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UHW cohomology pro Ondru

Project: Dissertation
Views: 102

Helper functions


... mainly for producing LaTeX 

def setup(CT, index_set):
    W = WeylGroup(CT)
    AS = W.domain()
    FW = AS.fundamental_weights()
    vFW = FW.map(lambda v: v.to_vector())
    RP = AS.root_poset(facade=True)
    nonparabolic_roots = [x.to_ambient() for x in AS.root_system.root_lattice().positive_roots_nonparabolic(index_set=index_set)]
    nRP = RP.subposet(nonparabolic_roots)
    rho = AS.rho()
    return W, AS, FW, vFW, rho, nRP

def inject_positive_integer_variables(names, n=1):
    """
    If names is a string (e.g. "A") this function injects into the workspace n variables named A till An.
    Otherwise it is assumed that names is a list of variables that will be injected.
    They are assumed to take only nonnegative integral values.
    """
    def inject_name(name, n):
        for i in range(n):
            if n != 1:
                var(name + str(i+1), "integer")
                eval("assume(%s%d >= 0)" %(name, i+1))
            else:
                var(name, "integer")
                eval("assume(%s >= 0)" %name)

    if isinstance(names, basestring):
        inject_name(names, n)
    else:
        for name in names:
            inject_name(name, n)

def setup_cone(cone_str):
    forget()
    print("Initial assumptions:")
    print(assumptions())
    print("Cone: %s with assumptions:" % cone_str)
    print(assumptions())
    exec cone_str in globals(), locals()

class RootWithScalarProduct:
    """
    Use this to relabel graphs of positive roots with scalar product with given weight v.
    """
    def __init__(self, r, v):
        self.root = r
        self.scalarproduct = v.dot_product(r.to_vector())

    def __latex__(self):
        return latex(self.scalarproduct)

    def __str__(self):
        return str(self.scalarproduct)

def latex_poset_scalar_product(poset, v, only_nonnegative=True):
    """
    Returns LaTeX code of poset of roots whose nodes were labeled by inner product of those roots with give weight v.
    """
    p = poset.relabel(lambda r: RootWithScalarProduct(r, v))
    if only_nonnegative:
        p = p.subposet([x for x in p if not(x.scalarproduct < 0)])
    return str(latex(p)).replace("\\texttt", "").replace("\\text", "").replace("*", "")

def fix_basis_latex(obj):
    """
    Basis of Ambient spaces are indexed by e_0, ..., e_{n-1} instead of conventional \epsilon_1, ..., \epsilon_n.
    This functions returns string with fixed LaTeX source. You can render its output in notebook by calling latex.eval(...)
    """
    import re
    def shift_number(matchobj):
        return "e_{%d}" % (int(matchobj.group(1)) + 1)

    index_re = re.compile("e_{(\d+)}")
    return index_re.sub(shift_number, str(latex(obj))).replace("e_{", "\epsilon_{")

Functions that calculate cohomology of unitarizable modules via Enright's formula 

def WG_action(w, v):
    """
    Action of weyl group element w on vector v.
    Workaround for subgroups not containing elements of the supergroup.
    """
    AS = v.parent()
    return AS.from_vector(w.matrix()*v.to_vector())

def get_length_function(positive_roots):
    positive_roots = set(positive_roots)
    @cached_function
    def l(w):
        return len([a for a in positive_roots if WG_action(w.inverse(), -a) in positive_roots])
    return l

def get_generating_roots(weight, index_set):
    """
    Returns a list of roots that generate the reflection subgroup which governs cohomology of unitarizable hihgest weight modules.
    First part of Enright's formula from his paper on u-cohomology.
    """
    AS = weight.parent() # the ambient space
    rho = AS.rho()
    Psi = [r for r in AS.positive_roots() if r.scalar(rho + weight) == 0]

    if AS.cartan_type()[0] in "BCG":
        long_root = any(not(x.is_short_root()) for x in Psi)
    else:
        long_root = False

    def test_root(r):
        n = r.associated_coroot().scalar(weight + rho) # TODO check coroot calculations
        if long_root:
            short = r.is_short_root()
        else:
            short = True
        #print r, long_root, short
        return n.is_integer() and n > 0 and short

    nonparabolic_roots = [x.to_ambient() for x in AS.root_system.root_lattice().positive_roots_nonparabolic(index_set=index_set)]
    parabolic_roots = [x.to_ambient() for x in AS.root_system.root_lattice().positive_roots_parabolic(index_set=index_set)]
    #print("Nonparabolic roots: %s" % sorted(nonparabolic_roots))
    Phi = [r for r in nonparabolic_roots if test_root(r) and all(r.scalar(s) == 0 for s in Psi)]
    return Phi, Psi, parabolic_roots, nonparabolic_roots

def generate_subgroup(generators):
    """
    Keep multiplying and taking inverses as long as new elements are constructed.
    Unfortunately, this routine takes too much time in practice.
    """
    new = set(a*b for (a,b) in cartesian_product([generators, generators])).union(set(g.inverse() for g in generators))
    if new == generators:
        return new
    else:
        return generate_subgroup(new)

def DyerN(w):
    W = w.parent()
    return [t for t in W.reflections() if (t*w).length() < w.length()]

def DyerCoxeterGenerators(H):
    return [w for w in H if set(DyerN(w)) == set([w])]

def get_subsystem_data(weight, index_set):
    AS = weight.parent()
    W = AS.weyl_group()
    generating_roots, Psi, parabolic_roots, nonparabolic_roots =  get_generating_roots(weight, index_set)
    reflections = W.reflections()
    generators = set(reflections[r] for r in generating_roots)

    #W_lambda = list(generate_subgroup(generators)) # subgroup generates H as a matrix group and we lose all the WeylGroupElement methods # too slow
    #W_lambda = [W.element_class(W, h) for h in W.subgroup(generators)] # too slow
    W_lambda = W.subgroup(generators)
    W_lambda_reflections = []
    for x in W_lambda:
        g = W.element_class(W, x)
        if g in reflections:
            W_lambda_reflections.append(g)

    # calculate Coxeter generators of the reflection subgroup
    # see [Deodhar] or [Dyer] for proof
    def DyerCoxeterGenerators(H_reflections):
        # optimized version
        #H_reflections = [W.element_class(W, x) for x in reflections if x in H] # WARNING switching H and reflections leads to empty set!
        # H_reflections = [W.element_class(W, x) for x in H if W.element_class(W, x) in reflections] # the previous stopped working in Sage 7.6 # refactored shortly thereafter to assume that we have only reflections at input
        W_length = get_length_function(AS.positive_roots())
        def DyerN(w):
            w = W.element_class(W, w)
            return set(t for t in H_reflections if  W_length(t*w) < W_length(w))

        return [w for w in H_reflections if DyerN(w) == set([w])]

    coxeter_generators = DyerCoxeterGenerators(W_lambda_reflections)

    lambda_positive_roots = [r for r in reflections.keys() if reflections[r] in W_lambda_reflections]
    lambda_simple_roots = [r for r in reflections.keys() if reflections[r] in coxeter_generators]
    lambda_parabolic_roots = [r for r in lambda_positive_roots if r in parabolic_roots]
    lambda_nonparabolic_roots = [r for r in lambda_positive_roots if r in nonparabolic_roots]

    # decompose coset representative according to their length
    from collections import defaultdict
    def is_dominant(v, positive_roots):
        return all(v.scalar(r) > 0 for r in positive_roots)
    lambda_W_c = defaultdict(list)
    rho = AS.rho()
    lambda_length = get_length_function(lambda_positive_roots)
    for w in W_lambda:
        if is_dominant(WG_action(w, rho), lambda_parabolic_roots):
            lambda_W_c[lambda_length(w)].append(w)

    return Psi, generating_roots, lambda_simple_roots, lambda_positive_roots, lambda_parabolic_roots, lambda_nonparabolic_roots, W_lambda, lambda_W_c

def examine(weight, index_set, cone=None):
    Psi, generating_roots, lambda_simple_roots, lambda_positive_roots, lambda_parabolic_roots, lambda_nonparabolic_roots, H, lambda_W_c = get_subsystem_data(weight, index_set)
    print("Weight: %s" % weight)
    print("Singular roots: %s" % sorted(Psi))
    print("Set of generating roots: %s" % sorted(generating_roots))
    print("Set of generated roots: %s" % sorted(lambda_positive_roots))
    print("Simple roots: %s" % sorted(lambda_simple_roots))
    print("Scalar products of pairs of distinct simple roots: %s" % set(u.scalar(v) for (u,v) in cartesian_product([lambda_simple_roots, lambda_simple_roots]) if u != v))

    if len(lambda_simple_roots) > 1:
        CM =  CartanMatrix([[Integer(2*ri.scalar(rj)/ri.scalar(ri)) for rj in lambda_simple_roots] for ri in lambda_simple_roots])
        print("Cartan matrix:\n%s" % CM)
        latex.eval(fix_basis_latex(latex(DynkinDiagram(CM, lambda_simple_roots))))
    print("Noncompact lambda-roots: %s" % lambda_nonparabolic_roots)

    AS = weight.parent()
    if cone is not None:
        v = weight.to_vector() + cone
    else:
        v = weight.to_vector()
    print("Cone: %s" % cone)
    nonparabolic_roots = [x.to_ambient() for x in AS.root_system.root_lattice().positive_roots_nonparabolic(index_set=index_set)]
    nRP = AS.root_poset(facade=True).subposet(nonparabolic_roots)
    print("Scalar products of the weight in the cone with noncompact roots:")
    latex.eval(latex_poset_scalar_product(nRP, v))

Explicit calculations 

def unitarizable_cohomology_An(p, q, pp, qq, l):
    n = p + q
    index_set = tuple([x for x in range(1, n) if x != p])
    print("Levi roots: %s" % str(index_set))
    CT = ["A", n - 1]

    W, AS, FW, vFW, rho, nRP = setup(CT, index_set)

    forget()
    print("Initial assumptions:")
    print(assumptions())
    inject_positive_integer_variables("a", n-1)
    cone_str = "Ca = " +  " + ".join("a{i}*(vFW[{i}] - vFW[{p}])".format(i=i, p=p) for i in range(pp, n-qq+1) if i != p)
    print("Cone: %s with assumptions:" % cone_str)
    print(assumptions())
    exec cone_str in globals(), locals()

    v = FW[pp] + FW[n-qq] - (n + l + 1 - pp - qq)*FW[p] + rho
    print("Vertex: %s" % v)
    examine(v, index_set, Ca)

p = 3
q = 3
pp = 2 # 1 <= pp <= p
qq = 2 # 1 <= qq <= q
# l = 1 # 1 <= l <= min(pp, qq)

for l in range(1, min(pp, qq) + 1):
    print("Level of reduction: %d" % l)
    unitarizable_cohomology_An(p, q, pp, qq, l)
    print("\n" + "-"*80 + "\n")
Level of reduction: 1 Levi roots: (1, 2, 4, 5) Initial assumptions: [] Cone: Ca = a2*(vFW[2] - vFW[3]) + a4*(vFW[4] - vFW[3]) with assumptions: [a1 >= 0, a2 >= 0, a3 >= 0, a4 >= 0, a5 >= 0] Vertex: (3, 2, 0, 3, 1, 0) Weight: (3, 2, 0, 3, 1, 0) Singular roots: [] Set of generating roots: [(1, 0, 0, -1, 0, 0), (1, 0, 0, 0, -1, 0), (1, 0, 0, 0, 0, -1), (0, 1, 0, -1, 0, 0), (0, 1, 0, 0, -1, 0), (0, 1, 0, 0, 0, -1), (0, 0, 1, 0, -1, 0), (0, 0, 1, 0, 0, -1)] Set of generated roots: [(1, -1, 0, 0, 0, 0), (1, 0, -1, 0, 0, 0), (1, 0, 0, -1, 0, 0), (1, 0, 0, 0, -1, 0), (1, 0, 0, 0, 0, -1), (0, 1, -1, 0, 0, 0), (0, 1, 0, -1, 0, 0), (0, 1, 0, 0, -1, 0), (0, 1, 0, 0, 0, -1), (0, 0, 1, -1, 0, 0), (0, 0, 1, 0, -1, 0), (0, 0, 1, 0, 0, -1), (0, 0, 0, 1, -1, 0), (0, 0, 0, 1, 0, -1), (0, 0, 0, 0, 1, -1)] Simple roots: [(1, -1, 0, 0, 0, 0), (0, 1, -1, 0, 0, 0), (0, 0, 1, -1, 0, 0), (0, 0, 0, 1, -1, 0), (0, 0, 0, 0, 1, -1)] Scalar products of pairs of distinct simple roots: set([0, -1]) Cartan matrix: [ 2 -1 0 0 0] [-1 2 -1 0 0] [ 0 -1 2 -1 0] [ 0 0 -1 2 -1] [ 0 0 0 -1 2]
Noncompact lambda-roots: [(0, 0, 1, -1, 0, 0), (0, 1, 0, -1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, -1, 0, 0), (0, 0, 1, 0, 0, -1), (0, 1, 0, 0, -1, 0), (0, 1, 0, 0, 0, -1), (1, 0, 0, 0, -1, 0), (1, 0, 0, 0, 0, -1)] Cone: (0, 0, -a2, a4, 0, 0) Scalar products of the weight in the cone with noncompact roots:
-------------------------------------------------------------------------------- Level of reduction: 2 Levi roots: (1, 2, 4, 5) Initial assumptions: [] Cone: Ca = a2*(vFW[2] - vFW[3]) + a4*(vFW[4] - vFW[3]) with assumptions: [a1 >= 0, a2 >= 0, a3 >= 0, a4 >= 0, a5 >= 0] Vertex: (2, 1, -1, 3, 1, 0) Weight: (2, 1, -1, 3, 1, 0) Singular roots: [(0, 1, 0, -1, 0, 0), (0, 0, 1, 0, -1, 0)] Set of generating roots: [(1, 0, 0, 0, 0, -1)] Set of generated roots: [(1, 0, 0, 0, 0, -1)] Simple roots: [(1, 0, 0, 0, 0, -1)] Scalar products of pairs of distinct simple roots: set([]) Noncompact lambda-roots: [(1, 0, 0, 0, 0, -1)] Cone: (0, 0, -a2, a4, 0, 0) Scalar products of the weight in the cone with noncompact roots:
-------------------------------------------------------------------------------- 
%latex
{\Huge $\mathrm{Sp}(n, \mathbb{R}) \sim C_n, n \geq 2$}

def unitarizable_cohomology_Cn(n, q, r, l):
    index_set = tuple([x for x in range(1, n)])
    print("Levi roots: %s" % str(index_set))
    CT = ["C", n]

    W, AS, FW, vFW, rho, nRP = setup(CT, index_set)

    inject_positive_integer_variables(["a%d" % i for i in range(r, n+1)])
    cone_str = "Ca = " +  " + ".join("a{i}*(vFW[{i}] - vFW[{p}])".format(i=i, p=n) for i in range(r, n))
    setup_cone(cone_str)

    v = FW[q] + FW[r] - (2 + n - 1/2*(r + q - l + 1))*FW[n] + rho
    print("Vertex: %s" % v)
    examine(v, index_set, Ca)

unitarizable_cohomology_Cn(4,3,2,1)
Levi roots: (1, 2, 3)
Error in lines 12-12 Traceback (most recent call last): File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "", line 8, in unitarizable_cohomology_Cn NameError: global name 'setup_cone' is not defined 
def setup_SOstar(n):
    print("Levi roots: %s" % str(index_set))
    CT = ["D", n]

    return setup(CT, index_set)

def setup_so_cone(n, p):
    inject_positive_integer_variables(["a%d" % i for i in range(p, n)])
    cone_str = "Ca = " +  " + ".join("a{i}*(vFW[{i}] - 2*vFW[{n}])".format(i=i, n=n) for i in range(p, n-1))
    if p < n-1:
        cone_str +=  " + "
    cone_str +=  "a{last}*(vFW[{last}] - vFW[{n}])".format(last=n-1,n=n)
    setup_cone(cone_str)

def setup_su_cone(n, q):
    inject_positive_integer_variables(["a1"] + ["a%d" % i for i in range(q+1, n)])
    cone_str = "Ca = a1*(vFW[1] - vFW[{n}])".format(n=n)
    if q < n-2:
        cone_str += " + "
    cone_str += " + ".join("a{i}*(vFW[{i}] - 2*vFW[{n}])".format(i=i, n=n) for i in range(q+1, n-1))
    if q <= n-2:
        cone_str +=  " + a{last}*(vFW[{last}] - vFW[{n}])".format(last=n-1,n=n)
    setup_cone(cone_str)

n = 6
index_set = tuple([x for x in range(1, n)])
W, AS, FW, vFW, rho, nRP = setup_SOstar(n)
FW
Levi roots: (1, 2, 3, 4, 5) Finite family {1: (1, 0, 0, 0, 0, 0), 2: (1, 1, 0, 0, 0, 0), 3: (1, 1, 1, 0, 0, 0), 4: (1, 1, 1, 1, 0, 0), 5: (1/2, 1/2, 1/2, 1/2, 1/2, -1/2), 6: (1/2, 1/2, 1/2, 1/2, 1/2, 1/2)} 
v = FW[2] - (2*n-2)*FW[n]
v
(-4, -4, -5, -5, -5, -5) 
examine(v, index_set)
Weight: (-4, -4, -5, -5, -5, -5) Singular roots: [] Set of generating roots: [(1, 1, 0, 0, 0, 0)] Set of generated roots: [(1, 1, 0, 0, 0, 0)] Simple roots: [(1, 1, 0, 0, 0, 0)] Scalar products of pairs of distinct simple roots: set([]) Noncompact lambda-roots: [(1, 1, 0, 0, 0, 0)] Cone: None Scalar products of the weight in the cone with noncompact roots: 
p = 3
l = 1
v = FW[p] - 2*(n-p+l)*FW[n]
v
(-3, -3, -3, -4, -4, -4) 
setup_so_cone(n, p)
Initial assumptions: [] Cone: Ca = a3*(vFW[3] - 2*vFW[6]) + a4*(vFW[4] - 2*vFW[6]) + a5*(vFW[5] - vFW[6]) with assumptions: [] 
Ca = a3*(vFW[3] - 2*vFW[5]) + a4*(vFW[4] - vFW[5])

