Permutations How to handle PERMUTATIONobjects Barred Permutations

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.

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: 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. The next routine is the inverse routine of the above routine: The following routine computes the reduced length of the barred permutation a. This routine computes a vector containing all the labelings of the classes of the group S2 ≀Sa: 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. The following routine computes the order of the centraliser of the class labeled by a. This routine multiplies a and b (as permutations: ''first b then a): This one computes the order of the class labeled by a. The next routine computes a random element of given length, so b becomes an element of S2 ≀Sa, while a is an INTEGERobject: Here is a routine that transforms a barred permutation a in list-notation into cycle notation The next routine transforms a barred permutation a in cycle-notation into list notation:
harald.fripertinger "at" uni-graz.at, May 26, 2011

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