Spectral Decomposition of square-tiled Surfaces
Luc Hillairet
1
November 10, 2005
Abstract
We consider N copies of a square S0 and define selfadjoint extensions of
the Euclidean laplacian acting on (C0∞ (S0 ))N by choosing some boundary
conditions that are parametrized by two unitary matrices H and V acting
on CN . Denoting by Sp(∆N , H, V) the spectrum of such an operator we derive
conditions on H and V so that the following spectral decomposition holds :
Sp(∆N , H, V) =
[
Sp(∆Ni , Hi , Vi ) with
k
X
Ni = N.
1
1≤i≤k
If H and V are permutation matrices this gives a spectral decomposition
of the spectrum of the square-tiled surface defined by the corresponding
permutations. We apply this to derive examples related to isospectrality
and to high multiplicity.
1
Luc Hillairet, UMPA ENS-Lyon, CNRS UMR 5569
46 allée d’Italie 69364 Lyon Cedex 7
Tel 04 72 72 84 18, e-mail: lhillair@umpa.ens-lyon.fr
1
Introduction
A square-tiled surface is a surface that is constituted by N identical squares
that are glued together so that the left (resp. top) side of each square is identified with the right (resp. bottom) side of one of the squares. These surfaces
are very particular examples of translation surfaces ; For instance, the beautiful result of Gutkin and Judge (see [13]) asserts that a translation surface
is tiled by parallelograms if and only if its Veech group is commensurable
with SL2 (Z). As for any translation surface, a square tiled surface can be
equipped with a singular Riemannian metric that is flat everywhere except
at a finite number of points where the metric has a cone-type singularity of
angle 2kπ. The main aim of this paper is the study of the spectrum of the
Laplacian that is associated with this metric. There are several motivations
for such a study : from a general point of view, there are many evidences
for intimate relations between the dynamical properties of the geodesic flow
on a Riemannian surface and the spectral properties of its laplacian. It is
thus interesting to see how this “quantum-classical” correspondence manifests itself in the rich dynamical setting of translation surfaces. For instance,
in [2] it is proved that the spectral statistics of a Veech translation surface
present in an intermediate behavior between that of an integrable system
and the one that is expected for a chaotic system. It thus gives a quantum
analogue to the fact that the dynamics on such a surface is usually referred
to as pseudo-integrable. More generally, since the dynamics on a translation
surface have been greatly studied, it would be interesting to take advantage
of this knowledge from the spectral point of view and to see what kind of
geometrical information is encoded in the spectrum. These motivations hold
for any translation surface and there are mainly two reasons why we will
restrict ourselves to square-tiled surfaces. The first one is because understanding square-tiled surfaces is an important step in the more general study
of the dynamical properties of translation surfaces (see [18, 14, 8]). The
second one is because, as in the classical case, when the surface is actually
square-tiled, we expect that the spectrum has particular features and that
particular techniques can be used to study it. For instance, since a squaretiled surface can be viewed as a branched covering of the torus, we will see
that our study can be related to the usual spectral theory of Riemannian
coverings (see [25, 1, 11]).
Content and Results
The definition of square-tiled surfaces, their Euclidean structure and their
spectrum will be recalled in the first section. In order to present the results,
2
let us simply say here that a square-tiled surface is given by N identical
squares S0 and two permutations h and v in SN that represent the gluing
pattern in the horizontal and vertical directions.
A simple argument (see remark 8) implies that the spectrum of M (denoted by Sp(M )) contains Sp(T ) : the spectrum of the underlying torus. The
first natural question consists in identifying the complement Sp(M )\Sp(T ) as
the spectrum of a self-adjoint operator. To answer it, we will introduce a class
N
ON of self-adjoint operators defined on [L2 (S0 )] . These will be extensions
of the Euclidian laplacian defined on [C0∞ (S0 )]N with particular boundary
conditions that will be characterized by two N × N unitary matrices H and
V. Such an operator will thus be described by (∆N , H, V ) and the Euclidean
laplacian on M is obtained when H and V are the permutation matrices
corresponding to h and v (see prop. 2 p.11). The following theorem that
gives a decomposition of the spectrum of a square-tiled surface is in fact a
corollary of the more general theorem (Thm. 5 p.13) that we prove in section
3. (The notion of sum of spectra is defined in def. 4 p.13.)
Theorem 1
Let M be the square-tiled surface defined by h and v, G the subgroup of SN
generated by h and v and ρ the representation of G in UN (C) by permutation
matrices. The following spectral decomposition then holds :
Sp(M ) =
k
X
Sp(∆Ni , Hi , Vi ),
(1)
i=1
where, for each i, the representation ρi of G in UNi (C) defined by ρi (h) =
Hi , ρi (v) = Vi is irreducible and
ρ = ρ1 ⊕ ρ2 ⊕ · · · ρk .
Actually, the central theorem (Thm. 5) should not be considered new.
Indeed, it is rather the specialization to our setting of the Peter-Weyl type
theorem (see [25]) occuring each time there exists a group action commuting
with an operator. However, we should also remark here that the natural
action of G on M does not commute with the laplacian (see prop. 1 p.8) so
that, at first sight, it is not completely clear that the usual theorem can be
used (although a trick allows us to overcome this difficulty -see[27] and section
3.2). The setting of square-tiled surfaces should also be compared with the
setting of [11], in particular, a square can be thought as a degenerate cross !
We will give a proof of theorem 5 which does not refer directly to the general
theory of Riemannian coverings but rather to the transplantations that were
3
introduced by Bérard and Buser (see [1, 5]). The main advantage of this
method is to give directly a concrete realization of the reduced operators
in terms of laplacians with boundary conditions (namely in the class ON ).
It is also worth remarking that since the definition of an operator in the
class ON involves some boundary conditions, the transplantation proposition
(see prop. 3) will become almost tautological. In [15], the relation between
boundary conditions and coverings is also exploited to derive new examples
of isospectral settings.
