A Proof of a Conjecture for the Number of Ramified Coverings of the Sphere by the Torus

Transcription

1 Journal of Cobnatoral Theory, Seres A 88, 4658 (999) Artcle ID jcta99999, avalable onlne at httpwwwdealbraryco on A Proof of a Conjecture for the Nuber of Rafed Coverngs of the Sphere by the Torus I P Goulden and D M Jackson Departent of Cobnatorcs and Optzaton, Unversty of Waterloo, Waterloo, Ontaro, Canada Councated by the Managng Edtors Receved Deceber 8, 998 An explct expresson s obtaned for the generatng seres for the nuber of rafed coverngs of the sphere by the torus, wth eleentary branch ponts and prescrbed rafcaton type over nfnty Ths proves a conjecture of Goulden, Jackson, and Vanshten for the explct nuber of such coverngs 999 Acadec Press INTRODUCTION Let X be a copact connected Reann surface of genus g0 A rafed coverng of S of degree n by X s a non-constant eroorphc functon f X S such that f & (q) =n for all but a fnte nuber of ponts q # S, whch are called branch ponts Two rafed coverngs f and f of S by X are sad to be equvalent f there s a hoeoorphs? X X such that f = f b? A rafed coverng f s sad to be sple f f & (q) =n& for each branch pont of f, and s alost sple f f & (q) =n& for each branch pont wth the possble excepton of a sngle pont, that s denoted by The preages of are the poles of f If,, are the orders of the poles of f, where }}}, then =(,, ) s a partton of n and s called the rafcaton type of f Let + (g) () be the nuber of alost sple rafed coverngs of S by X wth rafcaton type The proble of deternng an (explct) expresson for + (g) () s called the Hurwtz Enueraton Proble The purpose of ths paper s to prove the followng result for the torus, gvng an explct expresson for + () () for an arbtrary partton =(,, ) Theore was prevously conjectured by Goulden et al n [4] where t was proved for all parttons wth 6, and for the partcular partton ( ), for any Let C be the conjugacy class of the syetrc group S n on n sybols ndexed by the partton of n Copyrght 999 by Acadec Press All rghts of reproducton n any for reserved 46

2 RAMIFIED COVERINGS BY THE TORUS 47 Theore + () ()= C 4 n! (n+)! \ ` = ( &)!+\ n &n & & = (&)! e n &+ where e s the th eleentary syetrc functon n,, and e = +}}}+ =n Prevously Hurwtz [5] had stated that, for the sphere, + (0) ()= C (n+&)! n &3 n! \ ` = ( &)!+ () A proof was of ths sketched by Hurwtz [5] It was frst proved by Goulden and Jackson [] (see also Strehl [6]) The approach developed by Hurwtz s outlned n the next secton Very recently Vakl [7] has gven an ndependent proof of Theore He develops, by technques n algebrac geoetry, and solves a recurrence equaton that s copletely dfferent n character fro the one obtaned fro the dfferental equaton n ths paper HURWITZ'S COMBINATORIALIZATION OF RAMIFIED COVERINGS Hurwtz's approach was to represent a rafed coverng f of S, wth rafcaton type, by a cobnatoral datu (_,, _ r ) consstng of transpostons n S n, whose product? s n C, such that (_,, _ r ) acts transtvely on the set [,, n] of sheet labels and that r=n++ (g&), where =l(), the length of The latter condton s a consequence of the Reann-Hurwtz forula Under ths cobnatoralzaton he showed that + (g) ()= C n! c g(), where c g () s the nuber of such factorzatons of an arbtrary but fxed? # C He studed the effect of ultplcaton by _ r on _ }}}_ r& to derve a recurrence equaton for c g () The dffculty wth Hurwtz's approach s that the recurrence equatons for c g () are ntractable n all but a sall nuber of specal cases It appears that hs approach can be ade ore tractable by the ntroducton of cut operators and jon operators that have been developed for cobnatoral purposes by Goulden [], Goulden and Jackson [], and

3 48 GOULDEN AND JACKSON Goulden et al [4] These are partal dfferental operators n ndeternates p, p, that take account of the enueratve consequences of the ultplcaton by _ r on \=_ }}}_ r&, when sued over all such ordered transtve factorzatons There are two cases The acton of _ r on \ s ether to jon an -cycle and a j-cycle of \ to produce an + j-cycle, or to cut an + j-cycle of \ to produce an -cycle and a j-cycle In the frst case the operators are the jon operators p + j and p + p p j j\ p +\ p j+, and n the second case the operator s the cut operator p p j p + j A ``cut-and-jon'' analyss of the acton of _ r on \ therefore leads to a nonhoogeneous partal dfferental equaton n a countably nfnte nuber of varables (ndeternates) for the generatng seres 8 for c g () The type of 8 s deterned by the cobnatoral propertes of the cut-and-jon analyss The advantage of ths approach to the Hurwtz Enueraton Proble s that t facltates the transforaton of the dfferental equaton for 8 by an plct change of varables The seres that s nvolved wth ths transforaton s denoted by s=s(x, p) throughout, where p=(p, p, ), and appears to be fundaental to the proble 3 THE DIFFERENTIAL EQUATION Let p = p }}}p where =(,, ) Let 8(u, x, z, p)= n, g0 &n l()= C c g () u n++(g&) x n (n++(g&))! n! z g p, () the generatng seres for c g (), where &n sgnfes that s a partton of n It was shown n [4] that f =8(u,,z, p) satsfes the partal dfferental equaton f u =, j\ jp + jz f p p j +jp +j f f +(+j) p p j p p j f p +j+ (3)

4 RAMIFIED COVERINGS BY THE TORUS 49 By replacng p by x p for t s readly seen that f =8(u, x, z, p) satsfes (3) But 8(u, x, z, p)#q[u, z, p][[x]], and t s also readly seen that (3) has a unque soluton n ths rng Let F (x, p)=[z ] 8(, x, z, p) for =0,, where [z ] f denotes the coeffcent of z n the foral power seres f Then F 0 s the generatng seres for the nubers c 0 (), whch have been deterned by Hurwtz, so F 0 s known F s the generatng seres for c () The next result gves the lnear frst order partal dfferental equaton for F that s nduced by restrctng (3) above to ters of degree at ost one n z Lea 3 The seres f =F satsfes the partal dfferental equaton T 0 f &T =0, (4) where T 0 =x x + p T =, j F 0 & jp + j p p, j jp + j F 0 p p j Proof Clearly, fro (3), & p j (+j) p p j, j u u [z] 8=[z] \ x x + p p + 8 p +j, The result follows by applyng [z] to(3) K We now turn our attenton to solvng ths partal dfferental equaton Let G (x, p)= 4 n, &n l()= C \ ` = ( &)!+ _ &n & & \n = (&)! e n + & xn n! p (5) Snce (3) has a unque soluton n Q[u, z, p] [[x]], then (4) has a unque soluton n Q[p] [[x]] To establsh Theore t therefore suffces to show that f =G satsfes (4) (note that G has a constant ter of 0, so the ntal condton s satsfed)

5 50 GOULDEN AND JACKSON 4 THE GENERATING SERIES G To obtan a convenent for for G the followng lea s requred that expresses the eleentary syetrc functon e k (*) as the coeffcent n a foral power seres For a partton =(,, r ) let denote the nuber of occurrences of n, and we ay therefore wrte =( }}}r r) Let ()=> r =! Let P denote the set of all parttons wth the null partton adjoned Lea 4 For any nonnegatve nteger k and partton *, e k (*)= (*) k! [ p * ]( p + p +}}}) k # P p () Proof Frst # P p () =,, 0 p! p! }}} and ( p + p +}}}) k =,, 0 + +}}}=k If *=( j j ), then (*)= j j! j j! }}} so (*) k! [ p * ]( p + p +}}}) k # P p () = k! p p }}}!!}}},,,, 0\ j + +}}}=k + \ j + }}}, where the su s further constraned by + = j, + = j, Then (*) k! [ p * ]( p + p +}}}) k # P p () =[tk ](+t) j (+t) j (+3t) j 3 }}} =e k (( j j }}})) and the result follows Let K (x, p)= r r & a r p r x r, (6)

6 RAMIFIED COVERINGS BY THE TORUS 5 where s an nteger and a r =r r (r&)!, for r Let s#s(x, p) be the unque soluton of the functonal equaton s=xe 0 (s, p) (7) n the rng Q[p] [[x]] An explct seres expanson of s can be obtaned by Lagrange's Iplct Functon Theore (see [3, Sect ], for exaple) Let denote (s, p) The next result gves an expresson for G explctly n ters of s, and ndcates the fundaental portance of the seres s to the soluton of the partal dfferental equaton gven n (3) Theore 4 G (x, p)= 4 log(& ) & & 4 0 Proof For a partton =(,, ), let a =a }}}a and g(x, p)= n &n l()= n & () a p x n, a consttuent of the seres G gven n (5), snce ()=n! C It s easly shown that x g x = x n [t n ] e n 0 (t, p) n so, by Lagrange's Iplct Functon Theore, But, fro (7), and fro (6), x g x = & x s x = s (8) & s = s + Then x(g& 0 )x=0 and, snce g(0, p)=0, t follows that g(x, p)= 0

7 5 GOULDEN AND JACKSON Next we consder the ters of G n (5) that are not ncluded n g(x, p) Frst, note that % # P p % (%) =exp p, so, replacng p by ntp a for n Lea 4, we have () k! where =l() Then [ p t n ](e 0 (t, p) ) n k (t, p)=n&k a e k (), a n & k (k&)! n &k a e k () =()[ p t n ] \& k =()[ p x n ] & \ k(k&) k (t, p) + (e 0 (t, p) ) n & k k(k&) + k by Lagrange's Iplct Functon Theore, for n Then a n & k (k&)! n &k a e k ()=()[p x n ](+log(& ) & ) The result follows by cobnng the two expressons that have been obtaned and by usng the fact that G (0, p)=0 K 5 THE PROOF OF THEOREM The reanng porton of the paper s concerned wth the proof of Theore Proof Our strategy s to show that the expresson for G gven n Theore 4 satsfes the partal dfferental equaton gven n (4) We begn by consderng the dervatves that are requred n the deternaton of T 0 G &T Fro the functonal equaton (7) s = a k s k+ (9) p k k &

8 RAMIFIED COVERINGS BY THE TORUS 53 Then, for k, j =k j& a k s k + a k j+ s k (0) p k k & The only dervatves of F 0 that are needed are F 0 = a k p k k 3 sk & a k k s k+r a r p r r k+r, () fro Proposton 3 of [] and, fro (9) and (), s + j F 0 = a a j p p j j + j, () for, j For copleteness we note that, fro Proposton 3 of [], \ x x+ F 0 = 0 The dervatves of G that are needed are, fro (9), and, fro (0), for k, G = p k 4 a k 4\ x G x = s k & + 4 (& ) & & + (3) a k k sk \ (& ) & & + (4) Then fro Lea 3 and expressons (), (), (3), and (4) t follows that 4(& ) (T 0 G &T )= (+ 0 )& 0 (& )&(& ) A where A=, j a a j + j p + js + j, &(& ) B+(& ) C&( + &) D +( + &) E, (5) B=, j a a j j p + j s + j,

9 54 GOULDEN AND JACKSON C=, j, a a j a j( j+) p + jp s + j+ &, j (+ j) a + j p p j s + j, D=, j a a j j p + j s + j, E=, j, a a j a j( j+) p + jp s + j+ &, j a + j p p j s + j When the expresson (5) s transfored by replacng p s by q, for, t s edately seen to be a polynoal n q, q, of degree 3 wth ratonal coeffcents If U denotes the degree part of the expresson, for =,, 3, the transfored expresson can be wrtten n the for where 4(& ) (T 0 G &T )=U +U +U 3, U = & 0 &A&B+D, U = 0 ( + )+4 A+ B+C&( + ) D&E, U 3 =& A& C+( + ) E Then U # H [q, q, ], the set of hoogeneous polynoals of degree n q, q, Let,, H [q, q, ] [ Q[x, x, ] be the syetrzaton operaton defned by,, (q }}}q )= x?() }}}x,?()? # S extended lnearly to H [q, q, ] Then,, f =0 ples that f =0 for f # H [q, q, ] We therefore prove that U =0 by provng that,, U =0, for =,, 3 To deterne the acton of the syetrzaton operator on A,, E t s convenent to ntroduce the seres w=w(x) as the unque soluton of the functonal equaton w=xe w (6) n the rng Q[[x]] By Lagrange's Iplct Functon Theore we have w= n n n& n! x n

10 RAMIFIED COVERINGS BY THE TORUS 55 Now let w =w(x )andw ( j) =(x x ) j w Then, fro (6), w () = w &w, w () = w (&w ) 3, w(3) = w +w (&w ) (7) 5 The acton of the syetrzng operator on A,, E and ther products wth can be deterned n ters of these as follows It s readly seen that For (A), usng (7), we have ( )=w (+), & (A)= k x k k [xk ](w () ) = x 0 (w () ) dx x = w 0 w (&w ) 5 dw so, by rearrangeent Trvally, (A)= ((&w ) w (3) +w w () &w() ) (B)=w (3) w Next,, (C) s the syetrzaton of, j, wth respect to x and x Now a a j a j( j+) x+ j x & (+ j) a + j x x j, j, j, a a j a j( j+) x+ j x =w(3) a x j =w (3) a x But, fro (7), and ntegratng by parts, we obtan x a j j x 0 x j j+ w () x& dx x 0 w () x& dx = w 0 w e&w dw = a \ &x = & (&)! &x w +

11 56 GOULDEN AND JACKSON Thus, j, a a j a j( j+) x+ j x =w (3) \ =w (3) \ =w (3) \ x x &x & x x &x & x = x [t ] e t & (&)! x & w &w () x &x w &w &w w &w () + \\ & t & w + & +&w() + by the Lagrange Iplct Functon Theore Moreover, t s easly seen that, j + (+ j) a + j x x j =x w (3) &x w (3) x &x Thus, by syetrzng the ndcated lnear cobnaton of these sus, we have Trvally,, (C)=&w (3) w () &w () w (3) & w(3) w() &w() w(3) w &w (D)=w () w Fnally,, (E ) s obtaned n a fashon slar to, (C) The expresson s &w(), (E )=&w () w() &w() w() w() &w() w() w &w These results ay be cobned to gve expressons for the syetrzatons of U, U, U 3 as follows For the ter of degree one, (U )=w (3) &w() &((&w ) w (3) +w w () &w() )&w(3) w +w () w

12 RAMIFIED COVERINGS BY THE TORUS 57 For the ter of degree two, after rearrangeent,, (U )=(w () w(3) +w(3) w() )(&w &w )+w () w() (w +w ) & w(3) w() &w() w &w w(3) + w() w() &w() w() w &w When ultpled by w &w and a sutable power of (&w ) & and (&w ) & ths becoes a polynoal n w and w that s dentcally zero For the ter of degree three, after rearrangeent,,, 3 (U 3 )= w &w 3 (w () (w(3) w() 3 &w() w(3) 3 ) &(w () +w(3) )(w() w() 3 &w () w() 3 )) + (w () w &w (w(3) w () 3 &w() w(3) ) 3 3 &(w () +w(3) )(w() w() 3 &w () w() 3 )) + (w () 3 w &w (w(3) w () &w() w(3) ) &(w () 3 +w(3) 3 )(w() w() &w () w() )) &w () w() w(3) (&w 3 3)&w () w() 3 w(3) (&w ) &w () w() 3 w(3) (&w )&w () w() w() (w 3 +w +w 3 ) When ultpled by (w &w )(w &w 3 )(w &w 3 ) and a sutable power of (&w ) &,(&w ) & and (&w 3 ) & ths becoes a polynoal n w, w and w 3 that s dentcally zero It s quckly seen that (U ) s zero For both, (U )and,, 3 (U 3 ), however, the polynoal expressons were suffcently large that t was convenent to use Maple to carry out the routne splfcaton of ths stage Thus the syetrzaton of 4(& ) (T 0 G &T )=0 so T 0 G & T =0 It follows fro Lea 3 that F =G and ths copletes the proof of Theore K ACKNOWLEDGEMENTS Ths work was supported by grants ndvdually to IPG and DMJ fro the Natural Scences and Engneerng Research Councl of Canada

13 58 GOULDEN AND JACKSON REFERENCES I P Goulden, A dfferental operator for syetrc functons and the cobnatorcs of ultplyng transpostons, Trans Aer Math Soc 344 (994), 4440 I P Goulden and D M Jackson, Transtve factorzatons nto transpostons and holoorphc appngs on the sphere, Proc Aer Math Soc 5 (997), I P Goulden and D M Jackson, ``Cobnatoral Enueraton,'' Wley, New York, I P Goulden, D M Jackson, and A Vanshten, The nuber of rafed coverngs of the sphere by the torus and surfaces of hgher genera, Ann of Cobn, to appear, athag A Hurwtz, Ueber Reann'sche Fla chen t gegebenen Verzwegungspunkten, Math Ann 39 (89), 60 6 V Strehl, Mnal transtve products of transpostonsthe reconstructon of a proof by A Hurwtz, Se Lothar Cobn 37 (996), Art S37c 7 R Vakl, Recursons, forulas, and graph-theoretc nterpretatons of rafed coverngs of the sphere by surfaces of genus 0 and, athco9805