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
