SYMMETRICA-MANUAL -- Products of cycle indices

Products of cycle indices

Products of cycle indices

Let _GX and _HY be two finite group actions then the direct product G × H induces group actions on the sets X∪ Y, X × Y or Y^X and the wreath productH ≀_XG := {(ψ,g) | ψ∈ H^X, g∈ G} with a multiplication (ψ,g)(ψ', g')=(ψψ'_g, gg'), where ψψ'_g(x) := ψ(x)ψ'_g (x) and ψ'_g(x) := ψ'(g^-1x), acts in a natural way on the sets X × Y and Y^X.

In his famous article [13] Pólya demonstrated how to compute some of these cycle indices.

The direct sum of _GX and _HY acts on the disjoint union X DOTCUP Y by

(G × H) × (X DOTCUP Y)→ X DOTCUP Y,

where (g,h)(z)=gz for z∈ X and (g,h)(z)=hz for z∈ Y.

The direct productof _GX and _HY acts on the cartesian product X × Y by

(G × H) × (X × Y)→ X × Y

((g,h),(x,y))↦ (gx,hy).

The cycle indices of these actions can be computed from the cycle indices of the actions _GX and _HY.

The corresponding routines in SYMMETRICA are

INT zykelind_dir_summ(a,b,c)     OP a,b,c;
INT zykelind_dir_prod(a,b,c)     OP a,b,c;

In both cases a is Z(G,X), b is Z(H,Y), and c is the constructed cycle index. c must be different from a and b.

In order to compute the direct sum or the direct product of n copies of the same cycle index one can use

INT zykelind_hoch_dir_summ(a,b,c)  OP a,b,c;
INT zykelind_hoch_dir_prod(a,b,c)  OP a,b,c;

Here in this context b is an INTEGER object, which tells, how many copies of a shall be combined to compute c. a b and c must be different.

Furthermore there are

INT zykelind_dir_summ_apply(a,b)     OP a,b;
INT zykelind_dir_prod_apply(a,b)     OP a,b;

which compute b to be the direct sum or the direct product of the two cycle indices a and b.

The wreath product of G and H acts on the cartesian product X × Y by

H ≀_XG × (X × Y)→ X × Y

((ψ,g)(x,y))↦ (gx, ψ(x)y).

This action is called the compositionG[H] of G and H. In the case that X={1,...,n} and Y={1,...,m} there is a similar action of H ≀G on the set {1,...,nm}, which is called the plethysm H pleth G of G and H.

The following action of the wreath product on Y^X is called the exponentiationof H by G. It is given by

H ≀_X G × Y^X→ Y^X ((ψ,g),f)↦ ψ(⋅)f(g^-1⋅).

The cycle indices of these actions can be computed by

INT zykelind_kranz(a,b,c)                 OP a,b,c;
INT zykelind_plethysm(b,a,c)              OP a,b,c;
INT zykelind_exponentiation(a,b,c)        OP a,b,c;

In all these cases a is Z(G,X), b is Z(H,Y), and c is the cycle index of the wreath product action. c must be different from a and b. It should be mentioned that zykelind_kranz(a,b,c) equals zykelind_plethysm(b,a,c).

harald.fripertinger "at" uni-graz.at, May 26, 2011

Products of cycle indices