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# Find the longest word in a symmetric group which satisfies a certain property.

Let $A, B$ be two list of equal length $k$, where $A$ is weakly increasing. For example, $A=[1, 1, 2, 3, 3, 4]$, $B=[12, 9, 10, 15, 15, 14]$.

Define $m_{A,B}$ to be the multiset of pairs $[ A_i, B_i ]$, $i \in k$, (this is a multi-set of pairs of integers, not a list of pairs, so the order of pairs in $m_{A,B}$ does not matter).

By a result in symmetric group, there is a unique element with maximal length (length of the reduced word) in the symmetric group $S_k$ ($k$ is the length of $A$) such that $m_{A,w(sorted(B))} = m_{A,B}$, where $sorted(B)$ is to sorted B such that it is weakly increasing, and the action of $w=s_{i_1} \cdots s_{i_m}$ on a list $L$ is defined by: $s_j(L)$ means exchanging the jth and j+1th elements of $L$, and for $w,w' \in S_k$, $w w'(L) = w(w'(L))$.

There is a method to compute the longest word $w$ by checking all elements in $S_k$. But it takes a long time when $k$ is large. The following function works fine and returns correct result.

 def LongestPerm(A,Bsorted,B):
k = len(A)
S = set([(A[i],B[i]) for i in range(k)])
W = WeylGroup('A'+str(k-1), prefix = 's')

winner = W.one()
for w in W:
wstr = w.inverse().to_permutation_string()
if set([(A[int(wstr[i])-1],Bsorted[i]) for i in range(k)]) == S:
if w.length()>winner.length():
winner = w
return winner


In the example that

A=[1, 1, 2, 3, 3, 4]
B=[12, 9, 10, 15, 15, 14]


we have

w=s4*s5*s4*s2*s1


Is there some method to compute $w$ faster (without checking all elements of $S_k$)? Thank you very much.

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## 2 Answers

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You don't need to consider all permutations - it's enough to take one suitable permutation and multiply it by elements of the stabilizers of $A$ and $B$ from left and right, respectively.

Here is an example:

# returns list of cycles generating the stabilizer of L
def fixed_sets(L):
D = dict()
for p,l in enumerate(L):
D.setdefault(l,[]).append(p+1)
return [tuple(s) for s in D.values() if len(s)>1]

A=[1,1,2,3,3,4]
B=[12,9,10,15,15,14]

k = len(A)

S = SymmetricGroup(k)
SA = S.subgroup(fixed_sets(A))
SB = S.subgroup(fixed_sets(B))

perm = S( Word(B).standard_permutation() / Word(A).standard_permutation() )

# testing can be done like this:
#sortedB = sorted(B)
#assert set( zip(A,[sortedB[i-1] for i in perm.domain()]) ) == set( zip(A,B) )

W = WeylGroup('A'+str(k-1), prefix = 's')
winner = max( (W(a*perm*b) for a in SA for b in SB), key=lambda s: s.length() )

print( winner )

more

## Comments

@max, thank you very much!

( 2022-01-24 21:55:40 +0200 )edit

@max, I checked some more examples and found that some of the results given by your codes do not agree with the function LongestPerm in the question. I slightly modified your codes and wrote it as an answer below. Thanks again!

( 2022-01-25 09:11:54 +0200 )edit

I've corrected an error in my code.

( 2022-01-25 13:18:05 +0200 )edit

@max, thank you very much!

( 2022-01-27 06:31:54 +0200 )edit

I modified Max Alekseyev's answer. The result of the following codes agree with the result of the function LongestPerm in the post.

def LongestPermWInSBWInverseSA(W,A,B):
t1=StablizerOfTuple(A)
t2=StablizerOfTuple(sorted(B))
s=W.simple_reflections()
#print(t1,t2)
t3=[]
for i in t1:
t3.append(s[i])
t4=[]
for i in t2:
t4.append(s[i])
t5=find_permutation2(B,sorted(B))
w=W.one()
for i in t5:
w=w*s[i]
#print(t4,w,t3)
r2=LongestPermInDoubleCosetWeylGroup(W,t4,w,t3)
r=r2

return r

def find_permutation2(L1, L2):
perm = Word(L1).standard_permutation() / Word(L2).standard_permutation()
assert [L2[i-1] for i in perm] == L1
r=perm.reduced_word()
return r

def LongestPermInDoubleCosetWeylGroup(W,S1,w,S2):
g1=W.subgroup(S1)
g2=W.subgroup(S2)
winner = W.one()
for u1 in g1:
for u2 in g2:
t1=u1*w*u2
if t1.length()>winner.length():
winner=t1
r=winner

return r

def SymmetricGroupActionOnListSi(i,L): #  w=s_i,  L=[a1,a2,...,an]
r1=[]
for j in L:
r1.append(j)

t1=r1[i-1]
r1[i-1]=r1[i]
r1[i]=t1

r=r1

return r

def StablizerOfTuple(A): # A is a list, weakly increasing
k = len(A)
r=[]
for i in [1..k-1]:
t1=SymmetricGroupActionOnListSi(i,A)
if t1==A:
r.append(i)
return r

A=[1,1,2,3,3,4,6]
B=[12,9,12,15,15,14,12]
k = len(A)
W = WeylGroup('A'+str(k-1), prefix = 's')
t1=LongestPermWInSBWInverseSA(W,A,B)
print(t1)

more

## Comments

You can save some lines by using max() function like in my answer.

( 2022-01-25 13:20:28 +0200 )edit

@max, thank you very much for your nice answer!

( 2022-01-27 06:33:05 +0200 )edit

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Asked: 2022-01-24 09:35:15 +0200

Seen: 42 times

Last updated: Jan 25