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

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

sign (m):= 1 if m ÎM^{+}sign (m):=-1 if m ÎM^{-}.

The Involution PrincipleLetM=Mbe a disjoint decomposition of a finite set^{+}DOTCUP M^{-}Mand lettÎSbe a_{M}sign reversinginvolution:Then the the restriction of" m not ÎM_{ t}: sign ( tm)=- sign (m).ttoMis a bijection onto^{+}\M_{t}M. Moreover^{-}\M_{t}If in additionå_{m ÎM}sign (m)= å_{m ÎM t}sign (m).M, then_{ t}ÍM^{+}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 "at" uni-graz.at | http://www-ang.kfunigraz.ac.at/~fripert/ |

UNI-Graz | Institut für Mathematik |

UNI-Bayreuth | Lehrstuhl II für Mathematik |

The Involution Principle |