bruhat_catalan_stratification.sagews

def l_matrix(n,i,t):
    return elementary_matrix(n, row1=i-1, row2=i, scale=t)

def bruhat_to_catalan(n):
    for w in Permutations(n):
        print(w)
        M = identity_matrix(n)
        a = w.reduced_word()
        R = PolynomialRing(QQ, 'x', len(a))
        for i in range(len(a)):
            M = l_matrix(n,a[i],R.gens()[i])*M
        print(M)
        print()

bruhat_to_catalan(3)

[1, 2, 3] [1 0 0] [0 1 0] [0 0 1] [1, 3, 2] [1 0 0] [0 1 x] [0 0 1] [2, 1, 3] [1 x 0] [0 1 0] [0 0 1] [2, 3, 1] [ 1 x0 0] [ 0 1 x1] [ 0 0 1] [3, 1, 2] [ 1 x1 x0*x1] [ 0 1 x0] [ 0 0 1] [3, 2, 1] [ 1 x1 x0*x1] [ 0 1 x0 + x2] [ 0 0 1]

bruhat_to_catalan(4)

[1, 2, 3, 4] [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] [1, 2, 4, 3] [1 0 0 0] [0 1 0 0] [0 0 1 x] [0 0 0 1] [1, 3, 2, 4] [1 0 0 0] [0 1 x 0] [0 0 1 0] [0 0 0 1] [1, 3, 4, 2] [ 1 0 0 0] [ 0 1 x0 0] [ 0 0 1 x1] [ 0 0 0 1] [1, 4, 2, 3] [ 1 0 0 0] [ 0 1 x1 x0*x1] [ 0 0 1 x0] [ 0 0 0 1] [1, 4, 3, 2] [ 1 0 0 0] [ 0 1 x1 x0*x1] [ 0 0 1 x0 + x2] [ 0 0 0 1] [2, 1, 3, 4] [1 x 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] [2, 1, 4, 3] [ 1 x0 0 0] [ 0 1 0 0] [ 0 0 1 x1] [ 0 0 0 1] [2, 3, 1, 4] [ 1 x0 0 0] [ 0 1 x1 0] [ 0 0 1 0] [ 0 0 0 1] [2, 3, 4, 1] [ 1 x0 0 0] [ 0 1 x1 0] [ 0 0 1 x2] [ 0 0 0 1] [2, 4, 1, 3] [ 1 x0 0 0] [ 0 1 x2 x1*x2] [ 0 0 1 x1] [ 0 0 0 1] [2, 4, 3, 1] [ 1 x0 0 0] [ 0 1 x2 x1*x2] [ 0 0 1 x1 + x3] [ 0 0 0 1] [3, 1, 2, 4] [ 1 x1 x0*x1 0] [ 0 1 x0 0] [ 0 0 1 0] [ 0 0 0 1] [3, 1, 4, 2] [ 1 x1 x0*x1 0] [ 0 1 x0 0] [ 0 0 1 x2] [ 0 0 0 1] [3, 2, 1, 4] [ 1 x1 x0*x1 0] [ 0 1 x0 + x2 0] [ 0 0 1 0] [ 0 0 0 1] [3, 2, 4, 1] [ 1 x1 x0*x1 0] [ 0 1 x0 + x2 0] [ 0 0 1 x3] [ 0 0 0 1] [3, 4, 1, 2] [ 1 x1 x0*x1 0] [ 0 1 x0 + x3 x2*x3] [ 0 0 1 x2] [ 0 0 0 1] [3, 4, 2, 1] [ 1 x1 x0*x1 0] [ 0 1 x0 + x3 x2*x3] [ 0 0 1 x2 + x4] [ 0 0 0 1] [4, 1, 2, 3] [ 1 x2 x1*x2 x0*x1*x2] [ 0 1 x1 x0*x1] [ 0 0 1 x0] [ 0 0 0 1] [4, 1, 3, 2] [ 1 x2 x1*x2 x0*x1*x2] [ 0 1 x1 x0*x1] [ 0 0 1 x0 + x3] [ 0 0 0 1] [4, 2, 1, 3] [ 1 x2 x1*x2 x0*x1*x2] [ 0 1 x1 + x3 x0*x1 + x0*x3] [ 0 0 1 x0] [ 0 0 0 1] [4, 2, 3, 1] [ 1 x2 x1*x2 x0*x1*x2] [ 0 1 x1 + x3 x0*x1 + x0*x3] [ 0 0 1 x0 + x4] [ 0 0 0 1] [4, 3, 1, 2] [ 1 x2 x1*x2 x0*x1*x2] [ 0 1 x1 + x4 x0*x1 + x0*x4 + x3*x4] [ 0 0 1 x0 + x3] [ 0 0 0 1] [4, 3, 2, 1] [ 1 x2 x1*x2 x0*x1*x2] [ 0 1 x1 + x4 x0*x1 + x0*x4 + x3*x4] [ 0 0 1 x0 + x3 + x5] [ 0 0 0 1]