A multi-dimensional cycle index

A multi-dimensional cycle index

Whenever a group G is acting on sets X₁,..., X_n then G acts in a natural way on the disjoint union

X := ∪ _i=1ⁿ X_i.^._.

Replacing in such a situation the cycle index of G acting on X by a so-called n-dimensional cycle index we get more information about the permutation representation of G. The n-dimensional cycle index which uses for each set X_i a separate family of indeterminates x_i,1,x_i,2,... is given by

Z_n(G,X₁∪ ...∪ X_n) := ^._. 1

|G|
∑_{g∈ G} ^._. n
∏
i=1 (^._. |X_i|
∏
j=1 x_i,j^a_i,j(g)),^._.

where (a_i,1(g),...,a_{i,|X_i|}(g)) is the cycle type of the permutation corresponding to g acting on X_i.

Returning to the fullerene C₆₀ the groups R and S are acting on the disjoint union of the sets of all vertices, pentagonal edges, hexagonal edges, pentagonal faces, hexagonal faces and diagonals. When denoting the families of indeterminates for these actions by the following symbols v_i, e_i, E_i, f_i, F_i and d_i we compute:

Z₆(R)=^._. 1

60
( 24 v₅¹²e₅¹²E₅⁶f₁²f₅²F₅⁴d₅⁶+ 20 v₃²⁰e₃²⁰E₃¹⁰f₃⁴F₁²F₃⁶d₃¹⁰+ 15 v₂³⁰e₂³⁰E₁²E₂¹⁴f₂⁶F₂¹⁰d₁²d₂¹⁴+ v₁⁶⁰e₁⁶⁰E₁³⁰f₁¹²F₁²⁰d₁³⁰)^._.

and

Z₆(S)=^._. 1

2
Z₆(R)+ ^._. 1

120
( 24 v₁₀⁶e₁₀⁶E₁₀³f₂f₁₀F₁₀²d₅⁶+ 20 v₆¹⁰e₆¹⁰E₆⁵f₆²F₂F₆³d₃¹⁰+ 15 v₁⁴v₂²⁸e₁⁴e₂²⁸E₁⁴E₂¹³ f₁⁴f₂⁴F₁⁴F₂⁸d₁²d₂¹⁴+ v₂³⁰e₂³⁰E₂¹⁵f₂⁶F₂¹⁰d₁³⁰).^._.

From these cycle indices we deduce that the action on the sets of vertices and pentagonal edges have the same cycle type. The variables E_i, f_i and F_i determine the cycle index of the symmetry group of the icosahedron acting on its set of edges, vertices or faces. Using these 6-dimensional cycle indices we can compute the number of different simultaneous colourings of all vertices, pentagonal and hexagonal edges, pentagonal and hexagonal faces and diagonals with k₁,k₂,...,k₆ colours by replacing each variable v_i by k₁, e_i by k₂ and so on. For k₁= ...=k₆ =2 the number of R-different colourings is

109 700 303 821 413 736 143 664 612 170 571 163 303 931 905 179 435 773 317 873 664 ^._.

whereas the number of S-different colourings is

54 850 151 910 706 868 071 832 306 128 208 569 015 853 860 985 570 356 157 743 104. ^._.

This substitution into an n-dimensional cycle index can be computed using the SYMMETRICA routine polya_multi_const_sub(a,b,c).

harald.fripertinger "at" uni-graz.at, May 10, 2016

A multi-dimensional cycle index