In the rest of the paper, we will use the representation theory of symmetric
groups to construct square-tiled surfaces whose spectrum has some special
behaviour. Indeed, since our ultimate goal is to understand the spectrum
of square-tiled surfaces, it is quite interesting to first understand what kind
of particularities this spectrum may have. It will also come up that squaretiled surfaces are a nice setting to construct interesting examples from the
spectral point of view. This is essentially because we can actually choose any
finite group G generated by two elements and construct square-tiled surfaces
associated with G.
In section 4 we will derive from 1 applications that are related to the
theory of transplantation. A natural question is that of the existence of
self-transplantations on such surfaces. We refer to the definition 5 of selftransplantations only emphasizing here that these operators commute with
∆. Using theorem 5 we relate this problem to a group-theoretic question and
these so-called self-transplantations then arise as Hecke operators (see [21]).
In particular we prove the following :
Theorem 2
Let M be a square-tiled surface given by the permutations h and v, and G the
subgroup of SN generated by h and v. The vector space of self-transplantations
has dimension k where k is equivalently given by :
- k is the number of orbits of the diagonal action of G on { 1 · · · N }2 .
P 2
- k=
mλ , where the sum is over all the irreducible representations of
G and mλ is the multiplicity of the corresponding irreducible representation in the representation of G in GLN (C) by permutation matrices.
As a corollary we will construct surfaces having arbitrarily many self-transplantations.
(see the end of section 4)
The second application is related with isospectrality.
Theorem 3
Let M6 , M3 , T1 , T2 be the square-tiled surfaces defined page 24. The following identity holds :
Sp(M6 ) + 2 Sp(T1 ) = 2 Sp(M3 ) + Sp(T2 ).
4
This theorem thus gives an example of non-connected isospectral surfaces (actually families of surfaces since the shape of base parallelogram is
irrelevent). Isospectral square-tiled surfaces are already known since the
techniques of [5] can be used in this setting (see also [9]). However, our
example is based upon the study of S3 and thus only implies relatively simple group-theoretic arguments. It is also interesting to relate this theorem
to Kneser’s theorem (see [17, 19]) since it tells us that there exists a linear
relation between the spectra of of some two-dimensional torus coverings. In
section 5.1, we will give other examples where such a spectral identity holds.
In the last application we will consider the multiplicity of the spectrum.
We will first define the notion of structural multiplicity of an operator (see
def. 6). Roughly speaking, an operator will have d for structural multiplicity
if a large part of its spectrum has multiplicity d. We will then construct a
sequence Mn of square-tiled surfaces such that the structural multiplicity of
the spectrum grows to infinity. Actually we will compare the growth rate of
the structural multiplicity and that of the genus of Mn .
Theorem 4
There exists a sequence of square-tiled surfaces Mn such that the genus gn
goes to infinity and such that, for any ε, the following inequality holds for n
large enough :
1
−ε
s-m(Mn ) ≥ gn2
where s-m denotes the structural multiplicity.
√
It is quite interesting to note that the bound g is roughly the bound
that comes up when comparing the multiplicity of the first eigenvalue with
the chromatic number (see [23]). Another interesting point of the preceding proposition is the dependence with respect to the genus. Indeed, it is
well-known from the classical theory that translation surfaces are organized
in strata such that most of the known interesting dynamical features -such
as the Siegel-Veech constant for example - depend only on each (connected
component of the) stratum. However, the way such a constant changes when
the stratum does is an interesting challenge (see [7]). The preceding theorem thus gives us some information on -what we believe is- the quantum
counterpart of the latter problem.
1
Square-tiled surfaces
In this section we will give the definition of a square-tiled surface as well as
the basic geometric facts we will need about it. Most of what we will say in
5
this section is well-known, and much more is known about the geometry and
the dynamics of square-tiled surfaces (see [28] for example).
We will denote by S0 the square [0, 1]x × [0, 1]y . A square-tiled surface will
be a surface consisting of N squares (Si )1≤i≤N each one isometric to S0 ; we
will denote by (xi , yi ) the coordinates on Si . To define a square-tiled surface,
we need to identify pairwise the horizontal (resp : vertical sides) of the Si0 s.
Definition 1 (Square-tiled surface)
Consider N squares (Si )1≤i≤N , and two permutations h and v of SN then
the square-tiled surface associated with (h, v) is defined by
M=
N
[
Si / ∼,
i=1
with the following identification :
½
(xi , 1) ∼ (xv(i) , 0)
(1, yi ) ∼ (0, yh(i) )
We will denote by G the subgroup of SN generated by h and v.
-Remarks1. This way of representing a square-tiled surface was first remarked by
Kontsevich and Zorich
2. We will denote by p1 · · · pk the points on M corresponding to the vertices of the tiles. We will let M0 = M \{p1 , · · · , pk }.
3. With this definition, for any i, h(i) denotes the tile located on the right
of the i-th tile and v(i) the tile above. The permutations h and v thus
describe the gluing pattern in the horizontal (resp. vertical) direction
(see fig. 1).
v(i)
i
h(i)
Figure 1: Gluing pattern
6
4. Actually, the fact that S0 is a square is irrelevent for everything we will
say. Everything works for parallelogram-tiled surfaces.
Most of the geometric properties of the square-tiled surface M can be
directly deduced from the knowledge of h and v. For instance, M is connected
if and only if G acts transitively on {1 · · · N }.
The Euclidean metric on each square defines an Euclidean metric on M0 .
In the neighbourhood of pi , the surface M is obtained by gluing together
a certain number of tiles so that the metric is locally that of an Euclidean
cone. The following lemma gives the way of recovering the angles of the
conical singularities from h and v.
Lemma 1 Let c be the commutator of h and v (c = h−1 v −1 hv). In the
decomposition of c as a product of cycles, each cycle of length l corresponds
to one conical singularities of angle 2lπ.
Proof : starting from tile i, the tile c(i) is obtained by going up, then to
the right, then down, then to the left. If i is in the support of a l-cycle, then
l sequences up-right-down-left must be done before getting back to the i-th
tile.
¤
Using this lemma we can easily compute the genus of the surface.
Corollary 1 The genus of M is given by :
g=
N −k+2
,
2
(2)
where k is the cardinal of M \M0 .
Proof : we can compute the Euler characteristic using the tiling ; there
is one vertex per cycle in c.
¤
-RemarkThe artificial conical singularities (i.e those corresponding to fixed points of
c) must be counted in k.
The main object we will be interested in is the spectrum of the laplacian
that is associated with the Euclidean metric. Since the metric has conical
singularities, we must be slightly more precise. The vector fields ∂xi and ∂yi
defined on Si can be smoothly prolongated to vector fields ∂x and ∂y on M0 .
We define the quadratic form Q on C0∞ (M0 ) by :
Z
Q(u) =
|∂x u|2 + |∂y u|2 dxdy,
M
7
where dxdy is the Euclidean area. This quadratic form is positive and we
denote by ∆ the self-adjoint operator given by the Friedrichs procedure (see
[20] p. 176). By construction, the domain of ∆ is characterized by :
dom(∆) = {u ∈ L2 (M ) |∆∗ u ∈ L2 (M ) and Q(u) < ∞},
where ∆∗ denotes the Euclidean laplacian taken in the distributional sense
in C0∞ (M0 ). The only result we need to know about the spectral theory of
this operator is that the spectrum of ∆ is still purely ponctual (this derives
for instance from the Rellich-type theorem proved in [6]). The eigenvalues
will be denoted by 0 = λ0 < λ1 ≤ · · · ≤ λn ≤ · · · .
-RemarkLet ψn be the n-th eigenfunction of the torus T0 obtained by identifying
the parallel sides of S0 , and let φ be defined on M by copying ψn in each
Si . A straightforward computation based on Green’s formula implies that φ
belongs to dom(∆) so that ∆φ = µn φ where µn is the n-th eigenvalue of the
torus. Eventually, we have the following inclusion :
Sp(T0 ) ⊂ Sp(M ),
and the main aim of this paper is to understand the complement of Sp(T0 )
in Sp(M ), that we will call reduced spectrum.
The preceding fact is not surprising since we can make the group G act on
L (M ) in such a way that the corresponding invariant subspace is identified
with L2 (T ). The action is defined in the following way : we first identify
N
L2 (M ) with [L2 (S0 )] so that an element of L2 (M ) can be written
2
~u =
N
X
ui e~i , ui ∈ L2 (S0 ).
1
For any g in G we can now define an isometry Tg of L2 (M ) by :
Tg (u) =
N
X
ui~eg(i) .
1
However, the following proposition implies that the situation is not as
simple as it may seem.
Proposition 1 The following assertions are equivalent.
1. The operator Tg commutes with ∆ for all g in G.
8
2. The surface M is a torus.
Proof : if M is a torus, then ∂x , ∂y and ∆ are the usual operators, and
Th = exp (i∂x ), Tv = exp (i∂y ), so that any Tg commutes with ∆. Conversely,
if M isn’t a torus then h and v do not commute. Thus, there exists i such that
v(h(i)) 6= h(v(i)). Consider now a function u in C0∞ (M0 ) which is supported
around the top edge of the i-th square, letting Th act on this function we can
see that the domain of ∆ is not invariant under Th so that Th and ∆ don’t
commute (See fig. 2.)
¤
supp(u)
1
3 2
Th
1
3
2
h = (12)(3)
v = (1)(23)
Figure 2: Action of Th
-RemarkThis commutation problem is the reason why, in remark 1, we had to check
by hand (using Green’s formula) that φ belonged to dom(∆).
We now want to understand the reduced spectrum. For instance we would
like to know if it can be naturally identified with the spectrum of some selfadjoint operator. If we denote by NM (E) (resp NT (E)) the counting function
for the eigenvalues of M (resp. for T ), then Weyl’s law states that
NM (E) ∼ cN E and NT (E) ∼ cE,
so that, heuristically, if the complement is the spectrum of a self-ajoint operator, this operator should be defined on a surface tiled by N − 1 squares and
the first thing to do is to find good candidates for the reduced spectrum.
2
Self-adjoint extensions of the laplacian on
N squares
The aim of this section is to construct a class ON of operators such that
9
- the Euclidean laplacian on any square-tiled surface with N squares will
belong to ON ,
- the reduced spectrum will be the spectrum of an element of ON −1 .
Actually, only the first point will be proved in this section, the second one
will derive from theorem 5.
It is quite natural in this setting to expect the reduced operator as being
defined by some kind of boundary condition. Indeed, this is what happens for
instance when studying a domain with a symmetry (where one gets Dirichlet
and Neumann boundary conditions- see also [15]) or when studying the spectrum of a non-ramified torus covering (where one gets Floquet operators).
We denote by H 1 (S0 ) the usual Sobolev space i.e the set of v 0 s in L2 (S0 )
such that
Z
|∂x v|2 + |∂y v|2 dxdy ≤ ∞.
S0
The traces γ0 v l (resp. γ0 v r , γ0 v u , γ0 v d ) on the left edge (resp. right, upper,
1
lower) are well defined and continuous from H 1 (S0 ) in H 2 (0, 1) (see [12] for
example).
Consider H and V two elements of UN (C) the set of unitary matrices of
size N. For convenience, we will ask that the group generated by H and V
in UN (C) is finite even if this restriction is unnecessary for many statements.
N
We denote by D(H, V ) the set of all ~v in [H 1 (S0 )] satisfying the following
boundary condition (subscripts l and r are for left and right, u and d for up
and down) :
−
→
−−→
−
→
−−→
γ−
,
γ−
(3)
0 v l = H γ0 v r
0 v d = V γ0 v u
N
On [H 1 (S0 )]
we also define the quadratic form Q by :
QN (~u) =
N Z
X
1
|∂x ui |2 + |∂y ui |2 dxdy.
S0
The following definition provides us with a class ON of operators.
Definition 2 (Class ON ) Choose H and V in UN (C) such that the group
generated by H and V is finite. The quadratic form QN considered on
D(H, V ) defines a unique self-adjoint operator that is denoted by (∆N , H, V ).
