New York Journal of Mathematics. Proof of the Refined Alternating Sign Matrix Conjecture

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1 New York Journal of Mathemati New York J. Math. 2 ( Proof of the Refined Alternating Sign Matrix Conjeture Doron Zeilberger Abtrat. Mill, Robbin, and Rumey onjetured, and Zeilberger proved, that the number of alternating ign matrie of order n equal A(n := 1!4!7! (3n 2! n!(n + 1! (2n 1!. Mill, Robbin, and Rumey alo made the tronger onjeture that the number of uh matrie whoe (unique 1 of the firt row i at the r th olumn equal ( n+r 2 ( 2n 1 r A(n n 1 n 1 ( 3n 2. n 1 Standing on the houlder of A. G. Izergin, V. E. Korepin, and G. Kuperberg, and uing in addition orthogonal polynomial and q-alulu, thi tronger onjeture i proved. Content Introdution 59 Boiling It Down To a Determinant Identity 60 A Short Coure on Orthogonal Polynomial 62 A Lean and Lively Coure in q-calulu 63 q-legendre Polynomial 65 Denouement 66 Referene 68 Introdution An alternating ign matrix, or ASM, i a matrix of 0, 1, and 1 uh that the non-zero element in eah row and eah olumn alternate between 1 and 1 Reeived November 8, Mathemati Subjet Claifiation. Primary: 05A; Seondary: 33. Key word and phrae. enumeration, alternating ign matrie, quare ie, Izergin-Korepin formula, orthogonal polynomial, q-analyi, q-legendre polynomial, exatly olvable model. Supported in part by the NSF State Univerity of New York ISSN /96

2 60 Doron Zeilberger and begin and end with 1, for example: Mill, Robbin, and Rumey [MRR1] [MRR2] ([S], Conj. 1 onjetured, and I proved [Z], that there are A(n := 1!4!7! (3n 2! n!(n + 1! (2n 1! alternating ign matrie of order n. Another, horter, proof wa later given by Greg Kuperberg [K]. Kuperberg dedued the traight enumeration of ASM from their weighted enumeration by Izergin and Korepin [KBI]. In thi paper, I extend Kuperberg method of proof to prove the more general, refined enumeration, alo onjetured in [MRR1] [MRR2], and lited by Rihard Stanley [S] a the third of of hi Baker Dozen, that: Main Theorem. There are A(n, r :=A(n ( n+r 2 ( 2n 1 r n 1 n 1 ( 3n 2 n 1 n n alternating ign matrie for whih the (unique 1 of the firt row i at the r th olumn. A in Kuperberg proof, we are redued to evaluating a ertain determinant. Unlike the original determinant, it i not evaluable in loed form. We evaluate it by uing the q-analog of the Legendre polynomial over an interval, introdued, and greatly generalized, by Akey and Andrew [AA] and Akey and Wilon [AW]. All that i needed from the general theory of orthogonal polynomial and from q-alulu i reviewed, o a uffiient ondition for following the preent paper i having read Kuperberg paper [K] that inlude a very lear expoition of the Izergin-Korepin formula, and of it proof. Of oure, thi i alo a neeary ondition. In order to enourage reader to look up and read Kuperberg beautiful paper [K], and to ave myelf ome typing, I will ue the notation, and reult, of [K], without reviewing them. Boiling It Down To a Determinant Identity Let B(n, r be the number of ASM of order n whoe ole 1 of the firt (equivalently lat row, i at the r th olumn. In order to tand on Kuperberg houlder more omfortably, we will onider the lat row rather than the firt row. A in [K], define [x] :=(q x/2 q x/2 /(q 1/2 q 1/2, take q := e 2 1π/3, and onider Z(n ;2,..., 2, 2+a;0,...,0, where between the two emi-olon inide Z there are n 1 2 followed by a ingle 2+a. Here a i an indeterminate. Let look at an ASM of order n, whoe ole 1

