Algebraic Combinatorics -- Splitting Orbits

Splitting Orbits

Splitting Orbits

In the case when _GX is a finite action, we can apply the sign map e to bar (G), the permutation group induced by G on X. Its kernel

bar (G)⁺:= { bar (g) Îbar (G) | e( bar (g)) =1 }

is either bar (G) itself or a subgroup of index 2, as is easy to see. Denoting its inverse image by

G⁺:= {g ÎG | e( bar (g))=1 },

we obtain a useful interpretation of the alternating sum of fixed point numbers:

Lemma: For any finite action _GX such that G not = G⁺, the number of orbits of G on X which split over G⁺ (i.e. which decompose into more than one -- and hence into two -- G⁺-orbits) is equal to
(1)/( | G | ) å_{g ÎG} e( bar (g)) | X_g | =(1)/( | bar (G) | ) å_{bar (g) Îbar (G)} e( bar (g)) | X_{bar (g)} | .

Proof: As G not =G⁺, and hence | G⁺ | = | G | /2, we have

(1)/( | G | ) å_{g ÎG} e( bar (g)) | X_g | =(2)/( | G | ) å_{g ÎG⁺} | X_g | - (1)/( | G | ) å_{g ÎG} | X_g |

=(1)/( | G⁺ | ) å_{g ÎG⁺} | X_g | - (1)/( | G | ) å_{g ÎG} | X_g | = | G⁺ \\X | - | G \\X | .

Each orbit of G on X is either a G⁺-orbit or it splits into two orbits of G⁺, since | G⁺ | = | G | /2. Hence | G⁺ \\X | - | G \\X | is the number of orbits which split over G⁺. Finally the stated identity is obtained by an application of the homomorphism theorem.

Corollary: In the case when G not = G⁺, the number of G-orbits on X which do not split over G⁺ is equal to
(1)/( | G | ) å_{g ÎG}(1- e( bar (g))) | X_g | .

Note what this means. If G acts on a finite set X in such a way that G not =G⁺, then we can group the orbits of G on X into a set of orbits which are also G⁺-orbits. In the figure we denote these orbits by the symbol Ä. The other G-orbits split into two G⁺-orbits, we indicate one of them by Å, the other one by OMINUS , and call the pair { Å, OMINUS } an enantiomeric pair of G-orbits. Hence the Lemma above gives us the number of enantiomeric pairs of orbits, while corollary yields the number of selfenantiomeric orbits of G on X. The elements x ÎX belonging to selfenantiomeric orbits are called achiral objects, while the others are called chiral. These notions of enantiomerism and chirality are taken from chemistry, where G is usually the symmetry group of the molecule while G⁺ is its subgroup consisting of the proper rotations. We call _GX a chiral action if and only if G not = G⁺.

Enantiomeric pairs and selfenantiomeric orbits

Using this notation we can now rephrase aboves lemma and corollary in the following way:

Corollary: If _GX is a finite chiral action, then the number of selfenantiomeric orbits of G on X is equal to
(1)/( | G | ) å_{g ÎG}(1- e( bar (g))) | X_g | = 2 | G \\X | - | G⁺ \\X | ,
while the number of enantiomeric pairs of orbits is
(1)/( | G | ) å_{g ÎG} e( bar (g)) | X_g | = | G⁺ \\X | - | G \\X | .

The sign of a cyclic permutation is easy to obtain from the equation and the homomorphism property of the sign, described in the corollary:

(i₁ ...i_r) ÎA_n iff r is odd.

But in fact we need not check the lengths of the cyclic factors of p since an easy calculation shows (exercise) that, in terms of the number c( p) of cyclic factors of p, we have

e( p)=(-1)^{n-c( p)}, if pÎS_n .

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

Splitting Orbits