We denote by ON the set of all the operators that can be obtained this way.
-RemarkWe could have chosen H and V to be only invertible. This would have defined a greater set of operators. The reason for choosing H and V unitary is
10
N
the following. Using Green’s formula, an element ~u ∈ [H 1 (S0 )] such that
each ui is smooth up to the boundary will be in the domain of the operator corresponding to H and V if and only if the normal derivatives on the
edges satisfy some linear relation. When H and V are unitary, this relation
is obtained by changing in (3) the traces of ~u by the traces of the normal
derivatives (in the general case, H must also be replaced by (H ∗ )−1 and V
by (V ∗ )−1 ).
-ExampleIf N = 1, H (resp. V ) is the multiplication by eiα (resp. eiβ ) with α and β
in πQ. To study the corresponding operator on the square, we can separate
the variables and we are led to study two Floquet operators on (0, 1). These
are exactly the operators that come up when computing the spectrum of a
torus that covers a smaller one.
The symmetric group SN can be represented in UN (C) by asking that
σ.~ei = ~eσ(i) . Let h and v be two elements of SN , we denote by H and V their
images by this representation, the following proposition then holds.
Proposition 2 Let h and v be two permutations of SN and H and V the
corresponding permutation matrices then the associated element of ON is the
Euclidean Laplacian on the square-tiled surface M defined by h and v.
Proof : let ∆1 be the element of ON defined by H and V . By definition,
∆ is the unique self-adjoint extension of the Euclidean laplacian defined on
C0∞ (M0 ) whose domain is contained in H 1 (M ). Using Green’s formula and
the definition of D(H, V ), ∆1 is shown to satisfy both properties.
¤
For general H and V, giving a geometric meaning to the corresponding
operator in ON involves flat bundles over a punctured torus (see sec. 3.2). A
perhaps more concrete interpretation of the boundary condition can be given
in terms of the wave equation. Consider a wave travelling in the i-th square,
when it hits some side of the square, a part of this wave is then instantaneously transmitted to the other squares and a part of it is reflected. The
reflection and transmission coefficients will be computed from the matrices
H and V. If the wave hits a corner then a diffracted wave has to be added.
In the sequel, some particular types of boundary conditions will appear.
We describe them here
Definition 3
- We will denote by SN the subset of ON consisting of the operators for
which H and V are permutations matrices.
11
- We will denote by FN the operators of ON such that H and V satisfy
the following condition
∃h, v ∈ Sn , ∃(ωi ), (ζi ) ∈ UN |
H~ei = ωi~eh(i) , V ~ei = ζi~ev(i) .
These operators will be called of generalized Floquet type.
-RemarkIn the definition of the Floquet-type operator, the fact that the group generated by H and V is finite imposes conditions on the sets (ωi ), (ζi ). In
particular the ω’s and ζ’s must be roots of the unity.
-ExampleSuppose we can find H1 , V1 (resp. H2 , V2 ) in UN1 (resp. UN2 ) so that H
and V can be written :





H = 

H1
0

0 
 , V = 


H2
V1
0
0 


V2
then L2 (S0 )N can be decomposed in L2 (S0 )N1 ⊕ L2 (S0 )N2 . By construction
we will have D(H, V ) = D(H1 , V1 ) ⊕ D(H2 , V2 ), and the quadratic form Q̃N
will decompose orthogonally in Q̃N1 + Q̃N2 . Eventually we will have :
(∆N , H, V ) = (∆N1 , H1 , V1 ) ⊕ (∆N2 , H2 , V2 ).
The spectral decomposition then follows :
Sp(∆N , H, V) = Sp(∆N1 , H1 , V1 ) ∪ Sp(∆N2 , H2 , V2 ).
3
(4)
Spectral Decomposition in ON
The main aim of this section is to find conditions on H and V so that
a spectral decomposition like eq. (4) can hold. We first give a definition
generalizing the equality (4).
12
Definition 4 (Spectral sum)
Let (Ai )0≤i≤n be a collection of operators with pure point spectrum and (ai )1≤i≤n
a collection of integers, we will write
Sp(A0 ) =
n
X
ai Sp(Ai ),
(5)
i=1
if the following holds
∀µ ∈ C, m0 (µ) =
n
X
ai mi (µ),
1
where mi (µ) denotes the multiplicity of µ in the spectrum of Ai .
-Remarks- This definition allows us to do algebraic sums of spectra. We remark
however that the equality Sp(A) = Sp(B) − Sp(C) can hold only if
Sp(C) ⊂ Sp(B).
- With this definition, equation (4) can be rewritten :
Sp(∆N , H, V) = Sp(∆N1 , H1 , V1 ) + Sp(∆N2 , H2 , V2 ).
(6)
We now give a more group-theoretic flavour to the class ON . Let G be
a finite group generated by two elements h and v, and let ρ be a unitary
representation of G in CN . We define ∆ρ to be the element of ON such that
H = ρ(h) and V = ρ(v).
-Remarks1. The notation is slightly ambiguous since to define ∆ρ we need not only
ρ but also the generators h and v.
2. Any operator in ON can be written ∆ρ . Indeed, we just have to let G
be the group generated by H and V and since H and V are in UN (C)
we have already a unitary representation of G.
The theorem we will prove is the following.
Theorem 5
Let G be a finite group generated by two elements h and v. Let (ρi )0≤i≤n be
unitary
Prepresentations of G and (χi )0≤i≤n their corresponding characters. If
χ0 = ni=1 ai χi for some collection (ai ) of integers then the following spectral
decomposition holds :
Sp(∆ρ0 ) =
n
X
i=1
13
ai Sp(∆ρi ).
The proof we will give relies on the transplantation theory that was introduced by Bérard and Buser [1, 5] to explain the original result of Sunada
[25]. We will also sketch a second proof showing that we can actually use
the general theory of Riemannian coverings. The latter point is a priori not
obvious since we have seen that, for a square-tiled surface, the action of G
isn’t commuting with ∆.
