   The Involution Principle

### The Involution Principle

We look closer at actions of involutions. The following remark is trivial but very helpful: Let tÎSM 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 tÎSM be a sign reversing involution:
" m not ÎM t : sign ( tm)=- sign (m).
Then the the restriction of t to M+ \Mt is a bijection onto M- \Mt. Moreover
åm ÎM sign (m)= åm ÎM t sign (m).
If in addition M t ÍM+, then
åm ÎM sign (m)= | M t | = | M+ | - | M- | .

Proof: åm ÎM sign (m) is equal to

åm ÎM t sign (m) +åm ÎM+ \M t sign (m)+ åm ÎM- \M t sign (m)

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

 harald.fripertinger "at" uni-graz.at http://www-ang.kfunigraz.ac.at/~fripert/ UNI-Graz Institut für Mathematik UNI-Bayreuth Lehrstuhl II für Mathematik
last changed: January 19, 2005   The Involution Principle