Implementation in SYMMETRICA

Implementation in SYMMETRICA

All the cycle indices for the action of the symmetry group of the fullerene C₆₀ are implemented in SYMMETRICA. The 6-dimensional cycle indices for R and S are

INT zykelind_full60(a) OP a;

INT zykelind_full60_extended(a) OP a;

The cycle indices for the actions on the sets of vertices, faces, edges and diagonals are

INT zykelind_full60_vertices(a) OP a;

INT zykelind_full60_vertices_extended(a) OP a;

INT zykelind_full60_edges(a) OP a;

INT zykelind_full60_edges_extended(a) OP a;

INT zykelind_full60_faces(a) OP a;

INT zykelind_full60_faces_extended(a) OP a;

INT zykelind_full60_diagonals(a) OP a;

In all these cases the _extended versions are the cycle indices of the full symmetry group S.

Let me give a short description how to handle polynomials with variables in several alphabets. In SYMMETRICA there is a routine which allows to multiply two polynomials in disjoint sets of indeterminates. The corresponding routine is called

INT mult_disjunkt_polynom_polynom(a,b,c) OP a,b,c;

where a and b are the two polynomials that should be multiplied. c is the result. A POLYNOM object in SYMMETRICA consists of three parts:

A coefficient,
the so called self-part, which is a VECTOR of INTEGER objects that represent the exponents of the monomial summands,
and a next-part, which is the lexicographically next monomial, or zero, if there is no further monomial summand of the polynomial in question.

The routine for multiplication of two polynomials in disjoint sets of indeterminates works in the following way: At first the number of variables of the first polynomial a is evaluated. (Let this number be n.) Then for each monomial summand of a it is tested, if its self-part is of length less than n, and if this is so, then this self-part is changed into a VECTOR object of length n and all the new entries are set to zero. Then the self part of each monomial summand of b is appended to the self-part of a (of length n), forming a new self-part of a monomial summand of c. The corresponding coefficients of the monomials of a and b are multiplied to get the new coefficient of this monomial.

In order to work with these polynomials in two or more alphabets it is therefore important to know how many variables are in the first alphabet, in the second alphabet and so on. Or in other words, we must keep in mind at which index of the self-part of the monomial summands the different alphabets start. (The index where the i-th family starts, is the number of variables which have already occurred in the previous i-1 families.) Using a vector of INTEGER objects, where for each polynomial the position in the self-parts of the monomial summands is indicated, where the new alphabet starts, gives the whole information. For example consider two polynomials a and b in two different alphabets (a is a polynomial in x_i and b is a polynomial in y_i) where a has n variables. Then applying
mult_disjunkt_polynom_polynom(a,b,c)
makes c to be a polynomial in two families of variables and the corresponding vector of starting points would be [0,n]. A monomial summand of c can be interpreted as

s_po_k(c) Õ_i=0^n-1x_i+1^s_po_ii(c,i) Õ_i=n^s_po_li(c)-1y_i-n+1^s_po_ii(c,i).

For that reason a multi dimensional cycle index in SYMMETRICA consists of a VECTOR-part and a POLYNOM-part, which can be selected by

OP s_mz_v(a) OP a; OP s_mz_po(a) OP a;

The cycle indices for the actions on the sets of vertices, edges, etc. were computed from the 6-dimensional cycle indices by extracting and identifying some families of indeterminates. This extraction of some families of indeterminates of a multi dimensional cycle index can be done by

INT mz_extrahieren(a,b,c) OP a,b,c;

where a is a multi dimensional cycle index and b is a VECTOR object. Its length tells how many families shall be combined into the new cycle index c. The entries of b are INTEGER objects. If for instance a is a 6-dimensional cycle index and you want to extract the first and fifth family of indeterminates then b would be the VECTOR [1,5] of length 2. In the case you choose only one family to be extracted the result will be a POLYNOM object, otherwise it is a multi dimensional cycle index as described above.

For identifying different alphabets there is the routine

INT mz_vereinfachen(a,b) OP a,b;

which computes from a multi dimensional cycle index a a cycle index b in only one alphabet. For instance for computing the cycle index of the action on the set of faces of the fullerene, we first extract the two families corresponding to f_i and F_i and then identify f_i and F_i.

zykelind_full60(a);  /* 6-dimensional cycle index */
M_IL_V(2L,b);
M_I_I(4L,S_V_I(b,0L));
M_I_I(5L,S_V_I(b,1L));
mz_extrahieren(a,b,c); /* extracting the families 4 and 5 from a */
mz_vereinfachen(c,d);  /* identifying the two alphabets */

In addition to this it should be mentioned that the cycle indices of the symmetry groups of the tetrahedron, the cube and the dodecahedron are implemented in the same way in SYMMETRICA.

harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001

Implementation in SYMMETRICA