As it was mentioned in the previous post of this series, today we’re going to apply our discussions so far (about origamis, their automorphisms groups and the consequences to the study of the Kontsevich-Zorich cocycle) to two concrete cases: in the first part of this post we will study a family of quasi-regular origamis with automorphism group isomorphic to the symmetric group , and in the second part of this post we’ll study regular origamis with a rich representation theory (in the sense that real, complex and quaternionic representations appear at the same time inside the “interesting” part of homology of the origami).

**-A family of quasi-regular origamis-**

Fix an integer, and let be group consisting of all permutations of the set respecting its natural partition into even and odd numbers. Let and , so that

Define and , where is the set of even number in .

**Proposition.**

- and generate ;
- ;
- is the normalizer of and ;
- is a normal subgroup in of index .

**Proof.** Observe that , where is the set of odd integers in , so that the second item is clear. Also, it is clear that has order 2, i.e., is a subgroup of index 2 of , and hence is normal, so that the fourth item follows. For the third item, note that normalizes , and, as we just saw, is an index 2 normal subgroup of , so that is the normalizer of . Finally, for the first item, one notices that is the transposition . On the other hand, by induction on , one can check that is generated by the transpositions . Thus, by letting vary amongst the even numbers, we obtain that the elements generate . Hence, since and has index 2 in , we conclude that and generate , that is, and generate .

This proposition says that the data generates an origami with automorphism group . Moreover, as we saw in this previous post here, the fact that is normal and is Abelian is an *alternative* characterization of *quasi-regular origamis*. In other words, is a quasi-regular origami with automorphism group .

**Remark.** For the sake of comparison of with the case of *regular origamis* associated to , observe that the image of in the automorphism group of is (as, by definition, the image of is computed by looking at the action of on the even numbers in [in this case it is just the transposition ] and then renormalizing the even numbers in [by multiplication by ] in order to get a permutation of ), and thus is *not* the commutator of a pair of elements of (as such commutators are necessarily *even* permutations).

Let be a representation of . Since has index 2 in and , by this previous post here, the multiplicity of in the interesting part of the homology of is given by

On the other hand, because , we have that (after considering the action only on even numbers and renormalizing by multiplication by ) . Thus, the previous formula simplifies to

In order to render this formula more useful, we’ll briefly recall some aspects of the (very classical) representation theory of (along the lines of the classical book “Representation theory: a first course” of W. Fulton and J. Harris).

We start with the notion of Young diagrams. Given a list of integers such that, for some , for , we take boxes and we form a (Young) *diagram* by arranging them in a left-justified way in a manner that the first arrow consists of boxes, the second arrow consists of boxes, etc. For instance, we drew below the object/diagram corresponding to :

A natural operation on consists into taking its *dual* by considering the list . Geometrically, is the list associated to the diagram obtained from the diagram of after applying the reflection with respect to the anti-diagonal. For example, the dual of is depicted below:

It is well-known that conjugation classes of are described by the list of its cycles in non-increasing order, or, in other words, by Young diagrams associated to lists with . Furthermore, such Young diagrams allow to recover *all* irreducible representations of (see Fulton and Harris’ book), and, as a matter of fact, they are *all* real. Below we give some fairly easy examples of this correspondence between Young diagrams and irreducible representations of

**Example 1.** The list corresponds to the *trivial* representation .

**Example 2.** The dual of corresponds to the *alternating* (*signature*) representation .

**Example 3. **The list gives rise to the *standard* representation (such that is the usual permutation representation of ).

**Example 4. **The dual of is the representation obtained by taking the tensor product of the standard representation with the alternating representation .

**Remark.** In general, the representation associated to the dual of can be obtained by taking the tensor product of the representation corresponding to with the signature representation . In particular, Examples 2 and 4 above are concrete incarnations of this general fact.

The dimension of the irreducible representation can be computed with the aid of the *hook-length formula*:

where is the hook length of the box of number , that is, the number of boxes to the right in the same row of plus the number of boxes below in the same column of plus one (for the box itself).

In the concrete example of the list , we drew below the corresponding Young diagram and we filled each box with the number giving its hook length.

From this picture we deduce that .

Coming back to our concrete example of quasi-regular origami , we had that

where was a *transposition*. In particular, it follows that

.

In this formula, we already know the dimension of (thanks to the hook-formula), so that we need to know to determine . This is done with the aid of *Frobenius formula* (that we specialize to the case of a transposition ):

**Frobenius formula.** where .

Observe that . Since (by the hook-length formula) and , we have the following corollary of Frobenius formula:

**Corollary.** .

**Example 1′.** For , (this is coherent with the fact that is quasi-regular and hence the trivial representation has zero multiplicity in ).

