On 01/19/2007 11:18 AM, Martin Rubey wrote:
> Dear Ralf,
>
> I have found some more reading material... I share it with you, in case you
> are
> interested. What concerns myself, I won't code for a while now, I first have
> to
> understand all that properly...
Thank you, Martin, for all that references. I think there is lots of
things to understand. But I think after reading BLL Chp 4.3 several
times, I now know that the cycle index series is *not* the formal
addition of the cycle index polynomials of the actions
S_n x F[n] -> F[n]
Those cycle index polynomials would have a chance of including cycles
longer than n (since card(F[n]) might be bigger than n). An appropriate
(counter-)exampe is given in section BLL 1.2 where they consider the
species of involutions on U={a,b,c,d,e} and where the permutation
\sigma=(a b)(c d e) on U leads to a permutation F[\sigma] on F[U] with
cycles of length 6.
Rather the cycle index series is the sum of cycle index polynomials of
some stabilizer group (see BLL 4.3 Proposition 2 -- thanks Martin for
pointing to it).
I think, I can now start to put that knowledge into code and documentation.
I fear that substitution will cost me another head ache since it should
not be too inefficient. But at least addition and multiplication is more
or less trivial building on the formal power series we already have in
AC. (I'd be happy if there were someone who could implement the relaxed
power series multiplication in AC.)
Ralf
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