DeRhamComputation/sage/as_covers/group_action_matrices.sage

def group_action_matrices(space, list_of_group_elements, basis):
    n = len(list_of_group_elements)
    d = len(space)
    A = [matrix(F, d, d) for i in range(n)]
    for i, g in enumerate(list_of_group_elements):
        for j, omega in enumerate(space):
            omega1 = omega.group_action(g)
            v1 = omega1.coordinates(basis = basis)
            A[i][:, j] = vector(v1)
    return A

def group_action_matrices_holo(AS):
    n = AS.height
    generators = []
    for i in range(n):
        ei = n*[0]
        ei[i] = 1
        generators += [ei]
    return group_action_matrices(AS.holomorphic_differentials_basis(), generators, basis = AS.holomorphic_differentials_basis())
    
def group_action_matrices_dR(AS, threshold=8):
    n = AS.height
    generators = []
    for i in range(n):
        ei = n*[0]
        ei[i] = 1
        generators += [ei]
    basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis(), AS.de_rham_basis(threshold=threshold)]
    return group_action_matrices(basis[2], generators, basis = basis)
    
def group_action_matrices_old(C_AS):
    F = C_AS.base_ring
    n = C_AS.height
    holo = C_AS.holomorphic_differentials_basis()
    holo_forms = [omega.form for omega in holo]
    denom = LCM([denominator(omega) for omega in holo_forms])
    variable_names = 'x, y'
    for j in range(n):
        variable_names += ', z' + str(j)
    Rxyz = PolynomialRing(F, n+2, variable_names)
    x, y = Rxyz.gens()[:2]
    z = Rxyz.gens()[2:]
    holo_forms = [Rxyz(omega*denom) for omega in holo_forms]
    A = [[] for i in range(n)]
    for omega in holo:
        for i in range(n):
            ei = n*[0]
            ei[i] = 1
            omega1 = omega.group_action(ei)
            omega1 = denom * omega1
            v1 = omega1.coordinates(holo_forms)
            A[i] += [v1]
    for i in range(n):
        A[i] = matrix(F, A[i])
        A[i] = A[i].transpose()
    return A

def group_action_matrices_log(C_AS):
    F = C_AS.base_ring
    n = C_AS.height
    holo = C_AS.at_most_poles_forms(1)
    holo_forms = [omega.form for omega in holo]
    denom = LCM([denominator(omega) for omega in holo_forms])
    variable_names = 'x, y'
    for j in range(n):
        variable_names += ', z' + str(j)
    Rxyz = PolynomialRing(F, n+2, variable_names)
    x, y = Rxyz.gens()[:2]
    z = Rxyz.gens()[2:]
    holo_forms = [Rxyz(omega*denom) for omega in holo_forms]
    A = [[] for i in range(n)]
    for omega in holo:
        for i in range(n):
            ei = n*[0]
            ei[i] = 1
            omega1 = omega.group_action(ei)
            omega1 = denom * omega1
            v1 = omega1.coordinates(holo_forms)
            A[i] += [v1]
    for i in range(n):
        A[i] = matrix(F, A[i])
        A[i] = A[i].transpose()
    return A