examine(v, index_set, Ca)
Weight: (-3, -3, -3, -4, -4, -4) Singular roots: [(1, 0, 0, 1, 0, 0)] Set of generating roots: [(0, 1, 1, 0, 0, 0)] Set of generated roots: [(0, 1, 1, 0, 0, 0)] Simple roots: [(0, 1, 1, 0, 0, 0)] Scalar products of pairs of distinct simple roots: set([]) Noncompact lambda-roots: [(0, 1, 1, 0, 0, 0)] Cone: (1/2*a4, 1/2*a4, 1/2*a4, -a3 + 1/2*a4, -a3 - 1/2*a4, a3 + 1/2*a4) Scalar products of the weight in the cone with noncompact roots: 
a3*a4

v = FW[n-1] - (1+2*l)*FW[n]
v
(-1, -1, -1, -2) 
setup_so_cone(n, n-1)
Initial assumptions: [] Cone: Ca = a9*(vFW[9] - vFW[10]) with assumptions: [a9 >= 0] 
v =  - (2*l-1)*FW[n]
v
(-1/2, -1/2, -1/2, -1/2) 
q = 1
v = FW[1] + FW[q+1] - (2*n-q)*FW[n]
v
(-15/2, -17/2, -19/2, -19/2, -19/2, -19/2, -19/2, -19/2, -19/2, -19/2) 
setup_su_cone(n, q)
Initial assumptions: [] Cone: Ca = a1*(vFW[1] - vFW[10]) + a2*(vFW[2] - 2*vFW[10]) + a3*(vFW[3] - 2*vFW[10]) + a4*(vFW[4] - 2*vFW[10]) + a5*(vFW[5] - 2*vFW[10]) + a6*(vFW[6] - 2*vFW[10]) + a7*(vFW[7] - 2*vFW[10]) + a8*(vFW[8] - 2*vFW[10]) + a9*(vFW[9] - vFW[10]) with assumptions: [] 
v = FW[1] + FW[n-1] - (n+1)*FW[n]
v
(-4, -5, -5, -5, -5, -5, -5, -5, -5, -6) 
setup_su_cone(n, n-2)
Initial assumptions: [] Cone: Ca = a1*(vFW[1] - vFW[10]) + a9*(vFW[9] - vFW[10]) with assumptions: [] 
v = FW[1] - (n-1)*FW[n]
v
(-7/2, -9/2, -9/2, -9/2, -9/2, -9/2, -9/2, -9/2, -9/2, -9/2) 
setup_su_cone(n, n-1)
Initial assumptions: [] Cone: Ca = a1*(vFW[1] - vFW[10]) with assumptions: [] 
n = 6
CT = ["D", n]
index_set = tuple(range(2,n+1))
#####
W = WeylGroup(CT)
AS = W.domain()
FW = AS.fundamental_weights()

if n <= 6:
    _ = latex.eval(fix_basis_latex((AS.root_poset())))

for p in range(1, n-2):
    print("p = %d" % p)
    v = -(2*n - p -1)*FW[1] + FW[p+1]
    examine(v, index_set)
    print
p = 1 Weight: (-9, 1, 0, 0, 0, 0) Singular roots: [] Set of generating roots: [(1, 1, 0, 0, 0, 0)] Set of generated roots: [(1, 1, 0, 0, 0, 0)] Simple roots: [(1, 1, 0, 0, 0, 0)] Scalar products of pairs of distinct simple roots: set([]) Noncompact lambda-roots: [(1, 1, 0, 0, 0, 0)] Cone: None Scalar products of the weight in the cone with noncompact roots:
p = 2 Weight: (-8, 1, 1, 0, 0, 0) Singular roots: [] Set of generating roots: [(1, 1, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0)] Set of generated roots: [(1, 1, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0), (0, 1, -1, 0, 0, 0)] Simple roots: [(1, 0, 1, 0, 0, 0), (0, 1, -1, 0, 0, 0)] Scalar products of pairs of distinct simple roots: set([-1]) Cartan matrix: [ 2 -1] [-1 2]
Noncompact lambda-roots: [(1, 0, 1, 0, 0, 0), (1, 1, 0, 0, 0, 0)] Cone: None Scalar products of the weight in the cone with noncompact roots:
p = 3 Weight: (-7, 1, 1, 1, 0, 0) Singular roots: [] Set of generating roots: [(1, 1, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0), (1, 0, 0, 1, 0, 0)] Set of generated roots: [(1, 1, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0), (1, 0, 0, 1, 0, 0), (0, 1, -1, 0, 0, 0), (0, 1, 0, -1, 0, 0), (0, 0, 1, -1, 0, 0)] Simple roots: [(1, 0, 0, 1, 0, 0), (0, 1, -1, 0, 0, 0), (0, 0, 1, -1, 0, 0)] Scalar products of pairs of distinct simple roots: set([0, -1]) Cartan matrix: [ 2 -1 0] [-1 2 -1] [ 0 -1 2]
Noncompact lambda-roots: [(1, 0, 0, 1, 0, 0), (1, 0, 1, 0, 0, 0), (1, 1, 0, 0, 0, 0)] Cone: None Scalar products of the weight in the cone with noncompact roots: 
v = -(n+1)*FW[1] + FW[n-1]+ FW[n]
examine(v, index_set)
Weight: (-6, 1, 1, 1, 1, 0) Singular roots: [] Set of generating roots: [(1, 1, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0), (1, 0, 0, 1, 0, 0), (1, 0, 0, 0, 1, 0)] Set of generated roots: [(1, 1, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0), (1, 0, 0, 1, 0, 0), (1, 0, 0, 0, 1, 0), (0, 1, -1, 0, 0, 0), (0, 1, 0, -1, 0, 0), (0, 1, 0, 0, -1, 0), (0, 0, 1, -1, 0, 0), (0, 0, 1, 0, -1, 0), (0, 0, 0, 1, -1, 0)] Simple roots: [(1, 0, 0, 0, 1, 0), (0, 1, -1, 0, 0, 0), (0, 0, 1, -1, 0, 0), (0, 0, 0, 1, -1, 0)] Scalar products of pairs of distinct simple roots: set([0, -1]) Cartan matrix: [ 2 -1 0 0] [-1 2 -1 0] [ 0 -1 2 -1] [ 0 0 -1 2] 
w = -(n-1)*FW[1]  + FW[n]
examine(w, index_set)
Weight: (-9/2, 1/2, 1/2, 1/2, 1/2, 1/2) Singular roots: [(1, 0, 0, 0, 0, -1)] Set of generating roots: [(1, 0, 0, 0, 0, 1)] Set of generated roots: [(1, 0, 0, 0, 0, 1)] Simple roots: [(1, 0, 0, 0, 0, 1)] Scalar products of pairs of distinct simple roots: set([]) 
v = -(n-2)*FW[1]
examine(v, index_set)
Weight: (-4, 0, 0, 0, 0, 0) Singular roots: [(1, 0, 0, 0, -1, 0)] Set of generating roots: [(1, 0, 0, 0, 1, 0)] Set of generated roots: [(1, 0, 0, 0, 1, 0)] Simple roots: [(1, 0, 0, 0, 1, 0)] Scalar products of pairs of distinct simple roots: set([]) 
v = 0*FW[1]
if n < 8:
    _, _, _, lambda_positive_roots, _, _, _, _ = get_subsystem_data(v, index_set)
    print("Generated root system is the whole D%s: %s" % (n, set(lambda_positive_roots) == set(AS.positive_roots())))
else:
    print("Generating system is too big.")
Generated root system is the whole D6: True 
v = -(n-1)*FW[1]  + FW[n-1]
examine(v, index_set)
Weight: (-9/2, 1/2, 1/2, 1/2, 1/2, -1/2) Singular roots: [(1, 0, 0, 0, 0, 1)] Set of generating roots: [(1, 0, 0, 0, 0, -1)] Set of generated roots: [(1, 0, 0, 0, 0, -1)] Simple roots: [(1, 0, 0, 0, 0, -1)] Scalar products of pairs of distinct simple roots: set([]) 
n = 6
index_set = tuple(range(2,7))
#####
W = WeylGroup("E%d" % n)
AS = W.domain()
FW = AS.fundamental_weights()

