accidental degeneracies on necklace

[Wouter Meeussen]

hi Christian,

I found the three pairs of "accidental degeneracies" for the n/2 positive
and n/2 negative charges on a nacklace.

acci1=
{-1,-1,-1,-1,1,-1,1,1,-1,1,1,-1,-1,1,1,1}
{-1,-1,-1,-1,1,1,-1,1,-1,-1,1,1,1,-1,1,1}
acci2=
{-1,-1,-1,-1,-1,1,1,-1,1,-1,1,1,-1,1,1,1}}
{-1,-1,-1,-1,1,-1,1,1,-1,-1,1,1,1,1,-1,1}}
acci3=
{-1,-1,-1,1,-1,-1,1,1,-1,1,-1,1,-1,1,1,1}}
{-1,-1,-1,1,-1,1,-1,1,1,-1,-1,1,1,1,-1,1}}

pro memori:
accidental degeneracy means that these pairs of permutations can not be
converted into each other by reflection, rotation or sign change, or any
combination thereof. Nevertheless, they give the same result when
substituted in the adjacency matrix of 2n vertices in a ring.
They explain the difference between the 254 and the 257 for n=8.

Check:

with adjacency matrix as
n=16;
ar=Array[z,n];
(dis=Table[z[i]z[j]w[Min[Mod[n+i-j,n],Mod[-n-i+j ,n]]],{i,n},{j,n}])//
MatrixForm;

formu=Plus@@Flatten[dis];

In[41]:=(formu/.Thread[ar->#])& /@ acci1
Out[41]=
{16 w[0]-8 w[2]-8 w[4],
 16 w[0]-8 w[2]-8 w[4]}
In[42]:=(formu/.Thread[ar->#])& /@ acci2
Out[42]=
{16 w[0]-8 w[4]-8 w[6],
 16 w[0]-8 w[4]-8 w[6]}
In[43]:=(formu/.Thread[ar->#])& /@ acci3
Out[43]=
 {16 w[0]-8 w[1]-8 w[3]+8 w[4]-8 w[5]-8 w[7]+8 w[8],
  16 w[0]-8 w[1]-8 w[3]+8 w[4]-8 w[5]-8 w[7]+8 w[8]}


wouter.

Ah, les colliers ...


NV Vandemoortele Coordination Center
Oils & Fats Applied Research
Prins Albertlaan 79
Postbus 40
B-8870 Izegem (Belgium)
Tel: +/32/51/33 21 11
Fax: +/32/51/33 21 75
vdmcc at vandemoortele.be