# Dual Cells and Face Poset

Hello.

I would like to create a procedure in Sage to find the dual cell of a simplex $\sigma$ in a simplicial complex $K$. The dual cell $D(\sigma, K)$ of $\sigma$ is a subcomplex of the first barycentric subdivision of $K$. The vertex set is given by the barycentres of all cofaces of $\sigma$, and the simplices are joins of barycentres of the form $\widehat{\sigma_0} \widehat{\sigma_1} ... \widehat{\sigma_s}$ with $\sigma \leq \sigma_0 \leq ... \leq \sigma_s$.

My plan of attack is to view K as a poset, then find the maximal increasing chains $[\sigma_0,..., \sigma_s]$ in K which satisfy $\sigma_0 = \sigma$ . These chains would then be the maximal faces of the dual cell $D(\sigma, K)$.

Creating the poset and finding the maximal chains of simplices is fine. However, if I have a maximal chain $[\sigma_0,..., \sigma_s]$ a problem occurs when checking if $\sigma_0 = \sigma$ - Sage sees $\sigma_0$ as just an element of the poset and not as simplex and $\sigma$ as a simplex but not an element of the poset so the equality is never satisfied.

How can I correct this?

Thanks, Chris .

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What if you define the poset using 'facade=True'?

( 2012-11-28 13:01:33 -0500 )edit

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Okay, try this: change the last line of find_dual_cell to

return SimplicialComplex(dual_cell_facets)


It's always an option to create a simplicial complex without specifying the vertex set  it is then deduced from the list of facets  and in fact this will become the preferred way of doing things some time soon (after the patches here are merged). Anyway, with your original code, I get an error with Sage 5.4 (but not with a prerelease of Sage 5.5, which includes the relevant patches). By modifying that one line, it seems to work in Sage 5.4 as well.

• I guess then you can delete the function find_cofaces and the one call to it in find_dual_cell.
• I don't have the time to figure out why your original code fails, but at least I think we have a fix.
• Would it help to be able to do sage: K.face_poset(options...) with options including facade=True? That would be an easy thing to add to Sage.
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If I use your code with K being a 2-simplex, it seems to work (unless I misunderstand what the dual cell is):

sage: X = simplicial_complexes.Simplex(2)
sage: find_dual_cell(X, Simplex(X.vertices()[0]))
Simplicial complex with 4 vertices and 2 facets
sage: Y = find_dual_cell(X, Simplex(X.vertices()[0]))
sage: Y.facets()
{((0,), (0, 1, 2), (0, 2)), ((0,), (0, 1), (0, 1, 2))}


or

sage: Y = find_dual_cell(X, Simplex([0]))
sage: Y.facets()
{((0,), (0, 1, 2), (0, 2)), ((0,), (0, 1), (0, 1, 2))}


It also seems to work if K is $S^1$ or $S^2$. If I try it with a 3-sphere, I get this:

sage: X = simplicial_complexes.Sphere(3)
sage: Y = find_dual_cell(X, Simplex(X.vertices()[0]))
sage: Y
Simplicial complex with 15 vertices and 24 facets


Is this wrong? What sort of answer are you expecting?

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Thank you! Problem solved :). I think it would be helpful to do

sage: K.face_poset(options...)

with options including facade=True in Sage.

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I only actually want to find the dual cells of vertices. I added the code "facade = True" to the existing algorithm which finds the face poset of a simplicial complex K (a warning for below- I have not done much programming before)

def find_face_poset(K):

from sage.combinat.posets.posets import Poset
from sage.misc.flatten import flatten
covers = {}

all_cells = flatten([list(f) for f in K.cells().values()])
for C in all_cells:
if C.dimension() >= 0:
covers[tuple(C)] = []
for C in all_cells:
for face in C.faces():
if face.dimension() >= 0:
covers[tuple(face)].append(tuple(C))



I also created an algorithm to find the cofaces of a simplex $\sigma \in K$:

def find_cofaces(K, sigma):

m = sigma.dimension()
cofaces = []
cofaces.append(sigma)
for i in range(m+1, K.dimension()+1):
i_faces = K.n_faces(i)
for tau in i_faces:
if sigma.is_face(tau):
if tau.dimension() == i :
cofaces.append(tau)
else:
pass
else:
pass
return cofaces


and also an algorithm to find the dual cell of a vertex $\sigma$:

def find_dual_cell(K, sigma):

face_poset = find_face_poset(K)
maximal_chains = face_poset.maximal_chains()
dual_cell_vertices = find_cofaces(K, sigma)
dual_cell_facets = []
for tau in maximal_chains:
if sigma.tuple() == tau[0]:
dual_cell_facets.append(tau)
else:
pass
return SimplicialComplex(dual_cell_vertices, dual_cell_facets)


I played around with the example

K = simplicial_complexes.Sphere(3)

for sigma in K.n_faces(0): find_dual_cell(K, sigma)

When I get Sage to print dual_cell_vertices and dual_cell_facets (instead of returning SimplicialComplex(dual_cell_vertices, dual_cell_facets)), it seems to produce a correct result. But it never produces the correct simplicial complex! It gives me

Simplicial complex with vertex set (0, 1, 2, 3, 4) and 4 facets

Any help would be very much appreciated!

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This is completely the correct answer! How frustrating for me! When I try to run the example (on my Macbook Pro with Sage 5.4.1 and OS X 10.8.2)

sage: X = simplicial_complexes.Simplex(2)

sage: find_dual_cell(X, Simplex(X.vertices()[0]))

I get the error

sage: Traceback (click to the left of this block for traceback)...

sage: ValueError: The face ((0,), (0, 2), (0, 1, 2)) is not a subset of the vertex set.

I have created a worksheet at http://www.sagenb.org/home/pub/5038 that contains all of the above code (edit: public worksheets are currently disabled) . It still produces the same error!

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