AS.dynkin_diagram()
O 2 | | O---O---O---O---O 1 3 4 5 6 E6 
a = 1
b = 1
c = 0
v = a*FW[3] + b*FW[5] + c*FW[6] + (-2*a-2*b-c-8)*FW[1]
examine(v, index_set)
Reps the same? True Weight: (-1/2, 1/2, 1/2, 3/2, 3/2, 13/2, 13/2, -13/2) Singular roots: [] Set of generating roots: [(-1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2)] Set of generated roots: [(-1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1, 1, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0, 0, 0), (0, -1, 1, 0, 0, 0, 0, 0)] Simple roots: [(-1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1, 1, 0, 0, 0, 0, 0, 0), (0, -1, 1, 0, 0, 0, 0, 0)] Scalar products of pairs of distinct simple roots: set([0, -1]) Cartan matrix: [ 2 -1 0] [-1 2 -1] [ 0 -1 2]
Noncompact lambda-roots: [(-1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2)] 
a = 4
v = a*FW[6] + (-a-4)*FW[1]
examine(v, index_set)
Weight: (0, 0, 0, 0, 4, 4, 4, -4) Singular roots: [] Set of generating roots: [(-1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, 1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2)] Set of generated roots: [(-1, 1, 0, 0, 0, 0, 0, 0), (-1, 0, 1, 0, 0, 0, 0, 0), (-1, 0, 0, 1, 0, 0, 0, 0), (-1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, 1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1, 1, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0, 0, 0), (1, 0, 0, 1, 0, 0, 0, 0), (0, -1, 1, 0, 0, 0, 0, 0), (0, -1, 0, 1, 0, 0, 0, 0), (0, 1, 1, 0, 0, 0, 0, 0), (0, 1, 0, 1, 0, 0, 0, 0), (0, 0, -1, 1, 0, 0, 0, 0), (0, 0, 1, 1, 0, 0, 0, 0)] Simple roots: [(-1, 1, 0, 0, 0, 0, 0, 0), (-1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1, 1, 0, 0, 0, 0, 0, 0), (0, -1, 1, 0, 0, 0, 0, 0), (0, 0, -1, 1, 0, 0, 0, 0)] Scalar products of pairs of distinct simple roots: set([0, -1]) Cartan matrix: [ 2 0 -1 0 -1] [ 0 2 -1 0 0] [-1 -1 2 -1 0] [ 0 0 -1 2 0] [-1 0 0 0 2]
Noncompact lambda-roots: [(-1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, 1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2)]

