Exploring posets related to coxeter groupsΒΆ
[1]:
%display latex
[2]:
W=CoxeterGroup(["B",3])
[3]:
P = W.bruhat_poset()
[4]:
G = P.hasse_diagram()
[5]:
G.set_latex_options(format="dot2tex")
[6]:
view(G)
[7]:
W=CoxeterGroup(["B",4])
[8]:
c=W.coxeter_element()
[9]:
c.absolute_covers()
[9]:
[10]:
c.descents()
[10]:
[11]:
c.matrix()
[11]:
[12]:
c.weak_covers()
[12]:
[13]:
L=W.weak_lattice()
[14]:
print L.category()
Category of finite enumerated lattice posets
[15]:
L.is_supersolvable()
[15]:
[16]:
L.is_selfdual()
[16]:
[17]:
L.chains()
[17]:
The absolute order is not yet available for Weyl groups. In the mean time, we construct the same group as a reflection group.
[18]:
W = ReflectionGroup([2,1,3])
[ ]:
[19]:
W.cardinality()
[19]:
[20]:
P = W.absolute_poset()
[21]:
s = W.simple_reflections()
[22]:
s
[22]:
[23]:
s[1]
[23]:
[24]:
I = P.interval(W.one(), s[1]*s[2]*s[3])
[25]:
I
[25]:
[26]:
I = P.subposet(I)
[27]:
I.plot()
[27]:
[28]:
I.is_lattice()
[28]:
[29]:
I = LatticePoset(I)
[30]:
I.is_supersolvable()
[30]:
[ ]: