Monodromy of the SL_2 Hitchin fibration

0.37 MB·English

by  Laura P. Schaposnik

MONODROMY OF THE SL HITCHIN FIBRATION
2
LAURA P. SCHAPOSNIK
Abstract. We calculate the monodromy action on the mod 2 cohomology for SL(2,C)
1 Hitchin systems and give an application of our results in terms of the moduli space of
1 semistable SL(2,R) Higgs bundles.
0
2

v
Let Σ be a Riemann surface of genus g ≥ 3, and denote by K the canonical bundle. An
o
N SL(2,C) Higgs bundle as defined by Hitchin [7] and Simpson [8] is given by a pair (E,Φ),
0 where E is a rank 2 holomorphic vector bundle with det(E) = OΣ and the Higgs field Φ is
1 a section of End (E)⊗K, for End the bundle of traceless endomorphisms. A Higgs bundle
0 0
] is said to be semistable if for any subbundle F ⊂ E such that Φ(F) ⊂ F ⊗ K one has
G
deg(F) ≤ 0. When deg(F) < 0 we say the pair is stable.
A
Considering the moduli space M of S-equivalence classes of semistable SL(2,C) Higgs
.
h
bundles and the map Φ 7→ det(Φ), one may define the so-called Hitchin fibration [7]:
t
a
m h : M → A := H0(Σ,K2).
[

ThemodulispaceMishomeomorphictothemodulispaceofreductive representationsofthe
1
v fundamental group of Σ in SL(2,C) via non-abelian Hodge theory [7, 8, 4, 5]. The involution
0 on SL(2,C) corresponding to the real form SL(2,R) defines an antiholomorphic involution
5
5 on the moduli space of representations which, in the Higgs bundle complex structure, is
2
the holomorphic involution σ : (E,Φ) 7→ (E,−Φ). In particular, the isomorphism classes
.
1
of stable Higgs bundles fixed by the involution σ correspond to SL(2,R) Higgs bundles
1
1 (E = V ⊕V∗,Φ), where V is a line bundle on Σ, and the Higgs field Φ is given by
1
:
v 0 β
Φ =
Xi γ 0
(cid:18) (cid:19)
r
a for β : V∗ → V ⊗K and γ : V → V∗ ⊗K.
We shall denote by M the regular fibres of the Hitchin map h, and let A be the
reg reg
regular locus of the base, which is given by quadratic differentials with simple zeros. From
[7]oneknows thatthesmoothfibresaretoriofrealdimension 6g−6. There isasection ofthe
fibration fixed by σ and this allows us to identify each fibre with an abelian variety, in fact, a
Prym variety. The involution σ leaves invariant det(Φ) and so defines an involution on each
fibre. The fixed points then become the elements of order 2 in the abelian variety. Hence,
the points corresponding to SL(2,R) Higgs bundles give a covering space of A . This
reg
covering space is determined by the action of π (A ) on the first cohomology of the fibres
1 reg
November 11, 2011.
This work was funded by the Oxford University Press through a Clarendon Award, and by New College,
Oxford.
1
MONODROMY OF THE SL HITCHIN FIBRATION 2
2
with Z coefficients. In this paper we study this action, and thus obtain information about
2
the moduli space of SL(2,R) representations of π (Σ). Our main result is the following:
1
Theorem 1. The monodromy action on the first mod 2 cohomology of the fibres of the
Hitchin fibration is given by the group of matrices acting on Z6g−6 of the form
2
I A
(1) 2g ,
0 π
(cid:18) (cid:19)
where
• π is the quotient action on Z4g−5/(1,··· ,1) induced by the permutation action of the
2
symmetric group S on Z4g−5.
4g−4 2
• A is any (2g)×(4g −6) matrix with entries in Z .
2
Finally, we give an application of our result in terms of geometric properties of the moduli
space of semistable SL(2,R) Higgs bundles. In particular, we show the following:
Corollary 2. The numberof connectedcomponentsof the modulispaceof semistableSL(2,R)
Higgs bundles is 22g +g.
These connected components are known to be parametrized by the Euler class of the as-
sociated flat RP1 bundle. From our point of view this number k relates to the orbit of a
subset of 1,2,...,4g −4 with 2g −2k −2 points under the action of the symmetric group.
In a later work we shall extend this approach to the group SU(p,p).
Acknowledgements. The author is thankful to Prof. Nigel Hitchin for suggesting this
problem, and for many helpful discussions.
1. THE REGULAR FIBRES OF THE HITCHIN FIBRATION
Let us consider the Hitchin map h : M → A given by (E,Φ) 7→ det(Φ). From[7, Theorem
8.1] the map h is proper and surjective, and its regular fibres are abelian varieties. Moreover,
for any a ∈ A−{0} the fibre M is connected [6, Theorem 8.1]. For any isomorphism class
a
of (E,Φ) in M, one may consider the zero set of its characteristic polynomial
det(Φ−ηI) = η2 +a = 0,
where a = det(Φ) ∈ A. This defines a spectral curve S in the total space of K, for η the
tautological section of ρ∗K where ρ : S → Σ is the projection. Note that the curve S is
non-singular over A , and the ramification points are given by the intersection of S with
reg
the zero section. The curve S has a natural involution τ(η) = −η and thus we can define
the Prym variety Prym(S,Σ) as the set of line bundles M ∈ Jac(S) which satisfy
τ∗M ∼= M∗.
Proposition 3. [7, Theorem 8.1] The fibres of M are isomorphic to Prym(S,Σ).
reg
MONODROMY OF THE SL HITCHIN FIBRATION 3
2
To see this, given a line bundle L on S one may consider the rank two vector bundle
E := ρ L and construct an associated Higgs bundle as follows. Given an open set U ⊂ Σ,
∗
multiplication by the tautological section η gives a homomorphism
(2) η : H0(ρ−1(U),L) → H0(ρ−1(U),L⊗ρ∗K).
By the definition of a direct image sheaf, we have
H0(U,E) = H0(ρ−1(U),L).
Hence, equation (2) can then be seen as
(3) Φ : H0(U,E) → H0(U,E ⊗K),
defining the Higgs field Φ ∈ H0(Σ,End E ⊗ K). Note that the map Φ is traceless as it
0
satisfies its characteristic equation, which by construction is η2 +a = 0.
Any line bundle M on the curve S such that τ∗M ∼= M∗ satisfies Nm(M) = 0. From [1]
one has that
Λ2E = Nm(L)⊗K−1.
For K1/2 a choice of square root, M = L⊗ ρ∗K−1/2 is in the Prym variety and hence the
vector bundle E has trivial determinant:
Λ2E = Nm(M ⊗ρ∗K1/2)⊗K−1 = O.
2. A COMBINATORIAL APPROACH TO MONODROMY
The Gauss-Manin connection on the cohomology of the fibres of M → A defines the
reg reg
monodromy action for the Hitchin fibration. As each regular fibre is a torus, the monodromy
is generated by the action of π (A ) on H1(Prym(S,Σ),Z). The generators and relations
1 reg
of the monodromy action for hyperelliptic surfaces were studied from a combinatorial point
of view by Copeland in [3, Theorem 1.1]. Furthermore, by [9, Section 4] one may extend
these results to any compact Riemann surface:
Theorem 4 ([3, 9]). To each compact Riemann surface Σ of genus greater than 2, one may
associate a graph Γˇ with edge set E and a skew bilinear pairing < e,e′ > on edges e,e′ ∈ Z[E]
such that
(i) the monodromy representation of π (A ) acting on H (Prym(S,Σ),Z) is generated
1 reg 1
by elements σ labelled by the edges e ∈ E,
e
(ii) one can define an action of π (A ) on e′ ∈ Z[E] given by
1 reg
σ (e′) = e′− < e′,e > e,
e
(iii) the monodromy representation of the action of π (A ) on H (Prym(S,Σ),Z) is a
1 reg 1
quotient of this module Z[E].
MONODROMY OF THE SL HITCHIN FIBRATION 4
2
In order to construct the graph Γˇ Copeland looks at the particular case of Σ given by the
non-singular compactification of the zero set of y2 = f(x) = x2g+2−1. Firstly, by considering
ω ∈ A given by
2
dx
ω = (x−2ζ2)(x−2ζ4)(x−2ζ6)(x−2ζ8) (x−2ζj) ,
y
9≤j≤2g+2 (cid:18) (cid:19)
Y
for ζ = e2πi/2g+2, it is shown in [3, Section 7] how interchanging two zeros of the differential
provides information about the generators of the monodromy. Then, by means of the rami-
fication points of the surface, a dual graph to Γˇ for which each zero of ω is in a face could be
constructed. Copeland’s analysis extends to any element in A over a hyperelliptic curve
reg
[3, Section 23]. Moreover, by work of Walker [9, Section 4] the above construction can be
done for any compact Riemann surface.
Remark 5. Following [3, Section 6], we shall consider the graph Γˇ whose 4g−4 vertices are
given by the ramification divisor of ρ : S → Σ, i.e., the zeros of a = det(Φ).
As an example, for genus g = 3,5, and 10, the graph Γˇ is given by:
For g > 3 the graph Γˇ is given by a ring with 8 triangles next to each other, 2g − 6
quadrilaterals and 4g −4 vertices. In this case, we shall label its edges as follows:
Figure 1.
Considering the lifted graph of Γˇ in the curve S over Σ, Copeland could show the following:
MONODROMY OF THE SL HITCHIN FIBRATION 5
2
Proposition 6. [3, Theorem 11.1] If E and F are respectively the edge and face sets of Γˇ,
then there is an induced homeomorphism
R[E]
∼
(4) Prym(S,M) = ,
R[F]+ 1Z[E]
2
where the inclusion R[F] ⊂ R[E] is defined by the following relations involving the boundaries
(cid:0) (cid:1)
of the faces:
2g−2 2g−2
x˜ := l ; x˜ := u ;
1 i 2 i
i=1 i=1
X X
2g+2
x˜ := u − l + b ;
3 i i i
even≥6 odd≥5 i=1
X X X
2g+2
x˜ := l +l −u −u + u − l + b .
4 1 3 2 4 i i i
odd even i=1
X X X
Note that R[F]∩ 1Z[E] can be understood by considering the following sum:
2
2g+2
x˜ +x˜ +x˜ −x˜ = 2 l +l −u −u + b = 2x˜ .
3 4 1 2 1 3 2 4 i 5
!
i=1
X
Although the summands above are not individually in R[F] ∩ 1Z[E], when summed they
2
satisfy
1
x˜ ∈ R[F]∩ Z[E].
5
2
Remark 7. For g = 2 it is known that π (A ) ∼= Z ×π (S2), where S2 is the sphere S2
1 reg 1 6 6
with 6 holes (e.g. [3, Section 6]).
3. THE FIXED POINTS OF (E,Φ) 7→ (E,−Φ)
The direct image of the trivial bundle O in Prym(S,Σ) is given by ρ O = O⊕K. So for
∗
L = ρ∗K−1/2 one has
ρ ρ∗K−1/2 = K−1/2 ⊗ρ O = K−1/2 ⊕K1/2.
∗ ∗
It follows from Section 1 that the line bundle O ∈ Prym(S,Σ) has an associated Higgs
bundle (K−1/2 ⊕K1/2,Φ ), where the Higgs field Φ is obtained via Proposition 3:
a a
0 a
Φ = , for a ∈ H2(Σ,K2).
a 1 0
(cid:18) (cid:19)
Note that the automorphism
i 0
0 −i
(cid:18) (cid:19)
conjugates Φ to −Φ and so the equivalence class of the Higgs bundle is fixed by the
a a
involution σ. Thus, this family of Higgs bundles defines an origin in the set of fixed points
on each fibre.
MONODROMY OF THE SL HITCHIN FIBRATION 6
2
An infinitesimal deformation of a Higgs bundle is given by (A˙,Φ˙) where A˙ ∈ Ω01(End E)
0
and Φ˙ ∈ Ω10(End E). The holomorphic involution σ on M induces an involution on the
0
tangent space T of M at a fixed point of σ. Moreover, there is a natural symplectic form ω
defined on the infinitesimal deformations by
(5) ω((A˙ ,Φ˙ ),(A˙ ,Φ˙ )) = tr(A˙ Φ˙ −A˙ Φ˙ ).
1 1 2 2 1 2 2 1
ZΣ
As the trace is invariant under σ and σ(Φ ) = −Φ , the induced involution on the tangent
i i
space maps ω 7→ −ω. It follows that the ±1-eigenspaces T of this involution are isotropic
±
and complementary, and hence Lagrangian. Let us denote by Dh the derivative of the map
h, which maps the tangent spaces of M to the tangent space of A. As h is invariant under
the involution σ, the eigenspace T is contained in the kernel of Dh. Since the derivative
−
is surjective at a regular point, its kernel has dimension dim(M)/2 and thus it equals T .
−
Then, Dh is an isomorphism from T to the tangent space of the base. Since h is a proper
+
submersion on the fixed point set, it defines a covering space. The tangent space to the
identity in the Prym variety is acted as −1 by the involution σ and as the Prym variety is
connected, by exponentiation the action of σ on the regular fibres corresponds to x 7→ −x.
Hence, the points of order two in the fibres of M over the regular locus A correspond
reg reg
to stable SL(2,R) Higgs bundles.
By Proposition 6 one may describe the Prym variety as Prym(S,Σ) ∼= R6g−6/∧, where
1Z[E]
∧ := 2 .
R[F]∩ 1Z[E]
2
In particular, one has ∧ ∼= H (Prym(S,Σ),Z). Let us denote by P[2] the elements of order
1
2 in Prym(S,Σ), which are equivalent classes in R6g−6 of points x such that 2x ∈ ∧. Then,
P[2] is given by 1∧ modulo ∧ and as ∧ is torsion free,
2
P[2] ∼= ∧/2∧ ∼= H (Prym(S,Σ),Z ).
1 2
Moreover, H1(Prym(S,Σ),Z ) ∼= Hom(H (Prym(S,Σ),Z),Z ) and thus
2 1 2
H1(Prym(S,Σ),Z ) ∼= Hom(∧,Z ) ∼= ∧/2∧ ∼= P[2].
2 2
The monodromy action on H1(Prym(S,Σ),Z ) is equivalent to the action on P[2], the
2
space of elements of order 2 in Prym(S,Σ). Note that over Z , the equations for x˜ ,x˜ ,x˜ ,x˜
2 1 2 3 4
and x˜ are equivalent to
5
2g−2 2g−2
x := l ; x = u ;
1 i 2 i
i=1 i=1
X X
2g+2
x := l +l +u +u + u + l + b ;
3 1 3 2 4 i i i
even odd i=1
X X X
2g+2
x := l +l +u +u + u + l + b ;
4 1 3 2 4 i i i
odd even i=1
X X X
2g+2
x := l +l +u +u + b .
5 1 3 2 4 i
i=1
X
MONODROMY OF THE SL HITCHIN FIBRATION 7
2
Proposition 8. The space P[2] is given by the quotient of Z [E] by the subspace generated
2
by x ,x ,x and x .
1 2 4 5
4. THE ACTION ON Z [E]
2
It is convenient to consider Z [E] as the space of 1-chains C for a subdivision of the
2 1
annulus in Figure 1. The boundary map ∂ to the space C of 0-chains (spanned by the
0
vertices of Γˇ) is defined on an edge e ∈ C with vertices v , v as ∂e = v +v . Let Σ[4g−4] be
1 1 2 1 2
the configuration space of 4g−4 points in Σ. Then, there is a natural map p : A → Σ[4g−4]
reg
which takesaquadraticdifferential toitszero set. Furthermore, pinduces thefollowingmaps
π (A ) → π (Σ[4g−4]) → S ,
1 reg 1 4g−4
where S is the symmetric group of 4g−4 elements. Thus there is a natural permutation
4g−4
action on C and Copeland’s generators in π (A ) map to transpositions in S . Con-
0 1 reg 4g−4
cretely, these generators are defined as transformations of Z [E] as follows. The action σ
2 e
labelled by the edge e on another edge x is
(6) σ (x) = x + < x,e > e,
e
where < ·,· > is the intersection pairing. As this pairing is skew over Z, for any edge e one
has < e,e >= 0. Let G be the group of transformations of C generated by σ , for e ∈ E.
1 1 e
Then, one can see the following:
Proposition 9. The group G acts trivially on Z = ker(∂ : C → C ).
1 1 1 0
Proof. Let us consider a ∈ C such that ∂a = 0, i.e., the edges of a have vertices which occur
1
an even number of times. By definition, σ ∈ G acts trivially on a for any edge e ∈ E non
e 1
adjacent to a. Furthermore, if e ∈ E is adjacent to a, then ∂a = 0 implies that an even
number of edges in a is adjacent to e, and thus the action σ is also trivial on a. (cid:3)
e
We shall give an ordering to the vertices in Γˇ as in the figure below, and denote by E′ ⊂ E
the set of dark edges:
For (i,j) the edge between the vertices i and j, the set E′ is given by e := (4g − 4,1)
4g−4
together with the natural succession of edges e := (i,i+1) for i = 1,...,4g −5.
i
MONODROMY OF THE SL HITCHIN FIBRATION 8
2
Proposition 10. The reflections labelled by the edges in E′ ⊂ E generate a subgroup S′
4g−4
of G isomorphic to the symmetric group S .
1 4g−4
Proof. To show this result, one needs to check that the following properties characterizing
generators of the symmetric group apply to the reflections labelled by E′:
(i) σ2 = 1 for all i,
ei
(ii) σ σ = σ σ if j 6= i±1,
ei ej ej ei
(iii) (σ σ )3 = 1.
ei ei+1
By equation (6), it is straightforward to see that the properties (i) and (ii) are satisfied by
σ for all e ∈ E. In order to check (iii) we shall consider different options for edges adjacent
ei i
to e and e when e ,e ∈ E′. Let c ,c ,··· ,c ∈ E be the edges adjacent to e and e ,
i i+1 i i+1 1 2 n i i+1
where n may be 5,6 or 7. Taking the basis {c ,··· ,c ,e ,e }, the action σ σ is given
1 n i i+1 ei ei+1
by matrices B divided into blocks in the following manner:
0 0
. .
. .
I . .
 n 
B := 0 0 ,
 a ··· a 0 1 
 1 n 
 b ··· b 1 1 
 1 n 
 
where the entries a and b are 0 or 1, depending on the number of common vertices with the
i j
edges and their locations. Over Z one has that B3 is the identity matrix and so property
2
(iii) is satisfied for all edges e ∈ E′. (cid:3)
i
The subgroup S′ preserves Z [E′]. Furthermore, the boundary ∂ : C → C is compat-
4g−4 2 1 0
ible with the action of S′ on C and S on C . Thus, from Proposition 10, there is a
4g−4 1 4g−4 0
natural homomorphism
α : G −→ S ,
1 4g−4
which is an isomorphism when restricted to S′ . For N := kerα one has
4g−4
G = N ⋉S′ .
1 4g−4
Any element g ∈ G can be expressed uniquely as g = s · h, for h ∈ N ⊂ G and
1 1
s ∈ S′ ⊂ G . The group action on C may then be expressed as
4g−4 1 1
(7) (h s )(h s ) = h s h s−1s s ,
1 1 2 2 1 1 2 1 1 2
for s ,s ∈ S′ and h ,h ∈ N. In order to understand the subgroup N, we define
1 2 4g−4 1 2
E := E−E′ and let ∆ ∈ C denote the boundary of the square or triangle adjacent to the
0 e 1
edge e ∈ E in Γˇ. Note that each edge e ∈ E is contained in only one of such boundaries.
0 0
We shall write E˜ := {∆ for e ∈ E }. From Proposition 9 the boundaries ∆ are acted on
0 e 0 e
trivially by G .
1
By comparing the action labelled by the edges in E with the action of the corresponding
0
transposition in S′ one can find out which elements are in N. We shall denote by σ
4g−4 (i,j)
the action on C labeled by the edge (i,j). Furthermore, we consider s the element in
1 (i,j)
S′ such that α(σ ) = α(s ) ∈ S , where α(s ) interchanges the vertices of (i,j).
4g−4 (i,j) (i,j) 4g−4 (i,j)
Then, we have the following possibilities:
MONODROMY OF THE SL HITCHIN FIBRATION 9
2
• For a triangle with vertices i,i+1,i+2, the generators of S′ ⊂ S′ are given by
3 4g−4
σ and σ . Then, (i,i+ 1),(i+ 1,i+ 2) ∈ E′ and the action labelled by
(i,i+1) (i+1,i+2)
the edge (i,i+2) ∈ E can be written as:
0
s = σ σ σ .
(i,i+2) (i+1,i+2) (i,i+1) (i+1,i+2)
• For a square with vertices i,i + 1,i + 2,i + 3, the generators of S′ ⊂ S′ are
4 4g−4
σ ,σ and σ . In this case (i,i+1),(i+1,i+2),(i+2,i+3) ∈ E′
(i,i+1) (i+1,i+2) (i+2,i+3)
and the action labelled by the edge (i,i+3) ∈ E can be written as:
0
s = σ σ σ σ σ .
(i,i+3) (i+2,i+3) (i+1,i+2) (i,i+1) (i+1,i+2) (i+2,i+3)
Theorem 11. The action of σ labelled by e ∈ E on x ∈ C is given by
e 1
σ (x) = h ·s (x) ,
e e e
where s ∈ S′ is the element which maps under α to the transposition of the two vertices
e 4g−4
of e, and the action of h ∈ N is given by
e
x if e ∈ E′ ,
(8) h (x) =
e x+ < e,x > ∆ if e ∈ E .
e 0
(cid:26)
Proof. For e ∈ E′, one has σ ∈ S′ . Furthermore, one can see that the action of σ
e 4g−4 e
labelled by e ∈ E on an adjacent edge x is given by
0
σ (x) = x+ < x,e > e = s (x)+∆ .
e i e e
From Proposition 9, the boundary ∆ is acted on trivially by G and thus the above action
e 1
is given by h ·s (x) = h (σ (x)+∆ ) = h (x)+ < x,e > e +∆ = σ (x). (cid:3)
e e e e e e i i e e
Remark 12. Note that for e,e′ ∈ E0, the maps he and he′ satisfy
(9) hehe′(x) = x+ < x,e′ > ∆e′+ < e,x > ∆e .
In order to construct a representation for the action of π (A ) on C , we shall begin by
1 reg 1
studying the image B and the kernel Z of ∂ : C → C .
0 1 1 0
5. THE REPRESENTATION OF G
1
For y = (y ,...,y ) ∈ C , we define the linear map f : C → Z by
1 4g−4 0 0 2
4g−4
(10) f(y) = y .
i
i=1
X
Proposition 13. The image B of ∂ : C → C is formed by elements with an even number
0 1 0
of 1’s, i.e., B = kerf.
0
Proof. It is clear that B ⊂ kerf. In order to check surjectivity we consider the edges
0
e ∈ C given by e = (i,i+1) ∈ E′, for i = 1,...,4g −5. Given the elements Rk ∈ B for
i 1 i 0
k = 2,··· ,4g −4 defined as
R2 := ∂e = (1,1,0,...,0) ,
1
.
.
.
Rk := Rk−1 +∂e = (1,0,...,0,1,0,...,0),
k−1
MONODROMY OF THE SL HITCHIN FIBRATION 10
2
one may generate any distribution of an even number of 1’s. Hence, B = span{Rk} which
0
is the kernel of f. (cid:3)
Proposition 14. The dimensions of the image B and the kernel Z of ∂ : C → C are,
0 1 1 0
respectively, 5g −5 and 2g +3.
Proof. From Proposition 13 one has that dim(B ) = dim(kerf) = 4g −5. Furthermore, as
0
dimC = dimZ +dimB , the kernel Z of ∂ has dimension 2g +3. (cid:3)
1 1 0 1
Note that x ,x ,x ∈ Z and x ∈/ Z . From a homological viewpoint one can see that x
1 2 4 1 5 1 4
and ∆ for e ∈ E form a basis for the kernel Z . We can extend this to a basis of C by
e 0 1 1
taking the edges
β′ := {e = (i,i+1) for 1 ≤ i ≤ 4g −5} ⊂ E′ ,
i
whose images under ∂ form a basis for B , and hence a basis for a complementary subspace
0
V of Z . We shall denote by β := {β ,β′} the basis of C where β = {E˜ ,x }.
1 0 1 0 0 4
In order to generate the whole group G , we shall study the action of S′ by conjugation
1 4g−4
on N. Considering the basis β one may construct a matrix representation of the maps h
ei
for e ∈ E = {E ,E′}.
i 0
Proposition 15. For e ∈ E, the matrix [h ] associated to h in the basis β is given by
e e
I A
[h ] = 2g+3 e ,
e 0 I
4g−5
(cid:18) (cid:19)
where the (2g +3)×(4g −5) matrix A satisfies one of the following:
e
• it is the zero matrix for e ∈ E′,
• it has only four non-zero entries in the intersection of the row corresponding to ∆
e
and the columns corresponding to an adjacent edge of e for e ∈ E −{u ,l },
0 5 6
• it has three non-zero entries in the intersection of the row corresponding to ∆ and
e
the columns corresponding to an adjacent edge of e for e = u ,l .
5 6
Proof. As we have seen before, for e ∈ E the map h acts as the identity on the elements
0 e
of β . Furthermore, any edge e ∈ E − {u ,l } is adjacent to exactly four edges in β′. In
0 0 5 6
this case h has exactly four non-zero elements in the intersection of the row corresponding
e
to ∆ and the columns corresponding to edges in β′ adjacent to e. In the case of e = u ,l ,
e 5 6
the edge e is adjacent to exactly 3 edges in β′ and thus h has only 3 non-zero entries. (cid:3)
e
Let us recall that S′ preserves the space spanned by E′, hence also the subspace V,
4g−4
and acts trivially on Z . In the basis β the action of an element s ∈ S′ has a matrix
1 4g−4
representation given by
I 0
[s] = 2g+3 ,
0 π
s
(cid:18) (cid:19)
where π is the permutation action corresponding to s. Hence, for f ∈ E we may construct
0
the matrix for a conjugate of h as follows:
f
I 0 I A I 0 I A π−1
[s][h ][s]−1 = 2g+3 2g+3 f 2g+3 = 2g+3 f s .
f 0 π 0 I 0 π−1 0 I
(cid:18) s (cid:19)(cid:18) 4g−5 (cid:19)(cid:18) s (cid:19) (cid:18) 4g−5 (cid:19)