Before giving the proof, we apply this theorem to the square-tiled surfaces. Let M be a square-tiled surface defined by h and v. In this case G is
a subgroup of SN , and ∆M = ∆ρ where ρ is the representation of G by permutation matrices in UN (C). This representation is never irreducible since
P
the vector f~1 = N
ei is ρ(G)−invariant. We let F1 = f~1⊥ . The following
i=1 ~
spectral decomposition then holds :
Sp(M ) = Sp(∆1 , 1, 1) + Sp(∆N −1 , H1 , V1 ),
(7)
where H1 = H|F1 and V1 = V|F1 , and Sp(∆1 , 1, 1) is the spectrum of the torus.
In particular we have proved the following corollary.
Corollary 2 Let M be a square-tiled surface with N squares, then there
exists A in ON −1 such that :
Sp(∆M ) = Sp(T ) + Sp(A).
-RemarkWhen the restriction of ρ to F1 is irreducible, equation (7) cannot be refined.
In section 4 we will look for examples where a finer decomposition holds.
3.1
The transplantation method
We recall here the general philosophy of transplantations (see [5] or [1]). Consider two domains that are built of N isometric elementary bricks. Transplanting a function of the first domain into the second consists in restricting
the function to the elementary bricks and then in recombining the parts
thus obtained using some matrix A in GLN (C). This operation clearly commutes formally with any differential operator. The transplantation method
becomes fruitful in settings where this commutation is not only formal. It
is the difficult part of the usual theory to find such settings and it requires
quite sophisticated group theoretic arguments (See [5, 11]). From this general point of view it is natural to use transplantations to study the class ON .
Actually, since the operators in On are defined using boundary conditions,
the commutation property will be even easier to establish.
14
Proposition 3 Consider H1 , H2 , V1 , V2 elements of UN (C) and denote by
∆i the corresponding operators in ON . Let A be in GLN (C), such that
AH1 = H2 A,
and
AV1 = V2 A,
(8)
then we have :
A∆1 = ∆2 A.
(9)
Proof : since ∆∗ commutes with A, the only problem in proving (9) is to
check that the domains are consistent. We recall that
½
~v ∈ D(Hi , Vi ),
~v ∈ dom(∆i ) ⇔
.
(10)
∃M, ∀w
~ ∈ D(Hi , Vi ), |h∇~v |∇wi|
~ ≤ M kwk,
~
Equation (8) implies that AD(H1 , V1 ) = D(H2 , V2 ). Moreover, when A satisfies (8), so does A∗ −1 so that we also have A∗ −1 D(H1 , V1 ) = D(H2 , V2 ).
Let v~1 be in dom(∆1 ) then v~2 = Av~1 is in D(H2 , V2 ) and for any w~2 in
dom(∆2 ) there exists w~1 in D(H1 , V1 ) such that w~2 = A∗ −1 w~1 . The following
computation then holds :
h∇v~2 |∇w~2 i = h∇Av~1 |∇A∗ −1 w~1 i
= h∇v~1 |∇w~1 i.
Since v~1 ∈ dom(∆1 ), this last expression is bounded by M kw~1 k so that we
have eventually proved :
∀w~2 ∈ D(H2 , V2 ), |h∇v~2 |∇w~2 i| ≤ M̃ kw~2 k,
with M̃ = M kA∗ −1 k. This ends the proof.
¤
As usual in transplantation theory, this proposition is used to prove the
following corollary.
Corollary 3 Let ∆1 and ∆2 be two operators of ON such that there exists
A satisfying condition (8) then the following holds :
Sp(∆1 ) = Sp(∆2 ),
(counting multiplicities). In particular, if ρ1 and ρ2 are isomorphic representations of G, then
Sp(∆ρ1 ) = Sp(∆ρ2 ).
Proof : the former proposition implies that A conjugates ∆1 and ∆2 .
-Remarks15
¤
1. The transplantation A actually gives an explicit relation between eigenfunctions of ∆1 and ∆2 .
2. As we have already remarked, the preceding proposition is really simple
because we consider the whole class ON ; if we want to study isospectrality for square-tiled surfaces (i.e. in Sn ) the work is not finished yet,
since we have to know when H, V, AHA−1 , AV A−1 can be simultaneously permutations. However, there are examples where this condition
is satisfied (See [5, 9]).
Proof of theorem 5. We consider first the case when the ai ’s are positive. We
construct a finite-dimensional representation ρ of G by letting ρ(h) (resp.
ρ(v)) be the block diagonal matrix such that there are exactly ai blocks
ρi (h) (resp. ρi (v)) for 1 ≤ i ≤ n. It is obvious that the characters of ρ and ρ0
are equal so that the representation ρ and ρ0 are equivalent. The conclusion
of theorem 5 then follows from the preceding corollary and from example
p. 12. When the ai ’s are not necessarily positive, we begin by ordering them
so that the first k are
Pnegative and the
P last n − k are
Pnpositive and we rewrite
the condition χ0 =
ai χi as χ0 + ki=1 ai χi =
k+1 ai χi . The conclusion
then follows.
¤
3.2
The common cover method
We have already remarked that on a typical square-tiled surface, the action of
G isn’t commuting with the Euclidean laplacian. In this section we will show
that this difficulty can be overcome by passing to a covering. The operators
∆ρ then correspond to the laplacian on flat bundles over a punctured torus
so that theorem 5 will follow a standard construction.
Let G be a finite group generated by two elements h and v, we define
the square-tiled surface MG in the following way : for each element g in G
we take a square that we label by g. We now glue these squares so that the
square on the right of g is hg and that above g is vg. The corresponding
permutation are left-multiplication by h (resp. by v) in the horizontal (resp.
vertical) direction. This construction is closely related to the Cayley graph
of G and is already explained in [11].
On this surface, there are two natural actions of G on L2 (MG ). The first
one is given by the operators Tg . It corresponds to multiplication on the left,
and it does not commute with ∆ in general. The second one corresponds to
multiplication on the right. It is defined by
Ã
!
X
X
ug0 ~eg0 g−1 .
(11)
Sg
ug0 ~eg0 =
g0
g0
16
The domain of ∆ is clearly invariant under Sg so that Sg commutes with
∆. We are thus led to the general theory of an operator commuting with
˜ ρ for each
some group action. In this setting we can define an operator ∆
representation of G (see [25]). Using dim(ρ) squares as a fundamental domain
˜ ρ and ∆ρ . Theorem 5
for the action of G on [L2 (MG )] allows us to identify ∆
then follows from the general Peter-Weyl theorem of this theory (see [25]).
-RemarkAs a branched covering of the torus, a square-tiled surface is not necessarily
normal. However a non-normal covering can always be covered by a normal
one (see [27]). This is the construction we have sketched above.
Theorem 5 has just replaced our original question of decomposing the
spectrum of square-tiled surfaces by the usual question in representation
theory : “When is a given representation irreducible ?” The applications we
will present will thus mainly consist in showing that known facts of representation theory gives interesting properties of square-tiled surface. Before
this, we first translate some basic constructions from group theory into our
setting.
-RemarkThe formulation of theorem 5 relates our spectral problem to questions on
groups that are generated by two elements i.e. to quotients of F2 , the free
group generated by two elements. Putting things into this perspective thus
rises new questions such as adapting results from the general theory of Riemannian coverings ([4] for instance) when SL(2, R) is replaced by F2 . This
kind of questions also has to be related with the work of Brooks on regular
graphs ([3]) with the main difference that more geometry is involved in our
setting.
3.3
More group theory
The object of this section is to give some kind of dictionary between the
usual representation theory and spectral results. We will use [22] and [16] as
constant references for representation theory.
We start with a group G generated by two elements h and v and construct
the surface MG as in the preceding section. By construction the Euclidean
laplacian on MG is ∆ρ where ρ is the regular representation of G. Applying
theorem 5 then gives the following proposition (compare with [25] prop. 3)
17
Proposition 4 Let G be finite and generated by h and v, let ρλ be the set of
irreducible representations of G, and let dλ be the dimension of ρλ then :
X
Sp(∆MG ) =
dλ Sp(∆ρλ ).
irr. rep.
Proof : the decomposition of the regular representation in irreducible
ones is known to be :
X
χG =
dλ χλ .
irr. rep.
The proposition then follows from theorem 5.
¤
Let us now consider a general square-tiled surface M, we let G be the
corresponding group and H be the subgroup of G that stabilizes a square.
The surface M can actually be constructed in the following way : we take
one square for each coset gH and define the permutations in the horizontal
(resp. vertical) direction as the action of h (resp v) by left-multiplication on
G/H (i.e. h(gH) = (hg)H).
With this presentation the Euclidean laplacian on M corresponds to ∆ρ
where ρ is the quasi-regular representation associated to H (i.e. the representation of G that is induced by the trivial representation of H). If we take
a representation of G that is induced by ρ1 , a one-dimensional representation of H, the corresponding operator ∆ρ1 belongs to the Floquet class FN
where N = |G|/|H|. Combining Brauer’s theorem (see [22] Thm. 23 p.29)
and theorem 5 gives the following proposition.
Proposition 5 Let (∆, H, V ) be an operator in ON then the following spectral decomposition holds :
X
Sp(∆N , H, V) =
ai Sp(∆Ni , Hi , Vi ),
where each (∆Ni , Hi , Vi ) is in FNi .
This proposition is even more interesting if we give another interpretation
of the operators in FN . We recall that an operator A in FN is associated with
h, v and two collections of roots of the unity (see page 12) . It is very likely
that any operator in FN can be realized as a laplacian on the square-tiled
surface defined by h and v with magnetic flux lines going through its conical
points (see [24] for a definition of these operators in the plane).
We now show how other facts from representation theory have an interesting echo in the spectral setting.
18
4
Self-transplantations
As we have already pointed out in the text, the transplantation method
allows us to construct pairs of isospectral square-tiled surfaces by looking for
permutation matrices H1 , V1 , H2 , V2 such that there exists A such that :
AH1 = H2 A
AV1 = V2 A.
(see [9] for such an example). Furthermore, from the spectral point of view,
it is quite interesting to have operators commuting with the laplacian. This
naturally leads us to the following definition.
Definition 5 (Self-transplantations)
Let (∆N , H, V) be an operator in ON . Any element A of GLN (C) such that
AH = HA,
AV = V A,
will be called a self-transplantation of (∆N , H, V).
Let ρ be the representation such that (∆N , H, V) = ∆ρ , the set of selftransplantions coincide with the commutant set of the representation :
Com(ρ) = { A ∈ GLN (C) | ∀g ∈ G, Aρ(g) = ρ(g)A }.
-Remarks1. Since we consider only unitary representation, Com(ρ) is a linear subspace that is invariant under adjonction.
2. If (∆N , H, V) is in SN (i.e. if (∆N , H, V) is the laplacian on a squaretiled surface) then there are always two independent self-transplantations
that correspond to the identity I and to the matrix J that has all its
entries equal to 1 (J represents the orthogonal projector on f~1 in the
canonical basis).
The existence of many self-transplantations is closely related to the fact
that the spectrum of ∆ρ can be decomposed since the invariant subspaces of
a self-transplantation will be invariant by ∆ρ . The dimension of Com(ρ) can
be computed using the multiplicities of the irreducible representations of G.
Lemma 2 Let G be a finite group and (ρλ )λ∈Λ its irreducible representations.
Let ρ be a representation of G then
X
dim(Com(ρ)) =
m2λ ,
Λ
where mλ (ρ) is the multiplicity of ρλ in ρ.
19
Proof : we choose a basis such that the image of ρ is constituted by block
diagonal matrices and the lemma becomes an application of Schur’s lemma
(see [22] p.12).
¤
-Examples1. Let G be generated by h and v, and let us consider MG . The former
lemma combined with the fact that the corresponding representation
is the regular one gives
X
dim(Com(ρ)) =
d2λ = |G|.
