Barred Permutations
These partitions are elements of the wreath product S2
≀Sn, the Hyperoctahedral group. These elements are
objects of the type PERMUTATION, but now the kind of the
permutation is BAR or BARCYCLE. The permutation is of length n, and
the entries are between 1 and n, but here they can be positive
or negative.
The first routine we describe is that which computes the label
of the conjugacy class of the element a.
- NAME: class_bar
- SYNOPSIS: INT class_bar(OP a,b)
In order to use the next routine, you enter the label of a class,
and the routine computes an representative element b of that class:
- NAME: class_rep_bar
- SYNOPSIS: INT class_rep_bar(OP a,b)
Another routine computes the Lehmer code of a barred permutation a.
The result is a two-element VECTORobject, whose two entries are
INTEGER - VECTOR objects, whose lengths are the length of the
PERMUTATIONobject a. The first VECTOR is a 0-1-vector, the i-th
entry is one if the element i+1 is negative in the
PERMUTATIONobject a. The second VECTOR is the ordinary Lehmer code
of a permutation, but taken into account that we may have negative
entries.
- NAME: lehmercode_bar
- SYNOPSIS: INT lehmercode_bar(OP a,b)
- An EXAMPLE: If we are given the barred permutation
the result of an application of lehmercode_bar is
The next routine is the inverse routine of the above routine:
- NAME: lehmercode_vector_bar
- SYNOPSIS: INT lehmercode_vector_bar(OP a,b)
The following routine computes the reduced length of the barred
permutation a.
- NAME: length_bar
- SYNOPSIS: INT length_bar(OP a,b)
This routine computes a vector containing all the labelings of the
classes of the group S2 ≀Sa:
- NAME: makevectorof_class_bar
- SYNOPSIS: INT makevectorof_class_bar(OP a,b)
Another routine computes a vector with representatives of all the
conjugacy classes of the group S2 ≀Sa,
the ordering of classes is as in the function
makevectorof_class_bar.
- NAME: makevectorof_class_rep_bar
- SYNOPSIS: INT makevectorof_class_rep_bar(OP a,b)
The following routine computes the order of the centraliser of the
class labeled by a.
- NAME: ordcen_bar
- SYNOPSIS: INT ordcen_bar(OP a,b)
This routine multiplies a and b (as permutations: ''first b then
a):
- NAME: mult_bar_bar
- SYNOPSIS: INT mult_bar_bar(OP a,b,c)
This one computes the order of the class labeled by a.
- NAME: ordcon_bar
- SYNOPSIS: INT ordcon_bar(OP a,b)
The next routine computes a random element of given length, so b
becomes an element of S2 ≀Sa, while a
is an INTEGERobject:
- NAME: random_bar
- SYNOPSIS: INT random_bar(OP a,b)
Here is a routine that transforms a barred permutation a in
list-notation into cycle notation
- NAME: t_BAR_BARCYCLE
- SYNOPSIS: INT t_BAR_BARCYCLE(OP a,b)
- BUG: a and b must be different
The next routine transforms a barred permutation a in
cycle-notation into list notation:
- NAME: t_BARCYCLE_BAR
- SYNOPSIS: INT t_BARCYCLE_BAR(OP a,b)
- BUG: a and b must be different
harald.fripertinger "at" uni-graz.at, May 26,
2011
