[GAP Forum] Fw: icoshhedral group items maybe of interest?

[Rudolf Zlabinger]

I resend this mail, as there was obviously technical problems during sending 
the original message. Best regards, Rudolf Zlabinger


----- Original Message ----- 
From: "Rudolf Zlabinger" <Rudolf.Zlabinger at chello.at>
To: "Walter Becker" <w_becker at hotmail.com>
Cc: "GAP Forum" <forum at gap-system.org>
Sent: Wednesday, August 02, 2006 5:55 PM
Subject: Re: icoshhedral group items maybe of interest?


> Dear Walter Becker,
>
> I dealt with fullerene rotations as an exercise for regular permutation
> groups. In this case this is the regular permutation group for the
> icosahedral group A_5. The rotations of the 60 vertex fullerene is the
> regular permuation group of A_5. The remaining problem was, how to label 
> the
> vertices according to the distinct permutation group used. I used the GAP
> package GRAPE to construct a graph containing the adjacency matrix 
> showing,
> how to label these vertices.
>
> The remaining symmetries mentioned by you should be translations, i think, 
> i
> didnt deal with yet.
>
> I send you a log of a GAP session showing, how to proceed.
>
> Thank you for your information, best regards, Rudolf Zlabinger
>
>
> ----- Original Message ----- 
> From: "Walter Becker" <w_becker at hotmail.com>
> To: <Rudolf.Zlabinger at chello.at>
> Cc: <mckay at encs.concordia.ca>
> Sent: Wednesday, August 02, 2006 3:35 PM
> Subject: icoshhedral group items maybe of interest?
>
>
>>
>> Dear Drs Zlabinger and McKay:
>>
>> I ran across the following article on the icosahedral group  that may be
>> of interest. It was something I was interested in getting ---it contains
>> the symmetry adapted functions which one uses in allied work ---in ths
>> case dealing with the fullerene vibratonal studies of the C_(60)
>> mnolecule.
>>
>> Aticle appears in Spectroscopy Letters volume 21 issue 4 pp319-339 
>> (1988).
>>
>> there is a previous article dealing with the fullerenes in the same issue
>> (the previous article).
>>
>> Abstract of the article reads
>>
>> A complete set of the 174 symmetry coordinates for teh C_(60) molecular
>> model referred to as footballene is reported. The model is the truncated
>> icosahedron (symmetry I_h). Hence in addition to triple degenmeracy also
>> quadruple and qunituple degeneracies occur.
>>
>> Note the other 6 symmety functions here correspond to translations and
>> center of mass rotations of the whole molecule ---hence total of 60 time 
>> 3
>> or 180 total symmety adapted functions.
>>
>> Hoe this is not old information
>>
>>
>> Walter Becker
>>
>>
>>
> 
-------------- next part --------------
gap> a5:=AlternatingSubgroup(SymmetricGroup(5));;
gap> a5_60:=RegularActionHomomorphism(a5);;
gap> a5_60_home:=last;
<action epimorphism>
gap> a5_60:=Image(a5_60_home);; # group acting on the 60 vertices of fullerene isomorphic to the
gap> # icosahedron group A5
gap> # we want to draw now the graph containing the vertex adjacency matrix
gap> # in order to label the fullerene according to the group a5_60 found above
gap> fullerene:=EdgeOrbitsGraph(a5_60,[[1,2],[1,3],[1,4]]);;
gap> Adjacency(fullerene,1);
[ 2, 3, 4 ]
gap> Adjacency(fullerene,60);
[ 57, 58, 59 ]
gap> # the full adjacency matrix can also be drawn by:
gap> CollapsedAdjacencyMat(Group(()),fullerene);;

gap> quit;