1921.sagews

n=3;
variables=[]
for row in range(n):
    variables.append([]);
    for column in range(n):
        variables[row].append(var('z'+str(n-row)+str(column+1)))
variables

[[z31, z32, z33], [z21, z22, z23], [z11, z12, z13]]

p=[3,2,1]
for column in range (n):
    variables[p[column]-1][column]=1
    for column_prime in range (column+1,n):
        variables[p[column]-1][column_prime]=0
    for row in range (p[column],n):
        variables[row][column]=0

M = matrix(variables);M

[z31 z32 1] [z21 1 0] [ 1 0 0]

e=[]
zero_vector=list(0 for dummy in range(n))
for i in range (n):
    zero_vector[i]=1;
    e.append(vector(zero_vector));
    zero_vector[i]=0;
e

[(1, 0, 0), (0, 1, 0), (0, 0, 1)]

s=[]
for row in range (n):
    s.append(e[row]*(row+1));
    
N=matrix(s)
N

[1 0 0] [0 2 0] [0 0 3]

a=[];
for row in range(n):
    a.append([])
    for column in range(row+2):
        a[row].append(var('a'+str(row+1)+str(column+1)))
a

[[a11, a12], [a21, a22, a23], [a31, a32, a33, a34]]

X=[];
for i in range(n-2):
    X.append([]);
    X[i]=vector(zero_vector)
    for j in range (i+2):
        X[i]=X[i]+a[i][j]*M*e[j]
X

[(a11*z31 + a12*z32, a11*z21 + a12, a11)]

for j in range(n-2):
    for i in range(n):
        print(e[i]*N*M*e[j],e[i]*X[j])

(z31, a11*z31 + a12*z32) (2*z21, a11*z21 + a12) (3, a11)

for j in range(n-2):
    for i in range(n):
        print(e[i]*X[j],e[i]*N*M*e[j])

(a11*z31 + a12*z32, z31) (a11*z21 + a12, 2*z21) (a11, 3)

a11= 3; a11

3

a12 = 2*z21 -a11*z21; a12

-z21

ideal(a11*z31 + a12*z32- z31)

Principal ideal (-z21*z32 + 2*z31) of Symbolic Ring

a11*z31 + a12*z32

-z21*z32 + 3*z31

print 'The cell of 321 in the Peterson variety is the set of all matrices of the form'
print '[z31 z32   1]'
print '[z21   1   0]'
print '[  1   0   0]'
print 'where z31 = -z21*z32 + 3*z31.'

The cell of 321 in the Peterson variety is the set of all matrices of the form [z31 z32 1] [z21 1 0] [ 1 0 0] where z31 = -z21*z32 + 3*z31.