This is not surprising since we have seen that, on MG , G was commuting
with ∆. Actually, the elements of Com(ρ) are linear combination of the
Sg (see 16).
2. On the contrary, if dim(Com(ρ)) = 2, then the spectral decomposition
necessarily involves only two irreducible representations and we have a
spectral decomposition similar to eq. (1).
We will give now another way of computing the dimension of the set of
self-transplantations for a square-tiled surfaces. Let M be such a surface, we
have h and v elements of SN and H and V are defined by
H~ei = ~eh(i)
V ~ei = ~ev(i) .
Let B = (bij )1≤i,j≤N commute with H then necessarily bij = bh(i)h(j) , so that
if we consider the diagonal action of G on {1, · · · , N }2 then the entries of B
are constant on the orbits of this action. For each orbit Ok of this action,
(k)
we define B (k) by bij = 1 if and only if (i, j) belongs to Ok . Since the orbits
form a partition of {1, · · · , N }2 , the B (k) 0 s will be linearly independent, so
that we have proved the following proposition :
Lemma 3 Let h, v ∈ SN , G the group generated by h and v, and ρ the
representation of G in UN (C) by permutation matrices. Let (Ok )1≤k≤k0 be the
orbits of the diagonal action of G on {1, · · · N }2 . The matrices B (k) defined
above form a basis of Com(ρ). In particular we have dim(Com(ρ)) = k0 .
-RemarkThis proposition is a reformulation of exercise 2 p.30 of [22].
Theorem 2 is the combination of the two preceding lemmas.
-Examples20
1. If M is connected then G acts transitively on {1, · · · , N } so that the
diagonal is an orbit of the action we consider. In this case, we can take
B1 = I, and dim(Com(ρ)) = 2 is then equivalent to the fact that G
acts transitively on the off-diagonal terms.
2. If G = SN then dim(Com(ρ)) = 2, and we recover the well-known fact
that the representation of SN by permutation matrices, restricted to
F1 , is irreducible.
We have already seen that a surface of the form MG had a big set of
self-transplantation. We will now construct other examples of surfaces where
the dimension of the set of self-transplantations is greater than 2. This will
lead us to finer spectral decompositions than eq. (7)
-ExampleWe start by considering, for p prime, the group GA(Fp ) of invertible affine
transformations acting on Fp i.e. we have :
½
¾
f : Fp →
Fp
∗
GA(Fp ) =
, | a ∈ Fp , b ∈ Fp .
x 7→ ax + b
The group F∗p is cyclic with (p − 1) elements so that for each divisor d of
(p − 1) there is a group of index d in F∗p that we denote by Gd . This group
is generated by an element ad . We now let
Gd = {f ∈ GA(Fp ) | a ∈ Gd },
This group can be seen as a subgroup of Sp generated by two elements h
and v that correspond to :
fh : Fp → Fp
x 7→ x + 1
,
fv : Fp → Fp
.
x 7→ ad x
-Remarkis actually the dihedral group of order
The group corresponding to d = p−1
d
p−1
2p. More generally, if we denote q = d then Gd is isomorphic to Fp oZ/qZ.
The following lemma tells us that the corresponding translation surfaces
will have a non-trivial decomposition. Actually, using lemma 3 it gives the
dimension of the set of self-transplantations.
Lemma 4 The diagonal action of Gd on F2p has exactly d + 1 orbits.
21
Proof : the group Gd acts transitively on Fp so that the diagonal of F2p is one
orbit of the diagonal action. We pick now two off-diagonal points (x, y) and
(x0 , y 0 ) that are in the same orbit. There exist a ∈ Gd and b ∈ Fp such that :
x0 = ax + b, y 0 = ay + b.
This implies x0 − y 0 = a(x − y) so that x0 − y 0 and x − y are in the same class
modulo Gd . The converse is immediate so that the off-diagonal orbits are in
bijection with the number of classes modulo Gd
¤
This gives us examples with non-trivial spectral decomposition. In order
to find the reduced operators we introduce the notation ep (k) = exp( 2ikπ
),
p
we will also extend the definition of ~ei to i in Z by requiring that ~ej = ~ek if
j ≡ k [p]. For j ∈ Z we let


ep (j)
p

1 
1 X
 ep (2j) 
f~j = √ 
ep (jl))~el .
=

√
..
p
p l=1

.
ep (pj))
-RemarkWith this notation the Sp −invariant vector is f~0 and not f~1 (as it is elsewhere in the paper).
The family (f~j )1≤j≤p is an orthonormal basis of Cp . We now study how
H and V act on this basis. We recall that ad is an element of order d in F∗p ,
and we will denote by kd its inverse.
p
1 X
~
√
ep ((l − 1)j)~el = ep (−j)f~j ,
H fj =
N l=1
p
1 X
~
V fj = √
ep (kd l)~el = f~kd j .
N l=1
Eventually, the action of V groups together the f~j0 s where j belongs to the
same orbit modulo Gd . This implies that the spectrum of (∆p , H, V) can
be decomposed into the spectrum of the torus and the spectrum of d opsquares. For each of these, we can find a basis such that
erators on p−1
d
the Vr is the permutation matrix associated with a p−1
−cycle (we just have
d
22
to reorder the f~j0 s), and Hr is a diagonal matrix with p−th root of the unity.
For p = 5, d = 2, we get the following. We identify Fp with {1, . . . , 5}.
The only element of order 2 in F∗p is 4 so that we have :
h = (12345),
v = (14)(23)(5).
The surface M is of genus 3 and has one singularity of angle 10π.
The invariant subspaces are generated by f~5 , (f~1 , f~4 ), (f~2 , f~3 ). The corresponding matrices are :
µ
V1 = V2 =
0 1
1 0
¶
µ
, H1 =
ω 0
0 ω4
¶
µ
, H2 =
ω2 0
0 ω3
¶
,
where ω = exp( 2iπ
).
5
We then have the following decomposition :
Sp(M ) = Sp(T ) + Sp(∆N1 , H1 , V1 ) + Sp(∆N2 , H2 , V2 ).