**Example 2′.** By the corollary, for , we have .

**Example 5.** For and

we have , and thus . Also, by the corollary, for , we have .

**Example 6.** For

we have , and thus . Also, by the corollary, for , we have .

**Remark.** The examples above give the multiplicities of *all* irreducible representations for . For , it remains

**Remark.** Still concerning regular origamis associated to symmetric groups, we observe a theorem of O. Ore says that every element of is the commutator of two elements in . However, it is not obvious that, for a given , one can choose and with *and* generates or .

**-A family of regular origamis-**

Let where is prime. Note that , and

generate whenever .

**Notations.** , , is a generator of (a cyclic group of order ), and is the subgroup of consisting of elements with *norm* 1. Here, we recall that is a degree 2 extension of : this can be seen from the so-called Frobenius automorphism . Also, using the Frobenius automorphism, we can define a norm . In this language, . Moreover, we fix a generator of . Finally, we think of as a subset of by looking at the action of in a fixed basis .

**1. Conjugation classes in **

The following table presents the information we will need about the conjugation classes of , namely, it gives representatives for each “type” of class (in the first collumn), the number of elements on each class of a given “type” (in the second collumn), and the number of classes of a given “type” (in the third collumn).

representative | number of elements in a class | number of classes |

From this table, we see that the total number of conjugation classes is .

** 2. Irreducible representations**

Below we list the irreducible representations of .

**a.** the trivial representation .

**b.** the standard representation coming from the action of on (that is, we have a permutation representation of that we write as ). The character is given in the following table

**c. **let be a character with (there are possible choices of ), and let

be the usual Borel subgroup. Define the character and consider the induced representation of . One has , and the character is given in the following table

**d. **in the case of the character with , the induced representation is *reducible*: indeed, . Observe that implies that . The characters are given in the following tables.

For ,

and, for ,

**e.** let be a character with (there are possible choices of ). Using it is possible to construct a representation whose character is given by the following table

**f.** let be the character with . In this case, is reducible: indeed, . Observe that implies that . The characters are given in the following tables.

For ,

and, for ,

In resume, the complete list of irreducible representations is:

- (there are of them)
- (there are of them)

From the previous tables of characters, one sees that

- and are
*real* - and take real values
- are
*complex*when and*real*when (because in this case the characters take only real values, so that are either real or quaternionic, and the quaternionic possibility is easily ruled out as is*odd*and quaternionic representations always have*even*dimension) - are
*complex*when , and*real*or*quaternionic*when .

In order to distinguish whether are real or quaternionic (when ), we apply the following general criterion (based on the Frobenius-Schur indicator):

**Theorem.** Let be a character of an irreducible representation of a finite group . Then,

Applying this criterion (with the aid of the previous character tables), we can check that

- are
*quaternionic*when - for (with ), is
*real*when is even, and is*quaternionic*when is odd - for (with ), is
*real*when is even, and is*quaternionic*when is odd

In a nutshell, our discussions so far can be resumed as follows:

- for , there are
*real*representations,*quaternionic*representations, and*complex*representations. - for , there are
*real*representations,*quaternionic*representations, and*complex*representations.

This concludes our quick review of the representation theory of . Now, we pass to the study of the regular origami.

Recall that we have chosen

with . Their commutator is

and hence its trace is . The nature of the eigenvalues of are described by the quantity

It follows that

- if (this can only happen when as must be a square), then is conjugated to (and the order of is )
- if is a square, is conjugated to , and the order of satisfies and
- if is not a square, is conjugated to , and the order of satisfies and

As we saw in this post here, the genus of the regular origami is

Also, note that

and

Finally, note that the order of can be computed by the following recurrence equation

on (with , ). (This recurrence formula follows from the fact that for ).

For the first values of , one gets the following tables (where below is the Legendre symbol).

For :

For :

For :

For :

In the case of and (*parabolic*), one can use the character tables above (and the fact that, as has order , ) to deduce that

and hence

Of course, while this give the multiplicities of all irreducible representations in the interesting part of the homology of the regular origami associated to when and ), there are several interesting questions left open here: for instance, what are the signatures of the Hermitian forms associated to the quaternionic representations , and (when and for odd)?

It is likely that some of these questions will be dealt with in the forthcoming paper (by C.M., J.-C. Yoccoz and D. Zmiaikou), but J.-C. Yoccoz decided to stop here the comments on this project, so that the first “part” of the course ends now.

Next time, we will discuss simplicity of Lyapunov exponents of the Kontsevich-Zorich cocycle (along the lines of the works of A. Avila and M. Viana) in “general”, and then later we will restrict this discussion to the setting of square-tiled surfaces (by following the lines of a work in progress by C.M., M. Moeller and J.-C. Yoccoz).

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