3 of the lat row i at the r th olumn. It i readily een that the r 1 zero to the left of that 1 eah ontribute a weight of [2 + a], while the remaining n r zero, to the right of the aforementioned 1, eah ontribute a weight of [1 + a]. The 1 itelf ontribute q 1 a/2, whih i q a/2 time what it did before. Hene Z(n ;2,..., 2, 2+a;0,...,0=( 1 n q n q a/2 n r=1 We alo have, thank to [K], (or [Z]: jut plug in a = 0 above: Hene: Z(n ;2,...,2, 2; 0,...,0=( 1 n q n A(n. Z(n ;2,...,2, 2+a;0,...,0 Z(n;2,...,2, 2;0,...,0 = q a/2 A(n 61 B(n, r[2 + a] r 1 [1 + a] n r. n B(n, r[2 + a] r 1 [1 + a] n r. Sine {[2 + a] r 1 [1 + a] n r ;1 r n} are linearly independent, the B(n, r are uniquely determined by the above equation. Hene the Main Theorem i equivalent to: Z(n;2,...,2,2+a;0,...,0 Z(n; 2,...,2,2; 0,...,0 = q a/2 ( 3n 2 n 1 r=1 r=1 n ( ( n + r 2 2n 1 r n 1 n 1 [2 + a] r 1 [1 + a] n r. By replaing n by n + 1, and hanging the ummation on r to tart at 0, we get that it uffie to prove: Z(n +1;2,...,2,2+a;0,...,0 Z(n +1;2,...,2,2; 0,...,0 = ( q a/2 3n+1 n r=0 n ( ( n + r 2n r n n [2 + a] r [1 + a] n r. Let Z(n; x 1,...,x n ;y 1,...,y n denote the right hand ide of the Izergin-Korepin formula (Theorem 6 of [K]. Firt replae n by n + 1. Then, taking x i =2+iɛ, for i =1,...,n, x n+1 =2+a+(n+1ɛ, and y j = (j 1ɛ, j =1,...,n+ 1, yield, after anellation, Z(n +1;2+ɛ,...,2+nɛ, 2+a+(n+1ɛ;0, ɛ,..., nɛ = Z(n +1;2+ɛ,...,2+nɛ, 2+(n+1ɛ;0, ɛ,..., nɛ q a/2 n j=0 [2 + a +(n+1+jɛ][1 + a +(n+1+jɛ] [2+(n+1+jɛ][1+(n+1+jɛ] n 1 j=0 [(n +(n jɛ] det M n+1(a [a det M n+1 (0, where M n+1 (a =(m i,j, 0 i, j n i the (n +1 (n+ 1 matrix, defined a follow: { 1/([2+(i+j+1ɛ][1+(i+j+1ɛ] if 0 i n 1, 0 j n; m i,j = 1/([2 + a +(n+j+1ɛ][1 + a +(n+j+1ɛ] if i = n, 0 j n.

4 62 Doron Zeilberger Taking the limit ɛ 0, replaing q ɛ by, q a by X, etting w := e 1π/3, evaluating the limit whenever poible, and anelling out whenever poible, redue our tak to proving the following identity: { (1 n } det N n+1 (X = (Not Yet Done det N n+1 (1 (1 X n ( 3 n+2 w n n ( ( n + r 2n r n!(1 + X + X 2 n+1( 3n+1 w r (1 + wx r (1 w 2 X n r. n n n lim 1 r=0 Here the matrix N n+1 (X im n+1 (a divided by 3, to wit: N n+1 (X =(p i,j, 0 i, j n i the (n +1 (n+ 1 matrix, defined a follow. For the firt n row we have: p i,j = 1 i+j+1, (0 i n 1, 0 j n, 1 3(i+j+1 while for the lat row we have: p n,j = 1 Xn+j+1, (0 j n. 1 X 3 3(n+j+1 We are left with the tak of omputing the determinant of N n+1 (X, or at leat the limit on the left of (Not Yet Done. A Short Coure on Orthogonal Polynomial I will only over what we need here. Of the many available aount, Chapter 2 of [Wilf] i epeially reommended. For the preent purpoe, etion IV of [D] i mot pertinent. Theorem OP. Let T be any linear funtional ( umbra on the et of polynomial, and let i := T (x i be it o-alled moment. Let n n := det n n n n If n 0, for n 0, then there i a unique equene of moni polynomial P n (x, where the degree of P n (x i n, that are orthogonal with repet to the funtional T : T (P n (xp m (x=0 if m n. Furthermore, thee polynomial P n (x are given expliitly by: n P n (x = n+1 det n 1. n 1 n n 1 1 x x 2... x n (General Formula

5 63 Corollary 1. T (x n P n (x = n n 1 for n 1. Corollary 2. If S i another linear funtional, d i := S(x i, and Γ n := det n n n 1 n n 1 d 0 d 1 d 2... d n, then Γ n = S(P n(x n T (x n for n 1. P n (x For a long time it wa believed that Theorem OP wa only of theoretial interet, and that, given the moment, it wa impratial to atually find the polynomial P n (x, by evaluating the determinant. Thi onventional widom wa onveyed by Rihard Akey, bak in the late eventie, to Jim Wilon, who wa then tudying under him. Lukily, Wilon did not take thi advie. Uing Theorem OP led him to beautiful reult [Wil], whih later led to the elebrated Akey-Wilon polynomial [AW]. Jim Wilon independene wa later wholeheartedly endored by Dik Akey, who aid: If an authority in the field tell you that a ertain approah i worth trying, liten to them. If they tell you that a ertain approah i not worth trying, don t liten to them. The uelene of (General Formula wa till prolaimed a few year later, by yet another authority, Jean Dieudonné, who aid ([D], p. 11: La formule générale [(General Formula] donnant le... ont impratiable pour le alul expliite... There i another way in whih (General Formula and it immediate orollarie 1 and 2 ould be ueful. Suppoe that we know, by other mean, that a ertain et of expliitly given moni polynomial Q n (x are orthogonal with repet to the funtional T, i.e., T (Q n Q m = 0 whenever n m. Then by uniquene, Q n = P n. If we are alo able, uing the expliit expreion for Q n (x, to find T (x n Q n (x, then Corollary 1 give a way to expliitly evaluate the Hankel determinant n. Ifweare alo able to expliitly ompute S(Q n (x, then we would be able to evaluate the determinant Γ n. Thi would be our trategy in the evaluation of the determinant on the left of (Not Yet Done, but we firt need to digre again. A Lean and Lively Coure in q-calulu Until further notie, (a n := (1 a(1 qa(1 q 2 a...(1 q n 1 a. If I had my way, I would ban 1 Calulu from the Frehman urriulum, and replae it by q Calulu. Not only i it more fun, it alo deribe nature more

6 64 Doron Zeilberger aurately. The traditional alulu i baed on the fititiou notion of the real line. It i now known that the univere i quantized, and if you are at point x, then the point that you an reah are in geometri progreion q i x, in aordane with Hubble expanion. The true value of q i almot, but not quite 1, and i a univeral ontant, yet to be determined. The q-derivative, D q, i defined by The reader hould verify that D q f(x := f(x f(qx (1 qx. and the produt rule: D q x a = 1 qa 1 q xa 1, D q [f(x g(x] = f(x D q g(x+d q f(x g(qx. (Produt Rule The q-analog of integration, independently diovered by J. Thomae and the Rev. F. H. Jakon (ee [AA], [GR], i given by a f(x d q x := a(1 q 0 r=0 f(aq r q r, and over a general interval: f(x d q x := f(x d q x 0 0 f(x d q x. The reader i invited to ue teleoping to prove the following: Fundamental Theorem of q-calulu. D q F (x d q x = F (d F (. Combining the Produt Rule and the Fundamental Theorem, we have: q-integration by Part. If f(x or g(x vanih at the endpoint and d, then f(x D q g(x d q x = D q f(x g(qx d q x.

