The A 2 Rogers-Ramanujan identities revisited
aa r X i v : . [ m a t h . C O ] N ov THE A ROGERS-RAMANUJAN IDENTITIES REVISITED
SYLVIE CORTEEL AND TREVOR WELSHTo George Andrews for his 80 th birthday Abstract.
In this note we show how to use cylindric partitions to rederivethe A Rogers-Ramanujan identities originally proven by Andrews, Schillingand Warnaar. Introduction
The Rogers-Ramanujan identities were first proved in 1894 by Rogers and redis-covered in the 1910s by Ramanujan [14]. They are(1) X n ≥ q n ( n + i ) ( q ; q ) n = 1( q i ; q ) ∞ ( q − i ; q ) ∞ . with i = 0 , a, q ) ∞ = Q i ≥ (1 − aq i ) and ( a ; q ) n = ( a ; q ) ∞ / ( aq n ; q ) ∞ .There have been many attempts to give combinatorial proofs of these identitiesand the first one is due to Garsia and Milne [11]. Unfortunately, it is not simple, andno simple combinatorial proof is known. Recently in [6], the first author presenteda new bijective approach to the proofs of the Rogers-Ramanujan identities via theRobinson-Schensted-Knuth correspondence as presented in [13]. The bijection doesnot give the Rogers-Ramanujan identities but the Rogers-Ramanujan identitiesdivided by ( q ; q ) ∞ ; namely(2) 1( q ; q ) ∞ X n ≥ q n ( n +1) ( q ; q ) n = 1( q, q , q , q , q , q , q ; q ) ∞ where ( a , . . . , a k ; q ) ∞ = Q ki =1 ( a i ; q ) ∞ . This proof uses the combinatorics of cylin-dric partitions [10]. We interpret both sides as the generating function of cylindricpartitions of profile (3 ,
0) and the bijection is a polynomial algorithm in the sizeof the cylindric partition. The idea to use cylindric partitions is due to Foda andWelsh [12] in a more general setting: the Andrews-Gordon identities [2]. For k > ≤ i ≤ k , these identities divided by ( q ; q ) ∞ are1( q ; q ) ∞ X n ,...,n k q P kj =1 n j + P kj = i n j ( q ) n − n . . . ( q ) n k − − n k ( q ) n k = ( q i , q k +3 − i , q k +3 ; q k +3 ) ∞ ( q ; q ) ∞ . Foda and Welsh interpret the sum side as a generating function for (what theycall) decorated Bressoud paths and the product side is interpreted as the generatingfunction of cylindric partitions of profile (2 k + 1 − i, i ),and they provide a bijectionbetween the two objects. See [12] for more details. Date : November 27, 2019.
In this note, we take the idea of applying cylindric partitions to Rogers-Ramanujantype identities a step further, by using them to give an alternative proof of the A Rogers-Ramanujan identities due to Andrews, Schilling and Warnaar [3].
Theorem 1.1.
We have ∞ X n =0 2 n X n =0 q n + n − n n + