Some Identities and Asymptotics for Characters of the Symmetric Group

Some Identities and Asymptotics for Characters of the Symmetric Group

A. Klyachko* and E. Kurtaran

Department of Mathematics, Bilkent Uni¨ersity, Bilkent, 06533 Ankara, Turkey Communicated by Gordon James

Received December 17, 1996

1. INTRODUCTION

Ž w

This paper is motivated by the problem arising from quantum gravity 9, x.

2 of counting combinatorial types of triangulations ⌺ of a Riemann surface X with given degrees of vertices. Let us color 2-simplexes of the barycentric subdivision ⌺⬘ in black and white, so that adjacent simplexes are of different color, and consider a mapping ␲ : X ª ⺠¹ onto the Riemann sphere⺠¹ which send white simplexes onto the northern hemi- sphere, black simplexes onto the southern hemisphere, and centers k-sim- plexes of ⌺ into an equatorial point y , k s 0, 1, 2. Then degk ␲ s 3n, n is the number of 2-simplexes in ⌺, and␲ is unramified outside y , y , y . It0 1 2

is easy to see that points of X over y and y have ramification index 2_{1 2} Žrespectively, 3 , while rmaification indices of points over y. ₀ are just degrees of vertices of ⌺. So the problem of counting triangulations reduces to the problem of counting ramified coverings ␲ : X ª Y of Riemann surfaces with given ramification indices.

In Section 2 we recall a connection between ramified coverings␲ : X ª Y of a Riemann surface Y of genus gY and irreducible characters ␹ of the symmetric group S , nn s deg␲. The starting point is the following for-

w x Ž w x.

mula, essentially owing to Hurwitz 6 see also 10, 4, 2 :

< < < < < <

1 C₁ C₂ ⭈⭈⭈ C_k ␹ g ␹ g ⭈⭈⭈ ␹ g

Ž

₁

. Ž

₂

. Ž

_k

.

s ,

Ž

1.1

.

Ý

<Aut␲<

Ž

n!

.

2y2 gY

Ý

␹

Ž

_{␹ 1}

Ž . .

ky 2y2 gŽ Y.

␲ : XªY

*E-mail: klyachko@fen.bilkent.edu.tr.

413

0021-8693r98 $25.00

Copyright䊚 1998 by Academic Press All rights of reproduction in any form reserved.

where the first sum extends over all coverings of degree n with fixed conjugacy classes Ci; S of monodromy g g C around ramificationn i i

points yig Y, while the second sum runs over all irreducible characters of the symmetric group S . Following Serre we will refer to the first sum in_n Ž1.1 as Eisenstein number of coverings..

Ž .

We call the covering␲ : X ª Y and formula 1.1 elliptic, parabolic, or hyperbolic if the Euler characteristic 2y 2 g is positive, zero, or negative.X

By the Riemann᎐Hurwitz formula the type depends on the sign of the quantity

1 1 1 1

2y 2 g s q q ⭈⭈⭈ q y k q 2 y 2 g , 1.2

Ž

X

.

Y

Ž .

n l₁ l₂ l_k

where l is the mean value of cycle length of the monodromy g ._{i i}

In Section 3 we consider elliptic and parabolic coverings for which the

Ž .

transformations g consist of cycles of the same lengthi s l .i

In the elliptic case such a covering is a disjoint union of factorizations

⺠¹ª ⺠¹rG by a finite group of Mobius transformations G. For each such

¨

Ž .

group G the first sum in 1.1 may be easily evaluated. For example, the identity

␹ ␴ ␹ ␴ ␹ ␴

Ž

2

. Ž

3

. Ž

5

. Ž

30 n !2

.

^{30 n}

Ž

20 n !3

.

^{20 n}

Ž

12 n !5

.

^{12 n}

s

Ž

1.3

.

Ý

_{␹ 1} ⁿ

Ž . Ž

60 n !n! 60

. Ž .

␹

Ž .

comes from the icosahedral group Theorem 3.3 . Here ␴ g S_{d 60 n} is a permutation splitting in cycles of length d, and the sum extends over all irreducible characters of S_{60 n}.

Ž .

Under the same condition, parabolic formulas 1.1 relate to root systems of rank 2. They give identities like

3 3 n 3 ⬁

␹ ␴

Ž

3

.

s 3 n! coefficient at q inn

Ž

1y qk

.

_y1r3 ,

Ž

1.4

.

Ý

␹ 1

Ž . Ž

3n !

. ž

k

Ł

s1

/

␹gS3n

Ž .

which corresponds to root system A₂ Theorem 3.9 . In general under our Ž .

assumptions parabolic character sums 1.1 in essence coincide with Fourier coefficients of some negative power of Dedekind␩ function. There is an

w x

explicit Rademacher᎐Zukerman formula 11 for these coefficients, similar to that of for partition function p n .Ž .

Ž . Geometrically, these two types of identities correspond to coverings 1.1 of positive or zero Euler characteristic, but in the most interesting case of negative Euler characteristic, almost nothing is known about the corre- sponding character sums.

Of course in the hyperbolic case we have no chance to get an explicit

Ž . Ž .

formula similar to 1.3 or 1.4 . The main problem we keep in mind is an Ž .

asymptotic formula for the character sum in 1.1 . We have not yet succeeded completely in solving this problem, mainly because we do not know leading terms of the sum. Let us observe that in view of

Ž . Ž .

Riemann᎐Hurwitz formula 1.2 , the character sum in 1.1 may be written as

␹ g

Ž

1

.

␹ g

Ž

2

.

␹ g

Ž

k

.

Ž2y2 g rnX.

␹ 1

Ž .

⭈⭈⭈ .

Ž

1.5

.

Ý

_␹ ␹ 1

Ž .

1rl1 ␹ 1

Ž .

1rl2 ␹ 1

Ž .

1rlk

If genus gX of Riemann surface X is fixed, while nª ⬁, then the first Ž .^{Ž2y2 gX.r n}

factor ␹ 1 decreases or increases as a power of n. So in essence Ž . Ž .^{1r l} the problem reduces to studying for large n the ratio ␹ g r dim ␹ , where l is the mean value of cycles length gg S . This ratio may ben

Ž . Ž .

compared with or opposed to normalized character ␹ g rdim ␹ treated

w x

in 15, 16, 12 .

As a contribution to this problem, we give in Section 4 an asymptotic formula for characters of S_n under some restrictions. More specifically, consider a sequence of diagrams ␤: b G b G ⭈⭈⭈ G b such that the1 2 m

ratio birn s␤ is a constant, and a sequence of permutations ␴ g Si ␣ n

with constant relative multiplicity akrn s␣ of k cycles. The main resultk

Ž .

of this section is the asymptotic formula for nª ⬁ Theorem 4.2 , exp n

Ž

␻

.

Łi- j

Ž

1y exp

Ž

␶ y ␶j i

. .

␹ ␴ ;_␤

Ž

_␣

.

_{Žmy1 r2. m} ,

Ž

1.6

.

1rm Ý H ␶

' ^{Ž . Ž .}

2␲ n

Ž .

is1 i i

where we suppose␤ / ␤ and lengths of all cycles involved in ␴ g S arei j ␣ n

Ž .

coprime. Here ␻ s ␻ ␣, ␤ is minimum of the function

m

L␣ , ␤

Ž .

t s

Ý

␣ log exp kt q exp kt q ⭈⭈⭈ qexp ktk

Ž Ž

1

. Ž

2

. Ž

m

. .

y

Ý

␤ t ,i i

k is1

tg ⺢^m,

Ž

1.7

.

Ž . 5Ž ² . 5

and H_{i i} ␶ are principle minors of Hessian ⭸ r⭸ t ⭸ t L_{i j ␣ , ␤} at the point

m Ž

of minimum ␶ g ⺢ which exists, satisfies ␶ ) ␶ ) ⭈⭈⭈ ) ␶ , and is1 2 m

. Ž .

unique up to transformation ␶ ¬ ␶ q c . Note that ␤ ¬ ␻ ␣, ␤ is justi i

Ž . the Legendre transform of the first sum in 1.7 .

The asymptotic formula may be extended to diagrams ␤ with multiple rows by making use of an integral, which is a special case of Macdonald

Ž .

conjecture. The resulting formula differs from 1.6 only in pre-exponential factor, which becomes more complicated. To avoid technicalities we con-

sider only the most degenerate case of a rectangular diagram Proposi-Ž tion 4.4 ..

Ž .

The second restriction lengths of cycles are coprime is more subtle. Let q be the greatest common divisor of cycle lengths in ␴ . Then by Little-_␣

Ž . Ž .

wood theorem ␹ ␴ / 0 only if the diagram ␤ or character ␹_{␤ ␣ ␤} is divisible by q. In the last case

S_{nr q}

␹ ␴ s "Ind_␤

Ž

_␣

.

␹ = ␹ = ⭈⭈⭈ = ␹_␤1 _␤2 _␤q

Ž

␴_{␣ r q}

.

[ "␹␤ , ␤ , . . . ,␤1 2 q

Ž

␴␣ r q

.

,

< < < < < <

where diagrams ␤ , ␤ , . . . , ␤ , ␤ q ␤ q ⭈⭈⭈ q ␤ s nrq form so-1 2 q 1 2 q

called q quotients of ␤, and cycle lengths of ␴_{␣r q} are that of ␴ divided_␣

Ž w x . Ž .

by q see 7 for details . Our approach may be applied to␹␤ , ␤ , . . . ,␤1 2 q␴␣ r q

Ž . as well. As result we can see that for qN␤ the critical exponent of ␹ ␴_{␤ ␣} is given by the same formula ␻ s min L ._␣␤

In conclusion we give a useful estimation of the critical value ␻ ŽTheorem 4.6 ,.

H

Ž

␤

.

H

Ž

␤

.

F␻ F ,

l l_min

where l and l_min are mean value and minimal value of cycle lengths in Ž .

␴ , and H ␤ s yÝ ␤ log ␤ is an entropy function. It follows that for_␣ i i

large n,

1rl 1rlmi n

␹ 1_␤

Ž .

F␹ ␴ F ␹ 1_␤

Ž

_␣

.

_␤

Ž .

provided diagram ␤ is not rectangular and not all cycle lengths of ␴ are_␣ equal.

2. CONNECTION WITH CHARACTERS

As explained in the Introduction, the problem of counting triangulations is a particular case of counting ramified coverings with prescribed ramifi-

w x

cation indices 2 . In this section we recall some results on interrelation of

w x

the last problem with characters of symmetric groups 10, 4, 2 . This connection follows from two basic facts:

Fact 1. Topologically, coverings ␲ : X ª Y of degree n unramified outside k points yig Y are classified by conjugacy classes of homomor-

Ž  4

phisms ␲ ª S of the fundamental group ␲ s ␲ Y _ y , y , . . . , y ,1 n 1 1 1 2 k

which is known to be defined by the unique relation

w xw x

g g1 2 ⭈⭈⭈ g f , hk 1 1 f , h2 2 ⭈⭈⭈ f , h s 1,g g

where gs g is the genus of Y and the brackets denote the commutatorY

wf, hxs fhf h . Thus the coverings of Riemann sphere^{y1 y1} ␲ : X ª ⺠ of¹ given degree n and ramification indices are parametrized by solutions of the equation

g g_{1 2} ⭈⭈⭈ g s 1,_k g_ig C ,_i

Ž

2.1

.

up to conjugacy, where cycle lengths of the conjugacy class Ci; S aren

y1Ž .

equal to ramification indices of points in fibers ␲ y . For an arbitrary_i Riemann surface Y of genus g the number of coverings is equal to the number of solutions up to conjugacy of the equation

w x

g g1 2 ⭈⭈⭈ g f , h ⭈⭈⭈ f , h s 1,k 1 1 g g f , hi ig S , g g C .n i i

Ž

2.2

.

Fact 2. The Burnside theorem gives the number of solutions of the

Ž . Ž .

equations 2.1 and 2.2 for an arbitrary group G in terms of irreducible w x

characters 1 ,

 4

噛 g g ⭈⭈⭈ g s 1 N g g C1 2 k i i

< < < <C₁ C₂ ⭈⭈⭈ C< <k ␹ g ␹ g ⭈⭈⭈ ␹ g

Ž

1

. Ž

2

. Ž

k

.

s < <G

Ý

␹ ␹ 1

Ž .

ky2 ,

w x

噛 g g ⭈⭈⭈ g f , h ⭈⭈⭈ f , h s 1



1 2 k 1 1 g g

4

Ž

2.3

.

< < < <C1 C2 ⭈⭈⭈ C< <k ␹ g ␹ g ⭈⭈⭈ ␹ g

Ž

1

. Ž

2

. Ž

k

.

s < <G1y2 g

Ý

_␹ ␹ 1

Ž .

kq2 gy2 ,

where the sums extend over all irreducible characters ␹ of G and g g Ci i

are elements from fixed conjugacy classes C ._i

Combination of these two results gives the following formula for Eisen- w x

stein number of ramified coverings 4, 2 .

THEOREM 2.1. In the pre_¨ious notation the following formula for Eisen- stein number of co_¨erings ␲ : X ª ⺠¹ with prescribed ramification indices holds:

< < < < < <

1 C₁ C₂ ⭈⭈⭈ Ck ␹ g ␹ g ⭈⭈⭈ ␹ g

Ž

1

. Ž

2

. Ž

k

.

s .

Ž

2.4

.

Ý

¹ <Aut␲<

Ž

n!

.

2

Ý

␹

Ž

␹ 1

Ž . .

ky2

␲ : Xª⺠

Ž .

Proof. In view of the Burnside formula 2.3 it is sufficient to show that n!

 4

噛 g g ⭈⭈⭈ g s 1 N g g C ; S s^{1 2 k i i n}

Ý

_<Aut␲<.

Ž

2.5

.

␲ : Xª⺠1

 4

According to Fact 1, a solution g , g , . . . , g_{1 2 k} of the equation in the

Ž . ¹

left-hand side of 2.5 corresponds to a ramified covering␲ : X ª ⺠ and Aut␲ ( C g , g , . . . , g ,

Ž

_{1 2 k}

.

Ž .  4

where C g , g , . . . , g_{1 2 k} is the centralizer of the set g , g , . . . , g_{1 2 k} in S ._n

 4

Hence the number of solutions conjugate to g , g , . . . , g_{1 2 k} is equal to S : C g , g , . . . , g_n

Ž

_{1 2 k}

.

s n!

<Aut␲<

Ž .

and 2.5 follows.

The same arguments give a similar result for Eisenstein number of coverings of an arbitrary Riemann surface Y.

THEOREM2.2. In the pre_¨ious notation,

< < < < < <

1 C₁ C₂ ⭈⭈⭈ C_k ␹ g ␹ g ⭈⭈⭈ ␹ g

Ž

₁

. Ž

₂

. Ž

_k

.

s .

Ž

2.6

.

Ý

<Aut␲<

Ž

n!

.

2y2 gY

Ý

␹

Ž

␹ 1

Ž . .

ky 2y2 gŽ Y.

␲ : XªY

Theorems 2.1 and 2.2 may be used in both directions, i.e., information on coverings may be transformed into information on characters and vice versa. When the structure of the covering is known, it is easier to carry information on coverings to characters. In the next we give some examples.

3. EXPLICIT FORMULAE

There exist several remarkable cases in which the number of coverings can be evaluated explicitly. In essence these are the elliptic and parabolic

y1Ž . coverings␲ : X ª Y with equal ramification indices in each fiber ␲ y . Let us consider such a covering ␲ : X ª ⺠¹ of Riemann sphere unrami- fied outside y , y , . . . , y with ramification index m in points over y . To_{1 2 k i i}

Ž Ž ..

be elliptic or hyperbolic it should satisfy the inequality cf. 1.2

1 1 1

q q ⭈⭈⭈ q G k y 2.

m₁ m₂ m_k

It turns out that all such coverings may be explicitly described in terms of finite groups of Mobius transformations or affine Coxeter groups. Since

¨

Ž . the structure of these groups is known, explicit formulae for 2.4 can be obtained.

3.1. Elliptic Case

In this case we have strict inequality,

1 1 1

q q ⭈⭈⭈ q ) k y 2,

Ž

3.1

.

m₁ m₂ m_k

which admits the following values of m :_i Cyclic: ks 2, m s m s m;1 2

Dihedral: ks 3, m s m s 2, m s m;1 2 3

Tetrahedral: ks 3, m s 2, m s m s 3;1 2 3

Octahedral: ks 3, m s 2, m s 3, m s 4;1 2 3

Icosahedral: ks 3, m s 2, m s 3, m s 5.1 2 3

Remark 3.1. The fundamental group of a sphere with two punctures is

⺪. Hence in the case k s 2 there should be m s m , and we may1 2

Ž .

disregard other solutions of 3.1 . To each of these cases corresponds a unique covering that may be described in terms of a finite group of Mobius

¨

Ž . transformations we use to label solutions of 3.1 .

PROPOSITION 3.2. Let ␲ : ⺠¹ª ⺠¹ be a co_¨ering unramified outside y , . . . , y1 k with ramification index m in each point oi _¨er y . Theni ␲ is isomorphic to factorization␲ : ⺠_G ¹ª ⺠¹rG ( ⺠¹ by a finite group G of Mobius transformations.

¨

Ž . Proof. Let m , m , . . . , m be one of the above solutions of 3.1 . Then_{1 2 k}

Ž .

the group Gs G m , m , . . . , m defined by relations1 2 k

g1^m1s g2^m2s ⭈⭈⭈ s gk^mks g g ⭈⭈⭈ g s 11 2 k

is finite and isomorphic to the rotation group of the polyhedron we use for labeling the solution. Now observe that the only transitive permutation representation of G for which the element g splits into cycles of length_i m is regular. Hence the covering with ramification indices m is unique. It_{i i} may be constructed from realization of G as a group of Mobius transfor-

¨

mations.

In combination with Theorem 2.1 this implies THEOREM3.3. The following indentities hold:

2 n nr m

Cyclic:

Ý

_␹ ␹ ␴

Ž

m

.

s

ž /

m !m ;

2 nr2 2 nr m

␹ ␴

Ž

₂

.

␹ ␴

Ž

_m

. Ž

nr2 !2

. Ž

nrm !m

.

Dihedral:

Ý

␹ ␹ 1

Ž .

s n! 2 m

Ž .

nr2 m

Ž

nr2m !

.

; Tetrahedral:

2 nr3 2 nr2

␹ ␴

Ž

3

.

␹ ␴

Ž

2

. Ž

nr3 !3

. Ž

nr2 !2

.

s ;

Ý

_␹ ␹ 1

Ž .

n! 12

Ž .

nr12

Ž

nr12 !

.

Octahedral:

nr2 nr3 nr4

␹ ␴ ␹ ␴ ␹ ␴

Ž

2

. Ž

3

. Ž

4

. Ž

nr2 !2

. Ž

nr3 !3

. Ž

nr4 !4

.

s ;

Ý

_␹ ␹ 1

Ž .

n! 24

Ž .

nr24

Ž

nr24 !

.

Icosahedral:

nr2 nr3 nr5

␹ ␴ ␹ ␴ ␹ ␴

Ž

₂

. Ž

₃

. Ž

₅

. Ž

nr2 !2

. Ž

nr3 !3

. Ž

nr5 !5

.

s ,

Ý

_␹ ␹ 1

Ž .

n! 60

Ž .

nr60

Ž

nr60 !

.

where the summations are taken o_¨er all irreducible characters␹ of S and ␴n m

denotes a permutation consisting of nrm cycles of length m.

Proof. Let as before m , m , . . . , m be one of the above solutions of_{1 2 k} Ž3.1 , let G be the corresponding group of Mobius transformations, and let.

¨

␲ : X ª ⺠¹ be a covering with ramification indices m . Then each compo-_i nent X of X should be elliptic, i.e., isomorphic to_i ⺠¹, and by the previous proposition the restriction ␲ : X ª ⺠_{i i} ¹ is isomorphic to the factorization

1 1 < <

⺠ ª ⺠ rG. So for each n such that G divides n there exists only one such covering␲ and Aut ␲ is a wreath product of G and symmetric group

< <

S , mm s nr G , of permutation of the components of X. As result the Ž .

left-hand side of the formula 2.4 contains only one term,

1 n

, ms ,

< <m < <

m! G G

and the theorem follows.

3.2. Parabolic Case In this case we have

k 1

s k y 2

Ž

3.2

.

Ý

_m

is1 i

with the solutions

A1= A :1 ks 4, m1s m s m s m s 2;2 3 4

B :2 ks 3, m1s 2, m2s m s 4;3

G :2 ks 3, m1s 2, m2s 3, m3s 6;

A :2 ks 3, m1s m s m s 3.2 3

Ž .

Let as before Gs G m , m , . . . , m be a group defined by the relations1 2 k

g1^m1s g2^m2s ⭈⭈⭈ s gk^mks g g ⭈⭈⭈ g s 1,1 2 k

Ž .

where m , m , . . . , m is one of the solutions of 3.2 . It turns out that the_{1 2 k} w x

group G has a remarkable geometrical interpretation 3 as the group of

Ž .

even elements in affine Coxeter group ⌫ s ⌫ m , m , . . . , m generated1 2 k

by reflections in the sides of a plane k-gon with angles ␲rm . In otheri

words, ⌫ is an affine Weyl group of the root system used for labeling Ž .

solutions of 3.2 . In this interpretation the generators g are just rotations_i

Ž .

by angle 2␲rm around vertices of the k-gon. Let T ; G be the normali

subgroup of translations. T is the unique maximal torsion-free subgroup of

w x

G and index ␮ s G:T is finite.

PROPOSITION 3.4. Connected parabolic co_¨erings ␲ : X ª ⺠¹ with con-

y1Ž .

stant ramification indices m in singular fibers_i ␲ y , i_i s 1, 2, . . . , k, ha¨e

w x

degree ns m␮ di¨isible by ␮ s G:T . Such co¨erings are parametrized by

w x

conjugacy classes of translation subgroups H; T of index m s T:H . Proof. There exists a one-to-one correspondence between connected coverings ␲ : X ª ⺠¹ of degree n and conjugacy classes of subgroups in

Ž ¹  4.

␲ ⺠ _ y , . . . , y_{1 1 k} of index n. If the monodromy around y has order_i m , then we can deal with subgroups Hi ; G of G instead of␲ . We claim1

that the generators g split into cycles of the same length m in Gi i rH if and only if H; T, i.e., H consists of translations. Really

g splits into cycles of length m_{i i}m g_i^kgH/ gH, kk 0 m , ᭙g g H

Ž

_i

.

m g^y1gi^kgf H,

i.e., H contains no elements conjugate to g_i^k. But from the above geomet- rical interpretation of G we know that all elements of finite order are conjugate to some g . Hence H is torsion-free and the result follows._ik

PROPOSITION3.5. In the abo_¨e notation the Eisenstein number of parabolic connected co_¨erings␲ : X ª ⺠¹ of degree n␮ is equal to

1 1

s d,

Ý

_{<Aut␲< n␮}

Ý

1 dNn

␲ : Xª⺠

w x Ž .

where␮ s G:T , T is the translation subgroup of G s G m , . . . , m .1 k

Proof. We will need the following information:

Ž .i If G acts transitively on a set Y, then AutG

Ž

Y

.

( N G rG ,G

Ž

y

.

y

Ž .

where N GG y is the normalizer of stabilizer G of a point yy g Y.

Ž .ii Let H; T be a sublattice of index n in a lattice T of rank 2.

Then

w x

噛 H ; T N T :H s n s

 4 Ý

d

dNn

Žsee 14 .w x.

Using i and Proposition 3.4 we can write the Eisenstein number ofŽ . connected parabolic coverings as

1 1 1

s s G: N

Ž

H

.

,

Ž

3.3

.

Ý

_<Aut␲<

Ý

_{N H : H n␮}

Ý

^G

Ž .

1 H;G G H;G

␲ : Xª⺠

where summation on the right-hand side extends over conjugacy classes of torsion-free subgroups H; G of index n␮. Since such a subgroup H is contained in the translation group T, which is a lattice of rank 2, we can use ii and end the proof asŽ .

1 1 1

w x

G: N

Ž

H

.

s 噛 H ; T N T :H s n s

 4

d.

Ý

^G

Ý

n␮ _H;G n␮ n␮ _{d n<}

So we have a simple formula for the Eisenstein number of connected parabolic coverings. The number of all coverings may be easily deduced

Ž w x.

from here cf. 4 .

LEMMA 3.6. For co_¨erings ␲ : X ª Y with gi¨en constant ramification

y1Ž .

indices m in singular fibers_i ␲ y , the identity_i

q^deg␲ q^deg␲

s exp ,

Ž

3.4

.

Ý

^␲ _<^Aut␲_<

ž

^connected

Ý

^␲ _<^Aut␲_<

/

holds, where the first sum runs o_¨er all co_¨erings ␲ : X ª Y connected orŽ not , while the second sum extends only o. _¨er co_¨erings with connected sur- face x.

Proof. Let Xs " d X be a disjoint union of pairwise nonisomorphici i i

<

connected components X of multiplicity d and leti i ␲ s ␲ . Theni Xi

<Aut␲ s Ł Aut ␲< i< i<^diŽd ! and degi. ␲ s Ý d deg ␲ . Hence the left-handi i i

Ž .

side of 3.4 may be written as

q^dideg␲i q^ddeg␲ q^deg␲

s sexp .

Ý Ł

^{␲ i <Aut␲ d !i<}d_i ^{i connected}

Ł

^␲d

Ý

^{G0 <Aut␲ d!<}d

ž

^connected

Ý

^␲ <^Aut␲<

/

Combining this lemma with the previous proposition, we get the follow- ing formula.

COROLLARY 3.7. The Eisenstein number of elliptic co_¨erings␲ : X ª ⺠¹ of degree n with gi_¨en constant ramification indices m in singular fibers_i

y1Ž .

␲ y_i is equal to

1 ⬁ _y1r␮

n k␮

s coefficient at q in

Ž

1y q

.

,

Ý

_<Aut␲<

Ł

ks1

␲ : Xª⺠1

w x

where ␮ s G:T and T is the translation subgroup of G s

Ž .

G m , m , . . . , m ._{1 2 k}

Proof. Let Ns n␮, n G 1. Then

q^deg␲ 1 1

N N

s q s q d

Ý

_<Aut␲<

Ý Ý

_<Aut␲<

Ý

_N

Ý

connected␲ N deg␲sN N d n<

1 q^{␮m d} 1 _␮m

s _␮

Ý Ý

_d s y_␮

Ý

log 1

Ž

y q

.

.

Ž

3.5

.

mG1 dG1 mG1

In the second equality we use Proposition 3.5. The corollary follows by taking exponents and applying Lemma 3.6.

w x Ž .

Here are values of the index ␮ s G:T for each solution of 3.2 :

A1= A :1 ␮ s 2,

B :2 ␮ s 4,

G :2 ␮ s 6,

A :2 ␮ s 3,

Ž .

Remark 3.8. The right-hand side of 3.5 in essential is a negative power of Dedekind␩ function, so we may apply the Rademacher analytic

w x

formula 11 for its Fourier coefficients. As a simple corollary we get the

⬁ Ž following asymptotics for nonzero coefficients of the series ⌸ks11y

k␮.^y1r␮

q :

exp

Ž

␲r6␮ 24n

' .

'

an; 12␮ 3r4q1r4␮ , n' 0 mod

Ž

␮ .

. Ž

3.6

. Ž

24 n

.

Combining Corollary 3.7 with Theorem 2.2, we get a number of ‘‘strange’’

identities.

THEOREM3.9. The following identities hold:

A1= A :1

4 4 n 4 ⬁

␹ ␴

Ž

₂

.

2

Ž

n!

.

_{n k y1r2}

s coeff . at q in

Ž

1y q

.

,

Ý

_{␹ 1}

Ž .

2

Ž Ž

2 n !

. .

2

ž

ks1

Ł /

␹gS2 n

B :₂

2 6 n 2 ⬁

␹ ␴ ␹ ␴

Ž

2

. Ž

4

.

s2

Ž

2 n ! n!

. Ž .

coeff . at q inn

Ž

1yqk

.

_y1r4 ,

Ý

␹ 1

Ž . Ž

4 n !

. ž

k

Ł

s1

/

␹gS4 n

G :₂

␹ ␴ ␹ ␴ ␹ ␴

Ž

2

. Ž

3

. Ž

6

.

2^{4 n}3^{3 n}

Ž

3n ! 2 n !n!

. Ž .

Ý

_{␹ 1} s _{6 n !}

Ž . Ž .

␹gS6 n

⬁ y1r6

n k

= coeff. at q in

ž

ks1

Ł Ž

1y q

. /

,

3 3 n 3 ⬁

␹ ␴

Ž

3

.

3

Ž

n!

.

n k _y1r3

A :2 ␹gS

Ý

3n ␹ 1

Ž .

s

Ž

3n !

. ž

coeff . at q in k

Ł

s1

Ž

1y q

. /

,

where the permutation␴ splits into cycles of length m.m

Ž .

Remark 3.10. It is known that ␹ ␴ s 0 except the Young diagram ␭␭ m

is divisible by m. In the last case the value of character is given by the w x

hook formula 18

nrm !

Ž .

␹ ␴ s "_␭

Ž

m

.

,

Ž

3.7

.

Łmyhooks i jh rm

where hi j are hook lengths of ␭. The first identity just means that product of odd hooks 2

Ý

product of even hooks

domino diagrams

n k y1r2

s coeff. at q in

Ł Ž

1y q

.

,

k

where the sum is extended over all e_¨en diagrams of order 2 n i.e.,Ž diagrams which may be tiled by n dominos ..

Ž .

Let us observe that in view of estimation 3.6 the character sums of the theorem are of subexponential growth in n, while the dimension ␹ 1Ž .

Ž .^1r2 generically is superexponential. Hence the first identity implies that ␹ 1

Ž .

in general is a good estimation for ␹ ␴ . We may suppose that a similar2

Ž . Ž .

result is valid for ␹ ␴ , as suggested by formula 3.7 , and moreoverm 1rdq␧

<␹ g - ␹ 1

Ž .

<

Ž .

for ‘‘general’’ character ␹ and n ª ⬁. Here d is the mean value of cycle length of gg S . The characters we consider in the next section haven

exponential growth of dimension and thus are not general.

4. ASYMPTOTIC FORMULAE

Let␲ : X ª Y be a ramified covering of degree n, of surface Y of genus g_Y by surface X of genus g , ramified over k points y , . . . , y in Y. The_{X 1 k} Riemann᎐Hurwitz formula connecting the genus of X and Y may be written in the form

1 1 1

2 gXy 2 s n 2 y 2 q k yg Y y ⭈⭈⭈ y ,

Ž

4.1

.

l₁ l₂ l_k

y1Ž . where l is the mean value of ramification indices in the fiber_i ␲ y_i Žequal to the mean value of the cycle length of the monodromy g around_i y . Hence, Theorem 2.2 may be written as_i.

1

Ý

_<Aut␲<

␲ : XªY

< <C1 ⭈⭈⭈ C< <k Ž2y2 g rnX. ␹ g

Ž

1

.

␹ g

Ž

2

.

␹ g

Ž

k

.

s

Ž

n!

.

2y2 gY

Ý

_␹ ␹ 1

Ž .

␹ 1

Ž .

1rl1 ␹ 1

Ž .

1rl2 ⭈⭈⭈ ␹ 1

Ž .

1rlk.

Ž .^{Ž2ygX.r n}

If the genus g_X is fixed, then the first factor ␹ 1 decreases or increases at most as some power of n. So the main problem is in

Ž . Ž .^{1r l}

estimation of the quotient ␹ g r␹ 1 , where l is the mean value of cycle length of gg S , as n ª ⬁.n

Let us denote by

␤ : b1G b G ⭈⭈⭈ G b2 m

Ž

4.2

.

the Young diagram of ␹ s ␹ and by ␴ g S an element with cycle␤ ␣ n

structure

␣ : 1^a12^a2 ⭈⭈⭈ s^as.

Ž

4.3

.

In subsequent text, we restrict ourselves to sequences of characters ␹ and_␤ elements ␴ g S subject to the following conditions:_␣ n

Condition 1. Diagrams ␤ have a fixed number of rows and permuta- tions ␴ contain only cycles of bounded length._␣

Condition 2. Lengths of rows in ␤ increase linearly with n, i.e., bis␤ n.i

Condition 3. Multiplicities of cycles in ␴ g S increase linearly with n,_␣ n

i.e., aks␣ n.k

Ž . Under these conditions we will find asymptotics for ␹ ␴_{␤ ␣} and Ž . Ž .^{1r d}

␹ ␴ r␹ 1_{␤ ␣ ␤} . It follows that the last quotient exponentially increases for nª ⬁, provided not all rows of ␤ are equal and not all cycles of ␴␣

are of the same length.

4.1. Frobenius Formula

Let z , z , . . . , z_{1 2 m} be independent variables and let

␦ s m y 1, m y 2, . . . , 0 ,

Ž .

z^␤q␦s z₁^{b1qm y1}z₂^{b2qm y2} ⭈⭈⭈ z_m^bm,

s_js z₁^jq ⭈⭈⭈ qz_m^j,

⌬ s ⌬ z , . . . , z

Ž

^{1 m}

.

s

Ł Ž

zⁱy z .^j

.

i-j

In this notation we have the following Frobenius formula for irreducible characters:

␹ ␴ s coefficient at z␤

Ž

␣

.

^␤q␦ in⌬

Ł

s^aj^j.

Ž

4.4

.

j

Making use of the residue theorem we can rewrite it in integral form,

1 _{n⭈␻ Ž z.}

␹ g s␤

Ž .

^m

H

⭈⭈⭈

H

g z e

Ž .

dz,

Ž

4.5

.

< < < <

2␲ i

Ž .

z1s1 zms1

where

zs z , . . . , z

Ž

_{1 m}

.

g ⺓^m

⌬ z , . . . , z

Ž

1 m

.

zj

g z

Ž .

s ^{z z}1m 2my1 ⭈⭈⭈ zm s

Ł

i-j

ž

1y ^zi

/

,

Ž

4.6

.

n m

␻ z s

Ž . Ý

␣ log s yk k

Ý

␤ log z .i i

ks1 is1

4.2. Asymptotics of Characters w x

According to general principles 5 , in order to find asymptotics of the Ž .

integral 4.5 , we have to deform the surface of integration S in such a way that it passes through a critical point z of0 ␻ z with maximal value Re ␻.Ž .

Ž . Ž .

The critical points of ␻ z are given by the equation d␻ z s 0, which in coordinates looks like the system of nonlinear algebraic equations

z_i^k

k␣ s␤ , is 1, . . . , m.

Ž

4.7

.

Ý

k ^kz1kq ⭈⭈⭈ qzmk ⁱ

Ž .

The system 4.7 is homogeneous in z and thus by the Bezout theorem has a lot of complex solutions. The first problem is to understand which of them is responsible for the asymptotics. For this the following result is crucial.

Ž .

THEOREM4.1. For ␤ G ␤ G ⭈⭈⭈ G ␤ G 0 the system 4.7 has, up to1 2 m

Ž .

proportionality, unique positi_¨e real solution xs x , x , . . . , x_{1 2 m} and for this solution, x1G x G ⭈⭈⭈ G x G 0.2 m

Proof. Let us put

zis e^ti, tig ⺢, so that in variable t ,_i

n m

t_i

␻ e s

Ž . Ý

␣ log exp kt q ⭈⭈⭈ qexp kt^k

Ž Ž

¹

. Ž

^m

. .

y

Ý

␤ t^{i i}

ks1 is1

and the equation for critical points of w takes the form exp kt

Ž

i

.

k␣ s␤ , is 1, 2, . . . , m.

Ž

4.8

.

Ý

k ^k_{exp kt}

Ž

1

.

q ⭈⭈⭈ qexp kt

Ž

m

.

ⁱ

The proof may be divided into three steps.

Ž .

Step 1. ␻ is a convex function of t s t , . . . , t , i.e.1 m

⭸²␻

Hess

Ž

␻ s

. Ý

_{⭸ t ⭸ t} X X^{j i}G 0, ᭙X , X g ⺢.^{i j}

i j

i , j

Proof. Really, since

n ⭸²

Hess

Ž

␻ s

. Ý

␣^k

Ý

_{⭸ t ⭸ t} log exp kt

Ž Ž

¹

.

q ⭈⭈⭈ qexp kt

Ž

^m

. .

X X ,^{j i}

i j

ks1 i , j

it suffices to show that

⭸²

h t

Ž .

s

Ý

_{⭸ t ⭸ t} log exp t

Ž Ž

¹

.

q ⭈⭈⭈ qexp t

Ž

^m

. .

X X^{j i}G 0.

Ž

4.9

.

i j

i , j

A simple calculation gives

2 2

Ý exp t_i

Ž .

_i X_i Ý exp t_i

Ž .

_i X_i

h t

Ž .

s ^{Ý exp ti}

^{Ž .}

ⁱ y

ž

^{Ý exp ti}

^{Ž .}

ⁱ

/

G 0

Ž

4.10

.

and equality is possible only for X1s X s ⭈⭈⭈ s X .2 m

From Step 1 it follows that Ž .

Step 2. Restriction of Hess ␻ on the hyperplane Ý X s 0 is positivei i

and hence the mapping

⭸ f ⭸ f

⍀: t , . . . , t

Ž

_{1 m}

.

ª

ž

^{⭸ t1}, . . . , ^{⭸ tm}

/

s

Ž

␤ , . . . , ␤ ,_{1 m}

.

f t

Ž .

s

Ý

␣ log exp kt q ⭈⭈⭈ qexp kt^k

Ž Ž

¹

. Ž

^m

. .

,

k

is locally invertible on the hyperplane Ýtis 0.

Step 3. The mapping

⍀: log x , log x , . . . , log x

Ž

1 2 m

.

ª

Ž

␤ , ␤ , . . . , ␤ ,1 2 m

.

x1q x q ⭈⭈⭈ qx s 1,2 m xiG 0,

gives a homeomorphism between simplexes

 4

⌬ s x N x q x q ⭈⭈⭈ qx s 1, x G 0 ,x 1 2 m i

 4

⌬ s_␤ ␤ N ␤ q ␤ q ⭈⭈⭈ q␤ s 1, ␤ G 0 .1 2 m i

Proof. We will proceed by induction on m. Without loss of generality we may suppose restriction of ⍀ on a face of the simplex ⌬ ,x

⍀ : log x , . . . , log xi

Ž

1 iy1, 0, log xiq1, . . . , log xm

.

ª

Ž

␤ , . . . , ␤1 iy1, 0,␤iq1, . . . ,␤ ,m

.

to be a homeomorphism. Then ⍀ induces a homeomorphism of the boundaries ⭸ ⍀: ⭸⌬ ª ⭸⌬ , and by Brouwer theorem the mapping ⍀ isx ␤

surjective. Combining this with Step 2 we find out that ⍀ is an unramified covering of ⌬ . Since the simplex ⌬ is simply connected, ⍀ is in fact a_{␤ ␤} homeomorphism.

Ž .

The theorem follows from the Step 3, because ⍀ t , . . . , t_{1 m} is just the Ž .

left-hand side of 4.8 .

It turns out that the positive critical point x from Theorem 4.1 is Ž .

responsible for asymptotics of characters ␹ ␴ ._{␤ ␣}

THEOREM4.2. Suppose in addition to Conditions 1᎐3 that rows of Young diagrams␤ ha¨e distinct lengths␤ / ␤ and lengths of all cycles in_{i j} ¨ol_¨ed in

Ž .

␴ g S are coprime. Then, as n ª ⬁ the characters ␹ ␴_{␣ n ␤ ␣} ha_¨e the asymptotics

nw Ž x .

'

e m x_j 1

␹ ␴ ;␤

Ž

␣

. Ž

2␲ n

.

Žmy1 r2.

Ł

ⁱ-j

ž

1y ^xi

/ '

^Ýmis1 ^Hi i , where x is a positi_¨e critical point from Theorem 4.1,

n m

k k k

␻ x s

Ž . Ý

␣ log x q x q ⭈⭈⭈ qx^k

Ž

1 2 m

.

y

Ý

␤ log x ,i i

Ž

4.11

.

ks1 is1

and H is ith principle minor of the form_{i i}

m k 2 m k 2

n Ý_{is1 i}x dt_i Ý_{is1 i}x dt_i

Hess

Ž

␻ s

.

^k

Ý

^s1k2␣^k

ž

^{Ý xi} kⁱ y

ž

^Ým^{is1 ix}k

/ /

.

Ž

4.12

.

Proof. Step 2 below is crucial. It ensures that only critical points proportional to the positive one are essential for asymptotics. The rest is

w x

an exercise in the multidimensional saddle point method 5 , with a minor complication due to nonisolated critical points.

Step 1. Deformation of the surface of integration. By deformation of Ž .

the contour in integral 4.5 we can write

1 _{n␻ Ž z.}

␹ g s␤

Ž .

^m

H

⭈⭈⭈

H

g z e

Ž .

dz.

Ž

4.13

.

< < < <

2␲ i

Ž .

z1sx1 zmsxm

The surface of integration now passes through the positive critical point x.

Ž .

Step 2. The asymptotics of integral 4.13 depends on an arbitrary small

< <

neighborhood of the set of critical points zs␭ x, ␭ s 1 proportional to Ž Ž ..

the positive one x. It suffices to show that the maximum of Re ␻ z on

< < ^Ž␶i.

the contour zi s exp is attained only at ␭ x. In order to see this, write

< ^{k k k}< < <

Re

Ž

␻ z s

Ž . . Ý

␣ log z q z q ⭈⭈⭈ qz yk 1 2 m

Ý

␤ log z .i i

Ž

4.14

.

k i

Evidently,

<z1^kq z q ⭈⭈⭈ qz F z q z2^k m^k< < <1 ^k < <2^kq ⭈⭈⭈ q z< m<^ks x q x q ⭈⭈⭈ qx1^k 2^k m^k

and equality is possible only for collinear z_i^k. In the last case, zis␭␧ x ,i i ␧i^ks 1.

Ž .

This should be valid for all k that enter 4.14 with nonzero coefficient

␣ / 0, i.e., for all cycle lengths of ␴ g S which are supposed to bek ␣ n

coprime. As a result, we get ␧ s 1, and hence any critical point z withi

maximal value of Re␻ z should be proportional to x.Ž .

Step 3. The final formula. Let us write the positive critical point in the form

x , x , . . . , x s exp ␶ , exp ␶ , . . . , exp ␶

Ž

1 2 m

. Ž Ž

1

. Ž

2

. Ž

m

. .

Ž .

and put in integral 4.13 ,

zjs exp

Ž

␶ q it ,j j

.

y␲ F t F ␲ .j

Then

␲ ␲

1 exp

Ž

␶ q itj j

.

␹ ␴ s␤

Ž

␣

. ^Ž

^2␲

^.

^m

H

^y␲⭈⭈⭈

H

^{y␲ i-j}

Ł ž

1y ^exp

^Ž

^{␶ q iti i}

^. /

exp n

Ž

␻ t

Ž . .

dt,

Ž . Ž ^␶qit.

where ␻ t denotes ␻ e . Let

exp

Ž

␶ q itj j

.

F t

Ž .

s

Ł

^i-j

ž

1y ^exp

^Ž

^{␶ q iti i}

^. /

exp n

Ž

␻ t .

Ž . .

Observe that

F t , . . . , t

Ž

₁ m

.

s F t q a, . . . , t q a ,

Ž

1 m

.

᭙a g ⺢,

so that F t is a constant on lines parallel to the main diagonal tŽ . 1s t s2

⭈⭈⭈ s t of the m-dimensional cube y␲ - t - ␲. According to Step 2 them i

asymptotics of the integral depends only on an arbitrary small neighbor- hood of the main diagonal. Hence asymptotically

2␲ m

'

␹ ␴ ;␤

Ž

␣

.

^m

H

F t dt,

Ž .

2␲

Ž .

H_␧

where integration is over a neighborhood of origin in the hyperplane H:

Ý ti is 0 orthogonal to the diagonal, and the extra multiplier 2␲ m is

'

equal to the length of the diagonal. The asymptotics of the last integral were determined by the isolated critical point ts 0, so by the saddle point method we get

n␻ Ž x.

'

e m x_j 1

␹ ␴ ;␤

Ž

␣

. Ž

2␲ n

.

Žmy1 r2.

Ł

i-j

ž

1y ^xi

/ '

^{det Hess}

Ž

␻

.

<H

Ž .

x

n␻ Ž x.

'

e m x_j 1

s

Ž

2␲ n

.

Žmy1 r2.

Ł

ⁱ-j

ž

1y ^xi

/ '

^Ýmis1^Hi i , Ž . where H are the principal minors of the Hessian Hessi i ␻ .

EXAMPLE 4.3. As a simple illustration of Theorem 4.2, let us deduce the asymptotic formula for dimension,

e^{n H Ž␤ .}Łi- j

Ž

1y␤ r␤j i

.

␹ 1 ;_␤

Ž .

_{Žmy1 r2.} ,

2␲ n ␤ ␤ ⭈⭈⭈ ␤

Ž . '

1 2 m

Ž .

where H ␤ s yÝ ␤ log ␤ is the entropy function.i i i