7 Corollary. If g(q i x vanih at the endpoint and d, for i =0,1,...,n 1, then f(x Dq n g(x d q x =( 1 n Dq n f(x g(q n x d q x. Even thoe who till believe in 1-Calulu an ue q-calulu to advantage. All they have to do i let q 1 at the end. The q-analog of xa dx =(1 a+1 /(a +1i 1 1 q 65 x a d q x= 1 a+1. (q-moment 1 qa+1 Now thi look familiar! Letting q := 3 in the definition of N n+1 (X, given right after (Not Yet Done, (thi new q ha nothing to do with the former Kuperberg q, we ee that the matrix N n+1 (1 i the Hankel matrix of the moment with repet to the funtional T (f(x := 1 f(xd q x. 1 q So all we need i to ome up with orthogonal polynomial with repet to the q-lebegue-meaure, over the interval [, 1]. q-legendre Polynomial The ordinary Legendre polynomial, over an interval (a, b may be defined in term of the Rodrigue formula n! (2n! Dn {(x a n (x b n }. The orthogonality follow immediately by integration by part. Thi lead naturally to the q-analog, Q n (x; a, b := (1 qn (q n+1 n D n q {(x a(x qa...(x aq n 1 (x b(x qb...(x bq n 1 }. Uing q-integration by part repeatedly, it follow immediately that the Q n (x; a, b are orthogonal with repet to q integration over (a, b. The laial ae a = 1, b = 1 goe bak to Markov. Akey and Andrew [AA] generalized thee to q-jaobi polynomial, and Akey and Wilon [AW] found the ultimate generalization. While at preent I don t ee how to apply thee more general polynomial to ombinatorial enumeration, I am ure that uh a ue will be found in the future, and all enumerator are urged to read [AA], [AW], and the modern lai [GR]. Going bak to the determinant N n+1 (X of (Not Yet Done, we alo need to introdue the funtional, defined on monomial by: and extended linearly. S(x j = 1 Xn+j+1 1 X 3 q n+j+1,

8 66 Doron Zeilberger Let X := q α/3. Then (reall that = q 1/3 : S(x j = 1 α+n+j+1 1 q n+j+α+1 = 1 1 q By linearity, for any polynomial p(x: S(p(x = 1 1 q Uing Corollary 2 of (General Formula, we get x α+n p(x d q x. x α+n x j d q x. det N n+1 (X det N n+1 (1 = xα+n P n (x d q x xn P n (x d q x, (Almot Done where P n (x inowtheq Legendre polynomial over [, 1], Q n (x;, 1 and = q 1/3. In other word: P n (x := Denouement (1 qn (q n+1 n D n q {(x 1(x q...(x q n 1 (x (x q...(x q n 1 }. It remain to ompute the right ide of (Almot Done. Let firt do the denominator. Propoition Bottom. 1 1 q x n P n (x d q x = qn2 (q 2 n(q n 2n+1 (q n+1 n (q n+1 n+1. Firt Proof. Ue q-integration by part, n time (i.e., ue the above orollary. The reulting q-integral i the famou q-vandermonde-chu um, that evaluate to the right ide. See [GR], or ue qekhad aompanying [PWZ]. Remark. Propoition Bottom, ombined with Corollary 1 of (General Formula give an alternative evaluation of Kuperberg determinant N n (1, needed in [K]. Seond Proof. Don t get off the houlder of Greg Kuperberg yet. Ue hi evaluation, and Corollary 1 of (General Formula. Propoition Top. Realling that X = q α/3, we have 1 1 q x n+α P n (x d q x = ( 1n (qx 3 n (q n+1 n ( 1n (qx 3 n (q n+1 n n 1 q k X 3k (q n+k q r (q n+k q r+1/3 r=0 n 1 q k+1/3 X 3k+1 (q n+k+1/3 q r (q n+k+1/3 q r+1/3. r=0

