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@kfunigraz.ac.at,
last changed: August 28, 2001

 The Involution Principle