oops, here's the code! I keep getting server erros when trying to attach
as a file, so I'm just including the text of the code file below:
class GroupAction(Parent):
def __init__(self, G, S, phi):
#phi a group action G\times S \rightarrow S
self.phi=phi
self.G=G
self.S=S
def __repr__(self):
return "Action of "+self.G.__repr__()+" on the set
"+self.S.__repr__()
def action(self, g, s):
"""
Gives the action of g on s.
"""
return self.phi(g,s)
def group(self):
"""
Group which acts.
"""
return self.G
def gset(self):
"""
Set on which the group acts.
"""
return self.S
def action_function(self):
"""
Function from `G\times S \rightarrow S`.
"""
return self.phi
def check_action(self, g, h, s):
"""
Checks whether g(hs)=(gh)s.
"""
return self.phi(g*h,s)==self.phi(g,self.phi(h,s))
def check_action_full(self, gens=None, contragradiant=False):
"""
Checks that this is actually a group action using a generating set
for
the group acting on the full set.
"""
assert self.S.is_finite(), 'Cannot check group action on an
infinite set.'
if gens==None:
#Should check if g has gens implemented.
gens=self.group().gens()
for g in gens:
for h in gens:
for s in S:
if contragradiant:
if not self.phi(g*h,s)==self.phi(g,self.phi(h,s)):
stringy=g.__repr__()+', '+h.__repr__()+'
'+s.__repr__()
assert False, 'Action fails on '+stringy
else:
if not self.phi(h*g,s)==self.phi(g,self.phi(h,s)):
stringy=g.__repr__()+', '+h.__repr__()+'
'+s.__repr__()
assert False, 'Action fails on '+stringy
return True
def orbit(self, s):
return Set([self.action(g,s) for g in self.group()])
def is_transitive(self):
if len(self.gset())==0: return True
s=self.gset()[0]
return orbit(s)==Set(self.gset())
def twist(self, endomorphism):
"""
Twists this representation by an endomorphism of the group.
"""
phi=self.action_function()
kappa=lambda g,s: phi( endomorphism(g), s)
return GroupAction(self.G, self.S, kappa)
def character(self):
"""
Count fixed points for conjugacy class representatives.
"""
c=[]
for g in self.G.conjugacy_classes_representatives():
fix=0
for s in self.S:
if self.action(g,s)==s: fix+=1
c.append(fix)
return c
def cayley_graph(self, gens=None):
"""
Builds a cayley graph of the group action, using the specified
generating set.
"""
assert self.S.is_finite(), 'Cannot check group action on an
infinite set.'
if gens==None:
#Should check if g has gens implemented.
gens=self.group().gens()
G=DiGraph()
for g in gens:
for s in self.gset():
G.add_edge( s, self.action(g,s) )
return G
def product_action(self, B):
"""
Given a second group action B with the same group and set T,
generates
the product group action of self and B.
"""
assert self.group()==B.group(), 'Actions need to have same group
acting.'
T=B.gset()
U=CartesianProduct(self.gset(),T)
kappa=lambda g, u: U( [self.action_function()(g,u[0]),
B.action(g,u[1])] )
return GroupAction(G,U,kappa)
"""
#Example 1: Usual symmetric group action.
sage: G=SymmetricGroup(4)
sage: S=Set([1,2,3,4])
sage: phi = lambda g,s: g(s)
sage: A=GroupAction(G,S,phi)
sage: A.character()
[4, 2, 0, 1, 0]
#Example 2: Symmetric group acting on a set.
sage: rho=lambda g,s: Set([phi(g,t) for t in s])
sage: T=Subsets(S,2)
sage: B=GroupAction(G,T,rho)
sage: B.character()
[6, 2, 2, 0, 0]
#Example 3: Product action.
sage: C=A.product_action(B)
sage: C.character()
[24, 4, 0, 0, 0]
#Example 4: Twist by an automorphism.
sage: a=G.an_element()^2
sage: ai=a.inverse()
sage: auto=lambda g: a*g*ai
sage: At=A.twist(auto)
sage: y=G.simple_reflection(1)
sage: [A.action(y,s) for s in S]
[2, 1, 3, 4]
sage: [At.action(y,s) for s in S]
[1, 2, 4, 3]
"""
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