representations.sage

%display latex

def get_vecs(n):
    '''Get the orthonormal basis vectors of RR^n'''
    vecs = []
    for i in range(n):
        vecs.append([1 if x == i else 0 for x in range(n)])
    return vecs

def apply_permutation(lst, perm):
    return [lst[i-1] for i in perm]

def get_matr(p):
    return matrix(
        apply_permutation(
            get_vecs(len(p)),
            p
        ),
        sparse=True
    )

def get_isomorphism_classes(n):
    isomorphism_classes = dict()
    for p in Permutations(n):
        M1 = get_matr(p)
        G1 = MatrixGroup([M1])
        is_new = True  # is this a new isomorphism class?
        for k in isomorphism_classes.keys():
            M2 = get_matr(isomorphism_classes[k][0])
            G2 = MatrixGroup([M2])
            if G1.is_isomorphic(G2):
                is_new = False
                isomorphism_classes[k].append(p)
                continue
        if is_new:
            isomorphism_classes[str(p)] = [p]
    return isomorphism_classes