5
5.1
Other Applications
Isospectrality
We refer to [10] for an account on the long history of isospectrality ; we also
refer to [25, 1, 5, 11] for some results about it and to [15] for recent results
very close to what we are dealing with. In this section we will show that theorem 5 allows us to construct simply non-connected isospectral square-tiled
surfaces.
Example : Let M3 be the surface defined by h = (123), v = (12)(3).
Using the notations of the preceding section, this operator corresponds to
p = 3, d = 2. The group generated by h and v is S3 and the commutator of
h and v is a 3-cycle. The surface M is thus of genus 2 and has one conical
singularity of angle 6π.
Using the general result of the preceding section we have :
Sp(M3 ) = Sp(T ) + Sp(∆2 , H1 , V1 ),
with
µ
H1 =
j2 0
0 j
¶
µ
; V1 =
23
0 1
1 0
¶
,
.
h = (123)
1
2
3
v = (12)(3)
M3
4
5
h = (123)(456)
v = (14)(26)(35)
1
2
3
6
M6
2
h = (1)(2)
v = (12)
h = (1)
v = (1)
1
T2
T1
Figure 3: Isospectral configuration
24
The key point is to remark here that (∆2 , H1 , V1 ) is actually ∆ρ2 where ρ2
is the 2-dimensional irreducible representation of S3 . Let M6 be the surface
MG associated with M : M6 is given by h = (123)(456), v = (14)(26)(35).
The irreducible representations of S3 are known, there is ρ2 ,and two onedimensional representations denoted by t for the trivial one and by ε for the
signature. Using proposition 4 we have :
Sp(M̃ ) = 2 Sp(∆ρ2 ) + Sp(∆ε ) + Sp(∆t ).
The spectrum of ∆ε arises in the spectrum of the torus with two squares
constructed from M6 (see fig. 3). This finally gives the following identity :
Sp(M̃ ) = 2 Sp(M ) + Sp(T2 ) − 2 Sp(T ).
Theorem 3 is a rephrasing of this identity.
This procedure can be generalized to the groups G = Fp o Z/qZ that
were constructed in the preceding section. Recall that qd = p − 1 and that
we have constructed a surface M associated with this group such that
Sp(M ) = Sp(T ) +
d
X
Sp(ρi ),
1
where each ρi is of dimension q. First note that two ρi are not isomorphic since
ρi (h) is diagonal and the eigenvalues are different for different i. We now claim
that these ρi are irreducible. Indeed, we have proved that the dimension
P 2of
the set of self-transplantations was d + 1. The alternative expression ( mλ )
necessarily implies that each ρi is irreducible. We also have q one-dimensional
representations corresponding to the factor Z/qZ. These q one-dimensional
representations arise in the spectrum of the torus Tq with q squares defined
by H = Id and V a q-cycle. Since p p−1
= q + d( p−1
)2 a classical formula (see
d
d
[22] p. 18) tells us that we have found all the irreducible representations of
G. Comparing the spectrum of M , the spectrum of Tq and the spectrum of
MG , we get the following formula :
Sp(MG ) + q Sp(T ) = q Sp(M ) + Sp(Tq ).
5.1.1
(12)
High multiplicity surfaces
A spectral decomposition in the form of equation (4) implies that Sp(∆ρ )
has some multiplicity as soon as some ai > 1.
More precisely suppose we have the following identity :
Sp(∆ρ0 ) =
n
X
i=1
25
ai Sp(∆ρi )
with all the ai positive and a1 > 2. The spectrum of ∆ρ1 has then multiplicity
at least 2. More precisely, there are many eigenvalues that has multiplicity
at least 2. To give a quantitative meaning to this assertion we define the set
(m)
Sp(∆, m) = { λi }, where
(m)
∀i, ∃j(i) | λj
(m)
λ0
(m)
λi+1
= λj(i) ,
= min{λi , m(λi ) ≥ m}
= min{λj , m(λj ) ≥ m, λj ≥ λj(i)+m }.
We also introduce N (E, m) the counting function for Sp(∆, m). Using Weyl’s
law for ∆ρ0 and for ∆ρ1 we get that :
∃c > 0, E0 , | ∀E > E0 ,
N (E)
> c.
N (E, a1 )
(13)
This leads to the following definition :
Definition 6 (Structural multiplicity) The structural multiplicity of ∆
is the greatest a1 (possibly ∞) for which equation (13) holds. It is denoted
by s-m(∆).
The multiplicities ai of the irreducible representations in ρ accounts for
some multiplicity in the spectrum of ∆ρ . In particular s-m(∆) ≥ max(ai ).
However, even when ρλ is irreducible its spectrum may have some multiplicity
and the spectrum of two ∆ρλ may also intersect. Still, we expect that these
latter phenomena occur only exceptionnally.
Since for a surface MG the multiplicity of each irreducible representation
is its dimension, we get easily examples of surfaces with arbitrarily high
structural multiplicity. Comparing this number to the genus of the surface
MG gives the following result of which theorem 4 is a weakening.
Proposition 6 There exists a sequence of square-tiled surfaces Mn such that
the genus gn goes to infinity and such that the following inequality holds :
√
C0 exp(−c0 γ(3gn − 1)) gn ≤ s-m(∆Mn ),
where γ is the reciprocical function of the usual Γ function.
Proof : we choose h = (1 · · · n)(n + 1) and v = (1)(2) · · · (n − 1)(n n +
1). The group G generated by h and v is Sn and we construct MG . The
commutator c of h and v is a 3-cycle ; to get the genus of MG we address the
orbits of Sn under multiplication by c. Each orbit has 3 elements (g, cg, c2 g)
26
and thus there are
of MG is thus
n!
3
6π−conical points on MG . Using formula (2), the genus
n!
+ 1.
3
We now take advantage of the fact that the maximal dimension among the
irreducible repesentations of Sn is (almost) known (see [26]). More precisely
we have the following inequality :
√
√
exp(−c0 n) n! ≤ dn ≤ exp(−c1 n) n!.
gn =
This results in the proposition.
¤
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