In this previous post (about J.-C. Yoccoz 2011-2012 course at College de France), we reviewed the distinct points of view on origamis, and we started the discussion of the action of the automorphism group on the first absolute homology group of the origami. We quickly recall the main points of the previous post for the reader’s convenience.
Given an origami , we saw how to canonically describe it (through its monodromy group) by a finite group generated by two elements and , and a choice of subgroup such that , so that one has . Also, and act on the set of squares of by and . Furthermore, denoting by the normalizer of in , one has acting as , for .
Next, given a subfield , we decomposed and we reduced the problem of studying the action of on to understanding the action of on , and, to do so, we considered the set , and we proved that
as -module. In this way, since and , we were led to the study of the action of on . At this stage, we noticed that is the set of orbits of acting to the right on or equivalently the set of orbits of acting on by .
Nevertheless, we introduced the following notation: is the point of associated to , is the stabilizer of for the action of on , and is the order of the commutator .
Finally, if we denote by the smallest integer such that , we showed (in the very end of the previous post) the following lemma:
After this quick revision, we will enter (below the fold) the content of Yoccoz’s 2nd lecture (on January 18, 2012).
Analogously to the definition of , we denote by the smallest integer such that . Observe that, by construction, divides , and divides . For later use, in the next proposition we collect a series of elementary facts:
Proposition. It holds:
- is a cyclic group of order ;
- is a cyclic group of order ;
- the orbit of in has cardinality ;
Proof. The first two items follow from the definitions of and . The third item follows from the second one (as the size of an orbit is the order of the group divided by the size of the stabilizer and this last quantity is given by item 2.). The fourth item is a matter of counting correctly the involved objects: , so that the natural “reflex” is to count through and . However, we should be slightly careful here because the representation is not unique. To solve this (easy) problem we observe that two representations verify , so that we have an unique representation once the condition , and hence the fourth item follows. Finally, the fifth item is a direct consequence of the third and fourth items.
Following the “general plan” mentioned in the previous post, we will dedicate today’s discussion to the study of the action of on in the case , i.e.,
Standing Assumption. In the sequel, .
Recall that , so that it suffices to understand the action of on each piece.
For this reason, it is natural to introduce the character of the representation of (or ) on , and the character of the representation of on .
By the fifth item of the proposition above,
for . Here, is the characteristic function of .
Notation. We denote by the class determined by an element .
By the 2nd item of the proposition above,
Now, we consider a character of an irreducible representation (over ) of , and we denote by the multiplicity of in .
At this point, we observe that the action of on is “determined” by the knowledge of : indeed, morally speaking, the representation theory of finite groups (such as ) is “well-known” in the sense that irreducible characters are “determined”, at least when is a “classical” finite group (such as symmetric groups); therefore, the computation of permits (in principle) to calculate from character tables of finite groups.
In any event, recall that the multiplicity is given by
By using the formula of above, we can expand the right-hand side to get
where is the usual Kronecker’s delta.
Since is the character of a -representation, one has , and, hence,
It follows that
On the other hand, . By the second item of the proposition above, where , so that
Now, denoting by the degree of , we note that if are the eigenvalues of , then are the eigenvalues of . On the other hand, , so that and, hence,
where is the subspace of fixed vectors under the action of (a subspace of dimension , by definition).
By putting these identities together, we obtain the following statement:
Proposition. The multiplicity of in is
Next, we observe that is the sum of copies of the regular representation of . Since the regular representation contains (all) irreducible representations with a multiplicity equal to its degree, we have that the multiplicity of in is .
In particular, the multiplicity of in is . By the previous proposition, we can write
Observe that the right-hand side of this identity depends only on the class : indeed, , , implies and, a fortiori, , i.e., the elements and are conjugated by an element . Therefore, we can rewrite
Now we proceed to rewrite this formula by using induced representations. More precisely, we think of the irreducible representation of as a representation of of on a vector space , and we denote by the induced representation of on , where are representants of the classes and . Denote by the permutation of determined by the action of , i.e., . In this notation, we have, by definition, .