Experiments 

%latex
{\Huge $\mathrm{E}_7$}

CT = "E7"
W = WeylGroup(CT)
AS = W.domain()
FW = AS.fundamental_weights()
RP = AS.root_poset(facade=True)
nonparabolic_roots = [x.to_ambient() for x in AS.root_system.root_lattice().positive_roots_nonparabolic(index_set=(1,2,3,4,5,6))]
print(len(nonparabolic_roots))
nRP = RP.subposet(nonparabolic_roots)
nRP.plot()
27 
view(nRP)
Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/smc_sagews/sage_salvus.py", line 2649, in show s = show0(objs, combine_all=True) File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/smc_sagews/sage_salvus.py", line 2610, in show0 b = show0(a) File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/smc_sagews/sage_salvus.py", line 2635, in show0 sage.misc.latex.latex.eval(s) File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/smc_sagews/sage_salvus.py", line 1467, in latex0 sage.misc.latex.Latex.eval(sage.misc.latex.latex, s, **kwds) File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/sage/misc/latex.py", line 1130, in eval os.path.join(orig_base, filename + ".png")) File "/projects/sage/sage-7.6/local/lib/python/shutil.py", line 119, in copy copyfile(src, dst) File "/projects/sage/sage-7.6/local/lib/python/shutil.py", line 82, in copyfile with open(src, 'rb') as fsrc: IOError: [Errno 2] No such file or directory: '/projects/d97ca4df-6ebd-46ab-b7fb-40461f0f84e0/.sage/temp/compute2-us/11056/dir_I69ERH/tmpFKZ9ex.png' 
latex(nRP)
\begin{tikzpicture}[>=latex,line join=bevel,] %% \node (node_26) at (233.5bp,851.0bp) [draw,draw=none] {$-e_{6} + e_{7}$}; \node (node_24) at (140.5bp,7.0bp) [draw,draw=none] {$-e_{4} + e_{5}$}; \node (node_25) at (39.5bp,421.0bp) [draw,draw=none] {$e_{4} + e_{5}$}; \node (node_22) at (140.5bp,57.0bp) [draw,draw=none] {$-e_{3} + e_{5}$}; \node (node_23) at (111.5bp,367.0bp) [draw,draw=none] {$e_{3} + e_{5}$}; \node (node_20) at (140.5bp,107.0bp) [draw,draw=none] {$-e_{2} + e_{5}$}; \node (node_21) at (111.5bp,313.0bp) [draw,draw=none] {$e_{2} + e_{5}$}; \node (node_9) at (111.5bp,475.0bp) [draw,draw=none] {$\frac{1}{2}e_{0} - \frac{1}{2}e_{1} - \frac{1}{2}e_{2} - \frac{1}{2}e_{3} + \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_8) at (233.5bp,799.0bp) [draw,draw=none] {$-\frac{1}{2}e_{0} + \frac{1}{2}e_{1} + \frac{1}{2}e_{2} + \frac{1}{2}e_{3} + \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_7) at (422.5bp,421.0bp) [draw,draw=none] {$-\frac{1}{2}e_{0} + \frac{1}{2}e_{1} + \frac{1}{2}e_{2} - \frac{1}{2}e_{3} - \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_6) at (350.5bp,475.0bp) [draw,draw=none] {$-\frac{1}{2}e_{0} + \frac{1}{2}e_{1} - \frac{1}{2}e_{2} + \frac{1}{2}e_{3} - \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_5) at (111.5bp,529.0bp) [draw,draw=none] {$-\frac{1}{2}e_{0} + \frac{1}{2}e_{1} - \frac{1}{2}e_{2} - \frac{1}{2}e_{3} + \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_4) at (352.5bp,529.0bp) [draw,draw=none] {$-\frac{1}{2}e_{0} - \frac{1}{2}e_{1} + \frac{1}{2}e_{2} + \frac{1}{2}e_{3} - \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_3) at (114.5bp,583.0bp) [draw,draw=none] {$-\frac{1}{2}e_{0} - \frac{1}{2}e_{1} + \frac{1}{2}e_{2} - \frac{1}{2}e_{3} + \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_2) at (114.5bp,637.0bp) [draw,draw=none] {$-\frac{1}{2}e_{0} - \frac{1}{2}e_{1} - \frac{1}{2}e_{2} + \frac{1}{2}e_{3} + \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_1) at (259.5bp,259.0bp) [draw,draw=none] {$-\frac{1}{2}e_{0} - \frac{1}{2}e_{1} - \frac{1}{2}e_{2} - \frac{1}{2}e_{3} - \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_0) at (170.5bp,207.0bp) [draw,draw=none] {$-e_{0} + e_{5}$}; \node (node_19) at (111.5bp,259.0bp) [draw,draw=none] {$e_{1} + e_{5}$}; \node (node_18) at (140.5bp,157.0bp) [draw,draw=none] {$-e_{1} + e_{5}$}; \node (node_17) at (111.5bp,207.0bp) [draw,draw=none] {$e_{0} + e_{5}$}; \node (node_16) at (352.5bp,583.0bp) [draw,draw=none] {$\frac{1}{2}e_{0} + \frac{1}{2}e_{1} + \frac{1}{2}e_{2} + \frac{1}{2}e_{3} - \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_15) at (352.5bp,637.0bp) [draw,draw=none] {$\frac{1}{2}e_{0} + \frac{1}{2}e_{1} + \frac{1}{2}e_{2} - \frac{1}{2}e_{3} + \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_14) at (233.5bp,691.0bp) [draw,draw=none] {$\frac{1}{2}e_{0} + \frac{1}{2}e_{1} - \frac{1}{2}e_{2} + \frac{1}{2}e_{3} + \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_13) at (256.5bp,313.0bp) [draw,draw=none] {$\frac{1}{2}e_{0} + \frac{1}{2}e_{1} - \frac{1}{2}e_{2} - \frac{1}{2}e_{3} - \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_12) at (233.5bp,745.0bp) [draw,draw=none] {$\frac{1}{2}e_{0} - \frac{1}{2}e_{1} + \frac{1}{2}e_{2} + \frac{1}{2}e_{3} + \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_11) at (256.5bp,367.0bp) [draw,draw=none] {$\frac{1}{2}e_{0} - \frac{1}{2}e_{1} + \frac{1}{2}e_{2} - \frac{1}{2}e_{3} - \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \node (node_10) at (184.5bp,421.0bp) [draw,draw=none] {$\frac{1}{2}e_{0} - \frac{1}{2}e_{1} - \frac{1}{2}e_{2} + \frac{1}{2}e_{3} - \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$}; \draw [black,->] (node_6) ..controls (351.08bp,490.95bp) and (351.46bp,500.97bp) .. (node_4); \draw [black,->] (node_15) ..controls (314.74bp,654.5bp) and (283.84bp,668.0bp) .. (node_14); \draw [black,->] (node_25) ..controls (58.839bp,435.97bp) and (77.748bp,449.62bp) .. (node_9); \draw [black,->] (node_7) ..controls (400.32bp,438.02bp) and (383.5bp,450.16bp) .. (node_6); \draw [black,->] (node_10) ..controls (161.91bp,438.09bp) and (144.63bp,450.4bp) .. (node_9); \draw [black,->] (node_11) ..controls (234.32bp,384.02bp) and (217.5bp,396.16bp) .. (node_10); \draw [black,->] (node_22) ..controls (140.5bp,70.707bp) and (140.5bp,80.976bp) .. (node_20); \draw [black,->] (node_10) ..controls (238.22bp,438.83bp) and (283.65bp,453.06bp) .. (node_6); \draw [black,->] (node_19) ..controls (151.91bp,274.49bp) and (194.86bp,289.89bp) .. (node_13); \draw [black,->] (node_11) ..controls (310.22bp,384.83bp) and (355.65bp,399.06bp) .. (node_7); \draw [black,->] (node_18) ..controls (148.67bp,171.07bp) and (155.59bp,182.14bp) .. (node_0); \draw [black,->] (node_3) ..controls (114.5bp,598.95bp) and (114.5bp,608.97bp) .. (node_2); \draw [black,->] (node_9) ..controls (111.5bp,490.95bp) and (111.5bp,500.97bp) .. (node_5); \draw [black,->] (node_5) ..controls (112.36bp,544.95bp) and (112.94bp,554.97bp) .. (node_3); \draw [black,->] (node_21) ..controls (151.91bp,328.49bp) and (194.86bp,343.89bp) .. (node_11); \draw [black,->] (node_21) ..controls (111.5bp,327.63bp) and (111.5bp,339.75bp) .. (node_23); \draw [black,->] (node_19) ..controls (111.5bp,273.63bp) and (111.5bp,285.75bp) .. (node_21); \draw [black,->] (node_17) ..controls (111.5bp,220.95bp) and (111.5bp,232.26bp) .. (node_19); \draw [black,->] (node_18) ..controls (132.6bp,171.07bp) and (125.91bp,182.14bp) .. (node_17); \draw [black,->] (node_24) ..controls (140.5bp,20.707bp) and (140.5bp,30.976bp) .. (node_22); \draw [black,->] (node_23) ..controls (91.483bp,382.46bp) and (70.843bp,397.36bp) .. (node_25); \draw [black,->] (node_23) ..controls (131.11bp,381.97bp) and (150.28bp,395.62bp) .. (node_10); \draw [black,->] (node_13) ..controls (256.5bp,328.95bp) and (256.5bp,338.97bp) .. (node_11); \draw [black,->] (node_16) ..controls (352.5bp,598.95bp) and (352.5bp,608.97bp) .. (node_15); \draw [black,->] (node_1) ..controls (258.64bp,274.95bp) and (258.06bp,284.97bp) .. (node_13); \draw [black,->] (node_0) ..controls (154.15bp,221.85bp) and (138.27bp,235.32bp) .. (node_19); \draw [black,->] (node_12) ..controls (233.5bp,760.95bp) and (233.5bp,770.97bp) .. (node_8); \draw [black,->] (node_8) ..controls (233.5bp,815.12bp) and (233.5bp,825.1bp) .. (node_26); \draw [black,->] (node_3) ..controls (192.75bp,601.1bp) and (260.63bp,615.93bp) .. (node_15); \draw [black,->] (node_0) ..controls (194.89bp,221.7bp) and (218.18bp,234.79bp) .. (node_1); \draw [black,->] (node_4) ..controls (274.25bp,547.1bp) and (206.37bp,561.93bp) .. (node_3); \draw [black,->] (node_2) ..controls (152.26bp,654.5bp) and (183.16bp,668.0bp) .. (node_14); \draw [black,->] (node_6) ..controls (271.93bp,493.1bp) and (203.76bp,507.93bp) .. (node_5); \draw [black,->] (node_4) ..controls (352.5bp,544.95bp) and (352.5bp,554.97bp) .. (node_16); \draw [black,->] (node_20) ..controls (140.5bp,120.71bp) and (140.5bp,130.98bp) .. (node_18); \draw [black,->] (node_14) ..controls (233.5bp,706.95bp) and (233.5bp,716.97bp) .. (node_12); % \end{tikzpicture} 
%latex
\begin{tikzpicture}[>=latex,line join=bevel,]
%%
\node (node_26) at (233.5bp,851.0bp) [draw,draw=none] {$-e_{6} + e_{7}$};
  \node (node_24) at (140.5bp,7.0bp) [draw,draw=none] {$-e_{4} + e_{5}$};
  \node (node_25) at (39.5bp,421.0bp) [draw,draw=none] {$e_{4} + e_{5}$};
  \node (node_22) at (140.5bp,57.0bp) [draw,draw=none] {$-e_{3} + e_{5}$};
  \node (node_23) at (111.5bp,367.0bp) [draw,draw=none] {$e_{3} + e_{5}$};
