powers of Antisymmetric Signed binary Matrices

[Marc LeBrun]

 > wouter meeussen
 > it's becoming:
 > "invertable (-1,0,1) nXn matrices having (Tij = -Tji ;i<j) such that all 
T^k (k= 1..12) are also
 > (-1,0,1) matrices"
 > and comment:
 > "the matrix powers T^k reach identity I for k a divisor of 12. All are 
both invertible (non-zero
 > determinant) (-1,0,1)-matrices and "pseudo-anti-symmetric" (that is 
Tij=-Tji except for the main
 > diagonal, or, equivalently, the sum of an anti-symmetric and a diagonal 
matrix)."

"invertible"
        ^
If you say "all T^k for k=1..12 have Property X" then it includes both the 
cases k=1 and k>1 and so you don't have to append that repetitious "are 
also X" part.  (Also the T^0=I case apparently also satisfies X<;-).

Moreover, I agree, it's beginning to seem, from the proliferation of 
attributes and restrictions, that we ought to analyze what's going on a bit 
more before settling on a description and comment.

It's usually telling when we count something and all the instances that 
meet the criteria also meet some apparently stronger restrictions.

It's not clear exactly what combination of independent attributes 
constitutes Property X here, which of the nontrivial powers (k=-1, >+1) 
share all or some of these attributes comprising X, and which attributes 
are simply inevitable consequences of others.

For example, it seems significant that all T are "roots of unity".  Isn't 
being invertible a consequence of this?  If T^p=I then T^(-1)=T^(p-1).  And 
so mustn't they also all have determinant +1 or -1?

We might be counting some special subset embedded in a larger 
structure.  In particular, apparently they can all be expressed as powers 
of twelfth roots.  However any given one of these 12th roots itself might 
not qualify as fully having Property X--are there some that do (and hence 
are also included in the T count) and some that don't?

It might be a good idea to look at the entire graph of the explicit Ts, 
their underlying 12th roots and their powers for some small n and see if 
they form some recognizable structure.