Observe that is precisely the length of the cycle of containing , and the restriction of to is conjugated to the action of on . Thus, we get the following statement:
Theorem. The multiplicity is given by the formula
where is the dimension of the subspace of fixed by .
Even though this statement is a simple consequence of our discussion so far, we decided to call it a “theorem” because it has some interesting consequences. In fact, the rest of today’s post will be dedicated to the proof of the following “corollary” to this theorem:
Corollary 1. For any origami , and for any , one has .
We begin the discussion of the proof of the previous statement by showing the following result:
Corollary 2. The multiplicity is always greater than or equal to the multiplicity of the trivial representation.
Proof. By the theorem, . Since and for every , we conclude
so that the argument is complete.
Let us consider now the case of regular origamis (in the sense of D. Zmiaikou), i.e., , . In this context, the formulas for the multiplicities simplify to and .
Proposition. In the case of regular origamis:
- If then ;
- If then .
Proof. If , we have that is a homomorphism and hence as is a commutator. Therefore, in this case.
If , we have that , where denotes the representation with character . We claim that . Otherwise, , and hence the action of factorizes into the action of a commutative group (as and generate ). However, this would force (since commutative groups only have -dimensional irreducible representations), a contradiction proving our claim. Now, we observe that the claim and the fact that imply that has at least two eigenvalues , so that .
Partly motivated by this discussion, we introduce the following definition:
Definition. An origami is quasi-regular if the multiplicity of the trivial representation is zero.
Remark. By the previous proposition, any regular origami is quasi-regular (so that the nomenclature is coherent).
The following proposition allows to characterize quasi-regular origamis in terms of the commutator :
Proposition 1. is quasi-regular the normal subgroup generated by is contained in .
Proof. We know that . Thus, for all cycle acts trivially on .
Proposition 2. If is not quasi-regular, then for all .
Proof. By Corollary 2 above, it suffices to check that . Since and is not quasi-regular (i.e., ), we have that the permutation (associated to the action of ) is not the identity. On the other hand, since is a commutator, the permutation is even. In particular, is not a transposition, and, consequently, .
Before proceeding further, let us give an example of a quasi-regular but not regular origami.
Example. Let be a finite Heisenberg group. Observe that is generated by the two elements
We choose . Note that because
Hence, this data defines an origami . Since , is not a regular origami. On the other hand, the normalizer of is
is a normal subgroup (coinciding with the centralizer of ). Note also that
By Proposition 1, is a quasi-regular origami. However, this is not the most interesting example of quasi-regular origami (from the representation theory point of view) because is an Abelian group. In any case, this quasi-regular example shows two interesting features: is normal and . As we are going to see below, this is a “general phenomenon” for quasi-regular origamis.
Proposition. is quasi-regular if and only if is normal and is Abelian generated by two elements (i.e., is either cyclic group or a product of two cyclic groups).
Proof. If is normal and is Abelian, then the commutator is mapped into the identity under the natural map , and, by Proposition 1, is quasi-regular.
Conversely, if is quasi-regular, then . Thus, is generated by two elements satisfying inside , i.e., is an Abelian group (generated by two elements). Since is a subgroup of the Abelian group , it follows that is normal and is Abelian.
At this point, we are ready to complete today’s discussion by proving Corollary 1. We start by noticing that, by Proposition 2, it suffices to consider the case of quasi-regular origamis. The idea of the proof in this case is similar to the one in the case of regular origamis (see the proposition before Proposition 1), despite the fact that we need the following little trick.
Let be a quasi-regular origami and denote by and the orders of and in . Define and .
Note that is a complete system of representatives of . Define , for , , so that .
so that is the product of , , .
In this language, the Theorem implies that
Proposition. If is a quasi-regular origami, then .
On the other hand, since is a commutator of elements of , we obtain
We have the following possibilities:
- for all : here either for all and, a fortiori, , or for some , so that (by the condition on the determinant);
- for at least two pairs of indices : in this situation, for and, a fortiori, .
In any event, we find that . This completes the proof of Corollary 1.