  \node (node_20) at (140.5bp,107.0bp) [draw,draw=none] {$-e_{2} + e_{5}$};
  \node (node_21) at (111.5bp,313.0bp) [draw,draw=none] {$e_{2} + e_{5}$};
  \node (node_9) at (111.5bp,475.0bp) [draw,draw=none] {$\frac{1}{2}e_{0} - \frac{1}{2}e_{1} - \frac{1}{2}e_{2} - \frac{1}{2}e_{3} + \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$};
  \node (node_8) at (233.5bp,799.0bp) [draw,draw=none] {$-\frac{1}{2}e_{0} + \frac{1}{2}e_{1} + \frac{1}{2}e_{2} + \frac{1}{2}e_{3} + \frac{1}{2}e_{4} + \frac{1}{2}e_{5} - \frac{1}{2}e_{6} + \frac{1}{2}e_{7}$};
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Standard LaTeX file LaTeX Font Info: Redeclaring font encoding T1 on input line 48. )) (/usr/share/texlive/texmf-dist/tex/latex/soul/soul.sty Package: soul 2003/11/17 v2.4 letterspacing/underlining (mf) \SOUL@word=\toks22 \SOUL@lasttoken=\toks23 \SOUL@cmds=\toks24 \SOUL@buffer=\toks25 \SOUL@token=\toks26 \SOUL@spaceskip=\skip46 \SOUL@ttwidth=\dimen114 \SOUL@uldp=\dimen115 \SOUL@ulht=\dimen116 ) No file tmpTrH4aS.aux. \openout1 = `tmpTrH4aS.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 37. LaTeX Font Info: ... okay on input line 37. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 37. LaTeX Font Info: ... okay on input line 37. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 37. LaTeX Font Info: ... okay on input line 37. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 37. LaTeX Font Info: ... okay on input line 37. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 37. LaTeX Font Info: ... okay on input line 37. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 37. LaTeX Font Info: ... okay on input line 37. (/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii [Loading MPS to PDF converter (version 2006.09.02).] \scratchcounter=\count100 \scratchdimen=\dimen117 \scratchbox=\box28 \nofMPsegments=\count101 \nofMParguments=\count102 \everyMPshowfont=\toks27 \MPscratchCnt=\count103 \MPscratchDim=\dimen118 \MPnumerator=\count104 \makeMPintoPDFobject=\count105 \everyMPtoPDFconversion=\toks28 ) (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/pdftexcmds.sty Package: pdftexcmds 2011/11/29 v0.20 Utility functions of pdfTeX for LuaTeX (HO ) (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifluatex.sty Package: ifluatex 2010/03/01 v1.3 Provides the ifluatex switch (HO) Package ifluatex Info: LuaTeX not detected. ) (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifpdf.sty Package: ifpdf 2011/01/30 v2.3 Provides the ifpdf switch (HO) Package ifpdf Info: pdfTeX in PDF mode is detected. ) Package pdftexcmds Info: LuaTeX not detected. Package pdftexcmds Info: \pdf@primitive is available. Package pdftexcmds Info: \pdf@ifprimitive is available. Package pdftexcmds Info: \pdfdraftmode found. ) (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/epstopdf-base.sty Package: epstopdf-base 2010/02/09 v2.5 Base part for package epstopdf (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/grfext.sty Package: grfext 2010/08/19 v1.1 Manage graphics extensions (HO) (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvdefinekeys.sty Package: kvdefinekeys 2011/04/07 v1.3 Define keys (HO) )) (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty Package: kvoptions 2011/06/30 v3.11 Key value format for package options (HO) (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvsetkeys.sty Package: kvsetkeys 2012/04/25 v1.16 Key value parser (HO) (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/etexcmds.sty Package: etexcmds 2011/02/16 v1.5 Avoid name clashes with e-TeX commands (HO) Package etexcmds Info: Could not find \expanded. (etexcmds) That can mean that you are not using pdfTeX 1.50 or (etexcmds) that some package has redefined \expanded. (etexcmds) In the latter case, load this package earlier. ))) Package grfext Info: Graphics extension search list: (grfext) [.png,.pdf,.jpg,.mps,.jpeg,.jbig2,.jb2,.PNG,.PDF,.JPG,.JPE G,.JBIG2,.JB2,.eps] (grfext) \AppendGraphicsExtensions on input line 452. (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) ! LaTeX Error: Environment tikzpicture undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.38 \begin{tikzpicture} [>=latex,line join=bevel,] Your command was ignored. Type I <command> <return> to replace it with another command, or <return> to continue without it. ! Undefined control sequence. l.40 \node (node_26) at (233.5bp,851.0bp) [draw,draw=none] {$-e_{6} + e_{7}$}; The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.40 \node (node_ 26) at (233.5bp,851.0bp) [draw,draw=none] {$-e_{6} + e_{7}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. LaTeX Font Info: Try loading font information for U+msa on input line 40. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd File: umsa.fd 2013/01/14 v3.01 AMS symbols A ) LaTeX Font Info: Try loading font information for U+msb on input line 40. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd File: umsb.fd 2013/01/14 v3.01 AMS symbols B ) LaTeX Font Info: Try loading font information for U+rsfs on input line 40. (/usr/share/texlive/texmf-dist/tex/latex/jknapltx/ursfs.fd File: ursfs.fd 1998/03/24 rsfs font definition file (jk) ) ! Missing } inserted. <inserted text> } l.40 ...) at (233.5bp,851.0bp) [draw,draw=none] {$ -e_{6} + e_{7}$}; I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> $ l.40 ...t (233.5bp,851.0bp) [draw,draw=none] {$-e_ {6} + e_{7}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten \endgroup. l.40 ...1.0bp) [draw,draw=none] {$-e_{6} + e_{7}$} ; I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Undefined control sequence. l.41 \node (node_24) at (140.5bp,7.0bp) [draw,draw=none] {$-e_{4} + e_{5}$}; The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.41 \node (node_ 24) at (140.5bp,7.0bp) [draw,draw=none] {$-e_{4} + e_{5}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing } inserted. <inserted text> } l.41 ...24) at (140.5bp,7.0bp) [draw,draw=none] {$ -e_{4} + e_{5}$}; I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> $ l.41 ... at (140.5bp,7.0bp) [draw,draw=none] {$-e_ {4} + e_{5}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten \endgroup. l.41 ...7.0bp) [draw,draw=none] {$-e_{4} + e_{5}$} ; I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Undefined control sequence. l.42 \node (node_25) at (39.5bp,421.0bp) [draw,draw=none] {$e_{4} + e_{5}$}; The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.42 \node (node_ 25) at (39.5bp,421.0bp) [draw,draw=none] {$e_{4} + e_{5}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing } inserted. <inserted text> } l.42 ...5) at (39.5bp,421.0bp) [draw,draw=none] {$ e_{4} + e_{5}$}; I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> $ l.42 ... at (39.5bp,421.0bp) [draw,draw=none] {$e_ {4} + e_{5}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten \endgroup. l.42 ...21.0bp) [draw,draw=none] {$e_{4} + e_{5}$} ; I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Undefined control sequence. l.43 \node (node_22) at (140.5bp,57.0bp) [draw,draw=none] {$-e_{3} + e_{5}$}; The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.43 \node (node_ 22) at (140.5bp,57.0bp) [draw,draw=none] {$-e_{3} + e_{5}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing } inserted. <inserted text> } l.43 ...2) at (140.5bp,57.0bp) [draw,draw=none] {$ -e_{3} + e_{5}$}; I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> $ l.43 ...at (140.5bp,57.0bp) [draw,draw=none] {$-e_ {3} + e_{5}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten \endgroup. l.43 ...7.0bp) [draw,draw=none] {$-e_{3} + e_{5}$} ; I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Undefined control sequence. l.44 \node (node_23) at (111.5bp,367.0bp) [draw,draw=none] {$e_{3} + e_{5}$}; The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.44 \node (node_ 23) at (111.5bp,367.0bp) [draw,draw=none] {$e_{3} + e_{5}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing } inserted. <inserted text> } l.44 ...) at (111.5bp,367.0bp) [draw,draw=none] {$ e_{3} + e_{5}$}; I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> $ l.44 ...at (111.5bp,367.0bp) [draw,draw=none] {$e_ {3} + e_{5}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten \endgroup. l.44 ...67.0bp) [draw,draw=none] {$e_{3} + e_{5}$} ; I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Undefined control sequence. l.45 \node (node_20) at (140.5bp,107.0bp) [draw,draw=none] {$-e_{2} + e_{5... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.45 \node (node_ 20) at (140.5bp,107.0bp) [draw,draw=none] {$-e_{2} + e_{5... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing } inserted. <inserted text> } l.45 ...) at (140.5bp,107.0bp) [draw,draw=none] {$ -e_{2} + e_{5}$}; I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> $ l.45 ...t (140.5bp,107.0bp) [draw,draw=none] {$-e_ {2} + e_{5}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten \endgroup. l.45 ...7.0bp) [draw,draw=none] {$-e_{2} + e_{5}$} ; I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Undefined control sequence. l.46 \node (node_21) at (111.5bp,313.0bp) [draw,draw=none] {$e_{2} + e_{5}$}; The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.46 \node (node_ 21) at (111.5bp,313.0bp) [draw,draw=none] {$e_{2} + e_{5}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing } inserted. <inserted text> } l.46 ...) at (111.5bp,313.0bp) [draw,draw=none] {$ e_{2} + e_{5}$}; I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> $ l.46 ...at (111.5bp,313.0bp) [draw,draw=none] {$e_ {2} + e_{5}$}; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten \endgroup. l.46 ...13.0bp) [draw,draw=none] {$e_{2} + e_{5}$} ; I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Undefined control sequence. l.47 \node (node_9) at (111.5bp,475.0bp) [draw,draw=none] {$\frac{1}{2}e_{... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.47 \node (node_ 9) at (111.5bp,475.0bp) [draw,draw=none] {$\frac{1}{2}e_{... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing } inserted. <inserted text> } l.47 ...) at (111.5bp,475.0bp) [draw,draw=none] {$ \frac{1}{2}e_{0} - \frac{1... I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> $ l.47 ...bp,475.0bp) [draw,draw=none] {$\frac{1}{2} e_{0} - \frac{1}{2}e_{1} -... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten $. \frac #1#2->{\begingroup #1\endgroup \@@over #2} l.47 ...bp,475.0bp) [draw,draw=none] {$\frac{1}{2} e_{0} - \frac{1}{2}e_{1} -... I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Undefined control sequence. l.48 \node (node_8) at (233.5bp,799.0bp) [draw,draw=none] {$-\frac{1}{2}e_... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.48 \node (node_ 8) at (233.5bp,799.0bp) [draw,draw=none] {$-\frac{1}{2}e_... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing } inserted. <inserted text> } l.48 ...) at (233.5bp,799.0bp) [draw,draw=none] {$ -\frac{1}{2}e_{0} + \frac{... I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> $ l.48 ...p,799.0bp) [draw,draw=none] {$-\frac{1}{2} e_{0} + \frac{1}{2}e_{1} +... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten $. \frac #1#2->{\begingroup #1\endgroup \@@over #2} l.48 ...p,799.0bp) [draw,draw=none] {$-\frac{1}{2} e_{0} + \frac{1}{2}e_{1} +... I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Undefined control sequence. l.49 \node (node_7) at (422.5bp,421.0bp) [draw,draw=none] {$-\frac{1}{2}e_... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.49 \node (node_ 7) at (422.5bp,421.0bp) [draw,draw=none] {$-\frac{1}{2}e_... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing } inserted. <inserted text> } l.49 ...) at (422.5bp,421.0bp) [draw,draw=none] {$ -\frac{1}{2}e_{0} + \frac{... I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> $ l.49 ...p,421.0bp) [draw,draw=none] {$-\frac{1}{2} e_{0} + \frac{1}{2}e_{1} +... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten $. \frac #1#2->{\begingroup #1\endgroup \@@over #2} l.49 ...p,421.0bp) [draw,draw=none] {$-\frac{1}{2} e_{0} + \frac{1}{2}e_{1} +... I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Undefined control sequence. l.50 \node (node_6) at (350.5bp,475.0bp) [draw,draw=none] {$-\frac{1}{2}e_... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.50 \node (node_ 6) at (350.5bp,475.0bp) [draw,draw=none] {$-\frac{1}{2}e_... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing } inserted. <inserted text> } l.50 ...) at (350.5bp,475.0bp) [draw,draw=none] {$ -\frac{1}{2}e_{0} + \frac{... I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> $ l.50 ...p,475.0bp) [draw,draw=none] {$-\frac{1}{2} e_{0} + \frac{1}{2}e_{1} -... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten $. \frac #1#2->{\begingroup #1\endgroup \@@over #2} l.50 ...p,475.0bp) [draw,draw=none] {$-\frac{1}{2} e_{0} + \frac{1}{2}e_{1} -... I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Undefined control sequence. l.51 \node (node_5) at (111.5bp,529.0bp) [draw,draw=none] {$-\frac{1}{2}e_... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. <inserted text> $ l.51 \node (node_ 5) at (111.5bp,529.0bp) [draw,draw=none] {$-\frac{1}{2}e_... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing } inserted. <inserted text> } l.51 ...) at (111.5bp,529.0bp) [draw,draw=none] {$ -\frac{1}{2}e_{0} + \frac{... I've inserted something that you may have forgotten. (See the <inserted text> above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. <inserted text> [...] 
print(fix_basis_latex(nRP))
\begin{tikzpicture}[>=latex,line join=bevel,] %% \node (node_26) at (233.5bp,851.0bp) [draw,draw=none] {$-\epsilon_{7} + \epsilon_{8}$}; \node (node_24) at (140.5bp,7.0bp) [draw,draw=none] {$-\epsilon_{5} + \epsilon_{6}$}; \node (node_25) at (39.5bp,421.0bp) [draw,draw=none] {$\epsilon_{5} + \epsilon_{6}$}; \node (node_22) at (140.5bp,57.0bp) [draw,draw=none] {$-\epsilon_{4} + \epsilon_{6}$}; \node (node_23) at (111.5bp,367.0bp) [draw,draw=none] {$\epsilon_{4} + \epsilon_{6}$}; \node (node_20) at (140.5bp,107.0bp) [draw,draw=none] {$-\epsilon_{3} + \epsilon_{6}$}; \node (node_21) at (111.5bp,313.0bp) [draw,draw=none] {$\epsilon_{3} + \epsilon_{6}$}; \node (node_9) at (111.5bp,475.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_8) at (233.5bp,799.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_7) at (422.5bp,421.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_6) at (350.5bp,475.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_5) at (111.5bp,529.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_4) at (352.5bp,529.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_3) at (114.5bp,583.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_2) at (114.5bp,637.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_1) at (259.5bp,259.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_0) at (170.5bp,207.0bp) [draw,draw=none] {$-\epsilon_{1} + \epsilon_{6}$}; \node (node_19) at (111.5bp,259.0bp) [draw,draw=none] {$\epsilon_{2} + \epsilon_{6}$}; \node (node_18) at (140.5bp,157.0bp) [draw,draw=none] {$-\epsilon_{2} + \epsilon_{6}$}; \node (node_17) at (111.5bp,207.0bp) [draw,draw=none] {$\epsilon_{1} + \epsilon_{6}$}; \node (node_16) at (352.5bp,583.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_15) at (352.5bp,637.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_14) at (233.5bp,691.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_13) at (256.5bp,313.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_12) at (233.5bp,745.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_11) at (256.5bp,367.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \node (node_10) at (184.5bp,421.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$}; \draw [black,->] (node_6) ..controls (351.08bp,490.95bp) and (351.46bp,500.97bp) .. (node_4); \draw [black,->] (node_15) ..controls (314.74bp,654.5bp) and (283.84bp,668.0bp) .. (node_14); \draw [black,->] (node_25) ..controls (58.839bp,435.97bp) and (77.748bp,449.62bp) .. (node_9); \draw [black,->] (node_7) ..controls (400.32bp,438.02bp) and (383.5bp,450.16bp) .. (node_6); \draw [black,->] (node_10) ..controls (161.91bp,438.09bp) and (144.63bp,450.4bp) .. (node_9); \draw [black,->] (node_11) ..controls (234.32bp,384.02bp) and (217.5bp,396.16bp) .. (node_10); \draw [black,->] (node_22) ..controls (140.5bp,70.707bp) and (140.5bp,80.976bp) .. (node_20); \draw [black,->] (node_10) ..controls (238.22bp,438.83bp) and (283.65bp,453.06bp) .. (node_6); \draw [black,->] (node_19) ..controls (151.91bp,274.49bp) and (194.86bp,289.89bp) .. (node_13); \draw [black,->] (node_11) ..controls (310.22bp,384.83bp) and (355.65bp,399.06bp) .. (node_7); \draw [black,->] (node_18) ..controls (148.67bp,171.07bp) and (155.59bp,182.14bp) .. (node_0); \draw [black,->] (node_3) ..controls (114.5bp,598.95bp) and (114.5bp,608.97bp) .. (node_2); \draw [black,->] (node_9) ..controls (111.5bp,490.95bp) and (111.5bp,500.97bp) .. (node_5); \draw [black,->] (node_5) ..controls (112.36bp,544.95bp) and (112.94bp,554.97bp) .. (node_3); \draw [black,->] (node_21) ..controls (151.91bp,328.49bp) and (194.86bp,343.89bp) .. (node_11); \draw [black,->] (node_21) ..controls (111.5bp,327.63bp) and (111.5bp,339.75bp) .. (node_23); \draw [black,->] (node_19) ..controls (111.5bp,273.63bp) and (111.5bp,285.75bp) .. (node_21); \draw [black,->] (node_17) ..controls (111.5bp,220.95bp) and (111.5bp,232.26bp) .. (node_19); \draw [black,->] (node_18) ..controls (132.6bp,171.07bp) and (125.91bp,182.14bp) .. (node_17); \draw [black,->] (node_24) ..controls (140.5bp,20.707bp) and (140.5bp,30.976bp) .. (node_22); \draw [black,->] (node_23) ..controls (91.483bp,382.46bp) and (70.843bp,397.36bp) .. (node_25); \draw [black,->] (node_23) ..controls (131.11bp,381.97bp) and (150.28bp,395.62bp) .. (node_10); \draw [black,->] (node_13) ..controls (256.5bp,328.95bp) and (256.5bp,338.97bp) .. (node_11); \draw [black,->] (node_16) ..controls (352.5bp,598.95bp) and (352.5bp,608.97bp) .. (node_15); \draw [black,->] (node_1) ..controls (258.64bp,274.95bp) and (258.06bp,284.97bp) .. (node_13); \draw [black,->] (node_0) ..controls (154.15bp,221.85bp) and (138.27bp,235.32bp) .. (node_19); \draw [black,->] (node_12) ..controls (233.5bp,760.95bp) and (233.5bp,770.97bp) .. (node_8); \draw [black,->] (node_8) ..controls (233.5bp,815.12bp) and (233.5bp,825.1bp) .. (node_26); \draw [black,->] (node_3) ..controls (192.75bp,601.1bp) and (260.63bp,615.93bp) .. (node_15); \draw [black,->] (node_0) ..controls (194.89bp,221.7bp) and (218.18bp,234.79bp) .. (node_1); \draw [black,->] (node_4) ..controls (274.25bp,547.1bp) and (206.37bp,561.93bp) .. (node_3); \draw [black,->] (node_2) ..controls (152.26bp,654.5bp) and (183.16bp,668.0bp) .. (node_14); \draw [black,->] (node_6) ..controls (271.93bp,493.1bp) and (203.76bp,507.93bp) .. (node_5); \draw [black,->] (node_4) ..controls (352.5bp,544.95bp) and (352.5bp,554.97bp) .. (node_16); \draw [black,->] (node_20) ..controls (140.5bp,120.71bp) and (140.5bp,130.98bp) .. (node_18); \draw [black,->] (node_14) ..controls (233.5bp,706.95bp) and (233.5bp,716.97bp) .. (node_12); % \end{tikzpicture} 
