A multi-dimensional cycle index

# A multi-dimensional cycle index

Whenever a group G is acting on sets X1,..., Xn then G acts in a natural way on the disjoint union
X:=Èi=1n Xi.
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 Xi a separate family of indeterminates xi,1,xi,2,... is given by
Zn(G,X1È...ÈXn):=(1)/(|G|)ågÎG Õi=1n(Õj=1|Xi|xi,jai,j(g)),
where (ai,1(g),...,ai,|Xi|(g)) is the cycle type of the permutation corresponding to g acting on Xi.

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 vi, ei, Ei, fi, Fi and di 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 Ei, fi and Fi 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 vi by k1, ei by k2 and so on. For k1= ...=k6 =2 the number of R-different colourings is
109700303821413736143664612170571163303931905179435773317873664
whereas the number of S-different colourings is
54850151910706868071832306128208569015853860985570356157743104.
This substitution into an n-dimensional cycle index can be computed using the SYMMETRICA routine `polya_multi_const_sub(a,b,c)`.
harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001

 A multi-dimensional cycle index