtained by imposing the inequalities given by k on the positive orthant P. The package also computes the extremal rays of the cone D. P:=posorthant(12): C:=P:
for i from 1 to 294 do
r[i]:=Row(k,i): C:= intersection(C, r[i]) od:
h:=hspaces(C): #It took my computer about 5 minutes to compute the facets of the cone C. H:=convert(h,Matrix): NumberOfReducedInequalities:=RowDimension(H);rc:=rays(C): RC:=convert(rc,Matrix): NumberOfRays:=RowDimension(RC);Since the cone D was given by a system of 306 inequalities and has 306 facets, all these inequalities are irredundant. Thus the system of linear inequalities given by the matrix K is irredundant. The cone D has 81 extremal rays.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M7R0
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