9 Proof. Let (1 qn F n (x := (q n+1 (x 1(x q...(x q n 1 (x (x q...(x q n 1, n o that P n (x =DqF n n (x. Sine F n (q i x vanih for i =0,...,n 1atbothx=1 and x =, we have by q-integrating by part n time (the above orollary, that x n+α P n (x d q x = x n+α D n q F n (x d q x =( 1 n Dq n {x n+α } F n (q n xd q x = ( 1n (q α+1 n (1 q n x α F n (q n xd q x. Now ue the definition of q-integration over [, 1] and replae q α by X 3, to omplete the proof. To ompute the right ide of (Almot Done, we only need to divide the expreion given by Propoition Top by the expreion given by Propoition Bottom. Doing thi, multiplying by (1 n =(1 q 1/3 n, and taking the limit q 1, we get that the left ide of (Not Yet Done i the following expreion. (Warning: Now we are afely bak in 1 land, o from now (a n := a(a +1...(a+n 1, the ordinary riing fatorial. ( 1 n (1 X 3 n (2n + 1! 3 n n! 3 ( n +1/3 2n+1 67 ( (k +1 n (k+ 2 3 nx 3k (k +1 n (k+ 4 3 nx 3k+1 After trivial anellation, equation (Not Yet Done boil down to ( 1 X3 ( 1 n (3n + 1! 1 X 2n+1 3 n+1 n! 3 (k +1 n (k+2/3 n X 3k ( n +1/3 2n+1 ( 1 X3 ( 1 n (3n + 1! 1 X 2n+1 3 n+1 n! 3 ( n +1/3 2n+1 =( 3 n n r=0 w r n ( n + r n ( 2n r n (k +1 n (k+4/3 n X 3k+1 (1 + wx r (1 w 2 X n r. (Done Thi wa given to EKHAD, the Maple pakage aompanying [PWZ]. EKHAD found a ertain linear homogeneou eond order reurrene in n that i atified by both um on the left of (Done (and hene by their differene, and alo by the right ide. It remain to prove that both ide of (Done agree at n =0,1, whih Maple did a well, even though it ould be done by any human. Link to the input and output file, and to EKHAD and qekhad, may be found at The input and output file are alled indone and outdone, repetively. Of oure, your omputer hould be able to reprodue the file outdone: One you have downloaded EKHAD and indone into a diretory, type: maple -q < indone > outdone After 380 eond of CPU time, outdone will be ready.

10 68 Doron Zeilberger Referene [AA] G. E. Andrew, and R. Akey, Claial orthogonal polynomial, Polynôme Orthogonaux et Appliation (Bar-Le-Du, 1984 (C. Brezinki et. al, ed., Leture Note in Mathemati, no. 1171, Springer-Verlag, Berlin, 1985, pp [AW] R. Akey and J. Wilon, Some Bai Hypergeometri Orthogonal Polynomial that Generalize Jaobi Polynomial, Memoir of the Amer. Math. So., no. 319, Amer. Math. So., Providene, [D] J. Dieudonné, Fration ontinuee et polynôme orthogonaux dan l oeuvre de E. N. Laguerre, Polynôme Orthogonaux et Appliation (Bar-Le-Du, 1984 (C. Brezinki et. al, ed., Leture Note in Mathemati, no. 1171, Springer-Verlag, Berlin, 1985, pp [GR] G. Gaper and M. Rahman, Bai Hypergeometri Serie, Enylopedia of Mathemati and it appliation, no. 35, Cambridge Univerity Pre, Cambridge, [KBI] V. E. Korepin, N. M. Bogoliubov and A. G. Izergin, Quantum Invere Sattering and Correlation Funtion, Cambridge Univerity Pre, Cambridge, [K] Greg Kuperberg, Another proof of the alternating ign matrix onjeture, Inter. Math. Re. Note (1996, [MRR1] W. H. Mill, D. P. Robbin, and H. Rumey, Proof of the Madonald onjeture, Invent. Math. 66 (1982, [MRR2] W. H. Mill, D. P. Robbin, and H. Rumey, Alternating ign matrie and deending plane partition, J. Combin. Theo. Ser. A 34 (1983, [PWZ] M. Petkovek, H. Wilf, and D. Zeilberger, A=B, A.K. Peter, Welleley, [S] R. P. Stanley, A baker dozen of onjeture onerning plane partition, Combinatoire Enumerative (G. Labelle and P. Leroux, ed., Leture Note in Mathemati, no. 1234, Springer-Verlag, Berlin, [Wilf] H. Wilf, Mathemati for the Phyial Siene, Dover, New York, [Wil] J. Wilon, Hypergeometri Serie, Reurrene Relation, and Some New Orthogonal Funtion, Ph.D. thei, Univ. of Wionin, Madion, [Z] D. Zeilberger, Proof of the alternating ign matrix onjeture, Eletroni J. of Combinatori 3(2 (1996. Department of Mathemati, Temple Univerity, Philadelphia, PA zeilberg@math.temple.edu zeilberg Typeet by AMS-TEX