108 Signed permutations

A signed permutation of [1..n] is a permutation of the set {-n,...,-1,1,...,n} which preserves the pairs [-i,i]. It is represented internally as the images of [1..n]. It is printed as a product of signed cycles.

A signed permutation can be represented in two other ways which may be convenient. The first way is to replace the integers {1,...,n,-n,...,-1} by {1,3,...,2n-1,2,4,...,2n} to have GAP3 permutations, which form the hyperoctaedral group (see CoxeterGroupHyperoctaedralGroup).

The second way is to represent the signed permutation by a monomial matrix with entries 1 or -1. If such a matrix m represents the signed permutation sp, then l*m is the same as Permuted(l,sp).

Subsections

  1. Permuted for signed permutations
  2. PermutationMat for signed permutations
  3. SignedPerm
  4. Cycles for signed permutations
  5. SignedPermListList
  6. SignedMatStab
  7. SignedPermMatMat

108.1 Permuted for signed permutations

Permuted( l, sp)

Permuted returns a new list n that contains the elements of the list l permuted according to the signed permutation sp. If sp is given as a list, then n[AbsInt(i^sp)] = l[i]SignInt(i^sp).

    gap> p:=SignedPerm([-2,-1,-3]);
    (1,-2)(3,-3)
    gap> Permuted([20,30,40],p);
    [ -30, -20, -40 ]

108.2 PermutationMat for signed permutations

PermutationMat( sp [,d])

This function returns the signed permutation matrix of the signed permutation sp. This is a matrix m such that Permuted(l,sp)=l*m for any list of numbers l. If an additional argument d is given the matrix is returned of that dimension.

    gap> p:=SignedPerm([-2,-1,-3]);
    (1,-2)(3,-3)
    gap> PermutationMat(p);
    [ [ 0, -1, 0 ], [ -1, 0, 0 ], [ 0, 0, -1 ] ]

108.3 SignedPerm

SignedPerm( sp [,d or sgns])

This function converts to a signed permutation a list, an element of the hyperoctaedral group, a signed permutation matrix, or a pair of a permutation and of a list of signs. If given an element of the hyperoctaedral group, the rank d of that group can be given as an argument, otherwise a representation of sp as a list is given corresponding to the smallest hyperoctaedral group to which it belongs.

    gap> SignedPerm([[0,-1,0],[0,0,-1],[-1,0,0]]);
    (1,-2,3,-1,2,-3)
    gap> SignedPerm((1,4,5,2,3,6));
    (1,-2,3,-1,2,-3)
    gap> SignedPerm((1,2,3),[-1,-1,-1]);
    (1,-2,3,-1,2,-3)
    gap> SignedPerm([-2,-3,-1]);
    (1,-2,3,-1,2,-3)

108.4 Cycles for signed permutations

Cycles( sp )

Returns the list of cycles of the signed permutation sp on {-n,...,-1,1,...,n}. If one cycle is the negative of another, only one of the two cycles is given.

    gap> Cycles(SignedPerm([-2,-3,-1]));
    [ [ 1, -2, 3, -1, 2, -3 ] ]
    gap> Cycles(SignedPerm([-2,-1,-3]));
    [ [ 1, -2 ], [ 3, -3 ] ]
    gap> Cycles(SignedPerm([-2,-1,3]));
    [ [ 1, -2 ] ]

108.5 SignedPermListList

SignedPermListList( list1, list2 )

SignedPermListList returns a signed permutation that may be applied to list1 to obtain list2, if there is one. Otherwise it returns false.

    gap> SignedPermListList([20,30,40],[-40,-20,-30]);
    (1,-2,3,-1,2,-3)
    gap> Permuted([20,30,40],last);
    [ -40, -20, -30 ]

108.6 SignedMatStab

SignedMatStab(M[, l])

Finds the stabilizer of M in the group of signed permutations.

    gap> uc:=UnipotentCharacters(ComplexReflectionGroup(6));
    UnipotentCharacters( G6 )
    gap> SignedMatStab(Fourier(uc.families[2]));
    Group( (2,19)(4,-14)(5,20)(7,12), (1,-1)(2,-2)(3,-3)(4,-4)(5,-5)(6,-6)\ 
    (7,-7)(8,-8)(9,-9)(10,-10)(11,-11)(12,-12)(13,-13)(14,-14)(15,-15)(16,\ 
    -16)(17,-17)(18,-18)(19,-19)(20,-20)(21,-21)(22,-22), (1,3)(2,19)(4,-1\ 
    4)(5,-5)(6,-18)(7,-7)(8,10)(11,15)(12,-12)(13,22)(16,21)(17,-17)(20,-2\ 
    0), (1,6)(2,-19)(3,-18)(4,14)(8,16)(9,-9)(10,21)(11,-13)(15,-22), (1,1\ 
    1)(3,15)(4,14)(6,-13)(7,-12)(8,-10)(9,-9)(16,-21)(18,22) )
    gap> Size(last);
    32

108.7 SignedPermMatMat

SignedPermMatMat( M , N [, l1, l2])

M and N should be symmetric matrices. PermMatMat returns a signed permutation p such that OnMatrices(M,p)=N if such a permutation exists, and false otherwise. If list arguments l1 and l2 are given, the permutation p should also satisfy Permuted(l1,p)=l2.

This routine is useful to identify two objects which are isomorphic but with different labelings. It is used in CHEVIE to identify Lusztig Fourier transform matrices with standard (classified) data. The program uses sophisticated algorithms, and can often handle matrices up to 80× 80.

    gap> f:=SubFamilyij(CHEVIE.families.X(12),1,3,(3+ER(-3))/2);
    Family("RZ/12^2[1,3]")
    gap> M:=Fourier(ComplexConjugate(f));;
    gap> uc:=UnipotentCharacters(ComplexReflectionGroup(6));
    UnipotentCharacters( G6 )
    gap> N:=Fourier(uc.families[2]);;
    gap> SignedPermMatMat(M,N);
    (1,13)(2,19,-2,-19)(3,22)(4,-4)(5,-5)(6,-11)(7,12)(8,21,-8,-21)(9,-9)(\ 
    10,16)(15,-18,-15,18)
    gap> OnMatrices(M,last)=N;
    true

Previous Up Next
Index

gap3-jm
24 Apr 2021