%latex
\begin{tikzpicture}[>=latex,line join=bevel,]
%%
\node (node_26) at (233.5bp,851.0bp) [draw,draw=none] {$-\epsilon_{7} + \epsilon_{8}$};
  \node (node_24) at (140.5bp,7.0bp) [draw,draw=none] {$-\epsilon_{5} + \epsilon_{6}$};
  \node (node_25) at (39.5bp,421.0bp) [draw,draw=none] {$\epsilon_{5} + \epsilon_{6}$};
  \node (node_22) at (140.5bp,57.0bp) [draw,draw=none] {$-\epsilon_{4} + \epsilon_{6}$};
  \node (node_23) at (111.5bp,367.0bp) [draw,draw=none] {$\epsilon_{4} + \epsilon_{6}$};
  \node (node_20) at (140.5bp,107.0bp) [draw,draw=none] {$-\epsilon_{3} + \epsilon_{6}$};
  \node (node_21) at (111.5bp,313.0bp) [draw,draw=none] {$\epsilon_{3} + \epsilon_{6}$};
  \node (node_9) at (111.5bp,475.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_8) at (233.5bp,799.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_7) at (422.5bp,421.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_6) at (350.5bp,475.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_5) at (111.5bp,529.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_4) at (352.5bp,529.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_3) at (114.5bp,583.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_2) at (114.5bp,637.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_1) at (259.5bp,259.0bp) [draw,draw=none] {$-\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_0) at (170.5bp,207.0bp) [draw,draw=none] {$-\epsilon_{1} + \epsilon_{6}$};
  \node (node_19) at (111.5bp,259.0bp) [draw,draw=none] {$\epsilon_{2} + \epsilon_{6}$};
  \node (node_18) at (140.5bp,157.0bp) [draw,draw=none] {$-\epsilon_{2} + \epsilon_{6}$};
  \node (node_17) at (111.5bp,207.0bp) [draw,draw=none] {$\epsilon_{1} + \epsilon_{6}$};
  \node (node_16) at (352.5bp,583.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_15) at (352.5bp,637.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_14) at (233.5bp,691.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_13) at (256.5bp,313.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} + \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_12) at (233.5bp,745.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} + \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_11) at (256.5bp,367.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} + \frac{1}{2}\epsilon_{3} - \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \node (node_10) at (184.5bp,421.0bp) [draw,draw=none] {$\frac{1}{2}\epsilon_{1} - \frac{1}{2}\epsilon_{2} - \frac{1}{2}\epsilon_{3} + \frac{1}{2}\epsilon_{4} - \frac{1}{2}\epsilon_{5} + \frac{1}{2}\epsilon_{6} - \frac{1}{2}\epsilon_{7} + \frac{1}{2}\epsilon_{8}$};
  \draw [black,->] (node_6) ..controls (351.08bp,490.95bp) and (351.46bp,500.97bp)  .. (node_4);
  \draw [black,->] (node_15) ..controls (314.74bp,654.5bp) and (283.84bp,668.0bp)  .. (node_14);
  \draw [black,->] (node_25) ..controls (58.839bp,435.97bp) and (77.748bp,449.62bp)  .. (node_9);
  \draw [black,->] (node_7) ..controls (400.32bp,438.02bp) and (383.5bp,450.16bp)  .. (node_6);
  \draw [black,->] (node_10) ..controls (161.91bp,438.09bp) and (144.63bp,450.4bp)  .. (node_9);
  \draw [black,->] (node_11) ..controls (234.32bp,384.02bp) and (217.5bp,396.16bp)  .. (node_10);
  \draw [black,->] (node_22) ..controls (140.5bp,70.707bp) and (140.5bp,80.976bp)  .. (node_20);
  \draw [black,->] (node_10) ..controls (238.22bp,438.83bp) and (283.65bp,453.06bp)  .. (node_6);
  \draw [black,->] (node_19) ..controls (151.91bp,274.49bp) and (194.86bp,289.89bp)  .. (node_13);
  \draw [black,->] (node_11) ..controls (310.22bp,384.83bp) and (355.65bp,399.06bp)  .. (node_7);
  \draw [black,->] (node_18) ..controls (148.67bp,171.07bp) and (155.59bp,182.14bp)  .. (node_0);
  \draw [black,->] (node_3) ..controls (114.5bp,598.95bp) and (114.5bp,608.97bp)  .. (node_2);
  \draw [black,->] (node_9) ..controls (111.5bp,490.95bp) and (111.5bp,500.97bp)  .. (node_5);
  \draw [black,->] (node_5) ..controls (112.36bp,544.95bp) and (112.94bp,554.97bp)  .. (node_3);
  \draw [black,->] (node_21) ..controls (151.91bp,328.49bp) and (194.86bp,343.89bp)  .. (node_11);
  \draw [black,->] (node_21) ..controls (111.5bp,327.63bp) and (111.5bp,339.75bp)  .. (node_23);
  \draw [black,->] (node_19) ..controls (111.5bp,273.63bp) and (111.5bp,285.75bp)  .. (node_21);
  \draw [black,->] (node_17) ..controls (111.5bp,220.95bp) and (111.5bp,232.26bp)  .. (node_19);
  \draw [black,->] (node_18) ..controls (132.6bp,171.07bp) and (125.91bp,182.14bp)  .. (node_17);
  \draw [black,->] (node_24) ..controls (140.5bp,20.707bp) and (140.5bp,30.976bp)  .. (node_22);
  \draw [black,->] (node_23) ..controls (91.483bp,382.46bp) and (70.843bp,397.36bp)  .. (node_25);
  \draw [black,->] (node_23) ..controls (131.11bp,381.97bp) and (150.28bp,395.62bp)  .. (node_10);
  \draw [black,->] (node_13) ..controls (256.5bp,328.95bp) and (256.5bp,338.97bp)  .. (node_11);
  \draw [black,->] (node_16) ..controls (352.5bp,598.95bp) and (352.5bp,608.97bp)  .. (node_15);
  \draw [black,->] (node_1) ..controls (258.64bp,274.95bp) and (258.06bp,284.97bp)  .. (node_13);
  \draw [black,->] (node_0) ..controls (154.15bp,221.85bp) and (138.27bp,235.32bp)  .. (node_19);
  \draw [black,->] (node_12) ..controls (233.5bp,760.95bp) and (233.5bp,770.97bp)  .. (node_8);
  \draw [black,->] (node_8) ..controls (233.5bp,815.12bp) and (233.5bp,825.1bp)  .. (node_26);
  \draw [black,->] (node_3) ..controls (192.75bp,601.1bp) and (260.63bp,615.93bp)  .. (node_15);
  \draw [black,->] (node_0) ..controls (194.89bp,221.7bp) and (218.18bp,234.79bp)  .. (node_1);
  \draw [black,->] (node_4) ..controls (274.25bp,547.1bp) and (206.37bp,561.93bp)  .. (node_3);
  \draw [black,->] (node_2) ..controls (152.26bp,654.5bp) and (183.16bp,668.0bp)  .. (node_14);
  \draw [black,->] (node_6) ..controls (271.93bp,493.1bp) and (203.76bp,507.93bp)  .. (node_5);
  \draw [black,->] (node_4) ..controls (352.5bp,544.95bp) and (352.5bp,554.97bp)  .. (node_16);
  \draw [black,->] (node_20) ..controls (140.5bp,120.71bp) and (140.5bp,130.98bp)  .. (node_18);
  \draw [black,->] (node_14) ..controls (233.5bp,706.95bp) and (233.5bp,716.97bp)  .. (node_12);
%
\end{tikzpicture}
Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1034, in execute_with_code_decorators print code_decorator.eval(code, locals=self.namespace), File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/smc_sagews/sage_salvus.py", line 1467, in latex0 sage.misc.latex.Latex.eval(sage.misc.latex.latex, s, **kwds) File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/sage/misc/latex.py", line 1130, in eval os.path.join(orig_base, filename + ".png")) File "/projects/sage/sage-7.6/local/lib/python/shutil.py", line 119, in copy copyfile(src, dst) File "/projects/sage/sage-7.6/local/lib/python/shutil.py", line 82, in copyfile with open(src, 'rb') as fsrc: IOError: [Errno 2] No such file or directory: '/projects/d97ca4df-6ebd-46ab-b7fb-40461f0f84e0/.sage/temp/compute2-us/11056/dir_rhlr3d/tmpZqIlQF.png' 
latex.eval(fix_basis_latex(nRP))
Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/smc_sagews/sage_salvus.py", line 1467, in latex0 sage.misc.latex.Latex.eval(sage.misc.latex.latex, s, **kwds) File "/projects/sage/sage-7.6/local/lib/python2.7/site-packages/sage/misc/latex.py", line 1130, in eval os.path.join(orig_base, filename + ".png")) File "/projects/sage/sage-7.6/local/lib/python/shutil.py", line 119, in copy copyfile(src, dst) File "/projects/sage/sage-7.6/local/lib/python/shutil.py", line 82, in copyfile with open(src, 'rb') as fsrc: IOError: [Errno 2] No such file or directory: '/projects/d97ca4df-6ebd-46ab-b7fb-40461f0f84e0/.sage/temp/compute2-us/11056/dir_xPtH0a/tmpmtWZyd.png' 
var('a')
assume('a', a >= 1)
assume('a', "integer")
v = a*FW[6].to_vector() + (-2*a-8)*FW[7].to_vector() + AS.rho().to_vector()
v
a (0, 1, 2, 3, a + 4, -a - 3, -9/2, 9/2)