Special routines for wreath products Orbit-construction in SYMMETRICA Generators of permutation groups Generators of induced actions

Generators of induced actions

Given two permutation groups G and H we can compute the direct sum or the direct product of these two permutation groups.

Let g∈ G≤ SX and h∈ H≤ SY then you can compute elements φ,ψ∈ G × H such that the restriction of φ onto X equals g, the restriction of φ onto Y is the identity, the restriction of ψ onto X is the identity and the restriction of ψ onto Y equals h for the direct sum, and such that for the direct product

φ(x,y)=(gx,y)        ψ(x,y)=(x,hy)
hold, by using   
INT dir_sum_perm(a,b,c,d)     OP a,b,c,d;
INT dir_prod_perm(a,b,c,d)    OP a,b,c,d;
In both cases a b c and d stand for the permutations g h g' and h'.

Given the permutation groups G and H by systems of generators you can compute the generators of the direct sum or the direct product by  

INT gen_dir_sum(a,b,c)     OP a,b,c;
INT gen_dir_prod(a,b,c)    OP a,b,c;
where a and b are the VECTORS of generators of G and H. c is the VECTOR of generators for the corresponding permutation representation of the direct product of the groups G and H.

The induced permutation representation of a PERMUTATION acting on 2-sets can be computed with the (misnamed) procedure  

INT m_perm_paareperm(a,b)    OP a,b;
where a is the given PERMUTATION and b is the induced PERMUTATION on the set of all pairs (for a certain labelling of the pairs).

For a given set of generators you can compute a system of generators of the induced action on the set of all 2-sets by  

INT gen_on2sets(a,b)   OP a,b;
where a is a system of generators (PERMUTATION objects) and b is the system of the induced PERMUTATIONs on the set of all 2-sets.
harald.fripertinger "at" uni-graz.at, May 26, 2011

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