The Involution Principle

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.

The Involution Principle