(This is a white flag sort of question.) I have a numeric conjecture that I was trying to verify. The conjecture is this:
- take a polyhedron P whose vertices are a poset
- extend this to a (non-reflexive) < operation on all faces by saying that F<g if="" max="" vertex="" of="" f="" is="" <br=""> smaller than min vertex of G
- take an algebra R of rational functions on generators x_F for all faces of P, plus extra generator t
- define the following endomorphism f of R:
- for a face F, let Ch(F) be the set of all chains in F, where for a chain C = F1 <... < Fn its total codimension is codim(F,C) = dimF - sum dim Fi
- for a chain C = F1 < ... < Fn, define its monomial m(C) by taking the product of either x_Fi, if dim F_i>0, or of x_Fi/(1-x_Fi), if Fi is a vertex, and then times t^{codim(F,C)}
- define f by sending x_F to the sum of m(C) for all chains C in Ch(C)
- additionally define an endomorhism I by sending x_F to -x_F and t to -t
CONJECTURE: very often f*I will be an involution.
The conjecture is proved for simplices, cubes, and all polygons. It's also disproved in some cases. A key example where I need to know the answer is the associahedron, with Tamari order on vertices.
I've asked several questions here and, thanks to tmonteil, wrote a code that checks the conjecture for the 3D associahedron -- and... my code can't run in any reasonable amount of time! Does the problem above sound like something computationally approachable, given that the number of chains in the 3-dimensional face of associahedron is equal to 22663? How would you write this code?
Attached, my (very bad) code:
---
import itertools
A = Polyhedron(vertices = [[1,2,6,1],[1,6,2,1],[2,1,6,1],[1,2,3,4],[1,4,1,4],[1,6,1,2],[2,1,3,4],[4,1,4,1],[4,3,2,1],[4,3,1,2],[3,1,2,4],[3,2,1,4],[4,1,2,3],[4,2,1,3]])
Fempty = list(A.face_generator())
F = []
for i in Fempty:
if i.dim() != -1:
F.append(i)
R = Frac(PolynomialRing(QQ,x,len(F)+1))
R.inject_variables()
face = {x:f for x,f in list(zip(R.gens(),F))}
x = {f:x for x,f in list(zip(R.gens(),F))}
V = VectorSpace(QQ,4)
W = Cone([[1,-1,0,0],[1,0,-1,0],[1,0,0,-1],[0,1,-1,0],[0,1,0,-1],[0,0,1,-1]])
elms = A.vertices()
rels = []
for i in A.faces(1):
Li = i.vertices()
if vector(list(Li[0]))-vector(list(Li[1])) in W:
rels.append([Li[0],Li[1]])
else:
rels.append([Li[1],Li[0]])
P = Poset((elms, rels), cover_relations = True, facade = True)
def flarger(f1,f2):
L1 = f1.vertices()
L2 = f2.vertices()
for v in L1:
for w in L2:
if (P.lt(w,v) or v == w) == False:
return False
return True
bigelms = A.faces(0)+A.faces(1)+A.faces(2)+A.faces(3)
bigrels = []
for i in bigelms:
for j in bigelms:
if (flarger(i,j) and i != j):
bigrels.append([i,j])
Q = Poset((bigelms, bigrels), cover_relations = False, facade = True)
elll = list(x[i] for i in bigelms)
relll = []
for i in bigrels:
relll.append([x[i[0]],x[i[1]]])
QQ = Poset((elll, relll), cover_relations = False, facade = True)
def inside(f1,f2):
L1 = face[f1].vertices()
L2 = face[f2].vertices()
for v in L1:
if v not in L2:
return False
return True
def subfaces(feis):
result = []
for item in elll:
if inside(item,feis):
result.append(item)
return result
def sfposet(feis):
Res = Poset(QQ.subposet(subfaces(feis)),facade = True)
return Res
def chains(myx):
Huj = sfposet(myx).chains()
result = []
for i in Huj:
if i != []:
result.append(i)
return result
def p(feis):
answer = feis
if face[feis].dim() == 0:
answer = feis/(1-feis)
return answer
def monprod(monlist):
answer = 1
for item in monlist:
answer = answer*p(item)
return answer
def chdim(ch):
dim = 0
for item in ch:
dim = dim+face[item].dim()
return dim
def fmap(feis):
answer = 0
dimf = face[feis].dim()
Flist = chains(feis)
for item in Flist:
power = dimf - chdim(item)
answer = answer + monprod(item)*(x45^power)
return answer
minuslist = []
for feis in F:
minuslist.append(-x[feis])
minuslist.append(-x45)
fmaplist = []
for feis in F:
fmaplist.append(fmap(x[feis]))
fmaplist.append(x45)
f = R.hom(fmaplist)
I = R.hom(minuslist)
f*I*f*I
