### The Involution Principle

We look closer at actions of
involutions. The following remark is trivial but very helpful:
Let * tÎS*_{M} be an involution which has the following reversion
property with respect to the subsets *T,U ÍM:*
*
m ÎT iff tm ÎU.
*

Then the restriction of * t* to *T* establishes a bijection between *T* and
*U*.
We shall apply this to disjoint
decompositions *M=M*^{+} DOTCUP M^{-} of *M* into
subsets *M*^{ ±}. Each such disjoint decomposition gives rise to a
*sign function* on *M*:
* sign (m):= 1 if m ÎM*^{+} sign (m):=-1 if m ÎM^{-}.

**The Involution Principle **
Let *M=M*^{+} DOTCUP M^{-} be a disjoint
decomposition of a finite set *M* and let * tÎS*_{M} be a *sign
reversing* involution:
* " m not ÎM*_{ t} : sign ( tm)=- sign (m).

Then the the restriction of * t* to *M*^{+} \M_{t}
is a bijection
onto *M*^{-} \M_{t}. Moreover
* å*_{m ÎM} sign (m)= å_{m ÎM t} sign (m).

If in addition *M*_{ t} ÍM^{+}, then
* å*_{m ÎM} sign (m)= | M_{ t} | = | M^{+} | -
| M^{-} | .

Proof:
* å*_{m ÎM} sign (m) is equal to
* å*_{m ÎM t} sign (m) +å_{m ÎM+ \M t} sign (m)+ å_{m ÎM- \M t} sign (m)

where the sum of the second and third sum equals 0 by
the formula.

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001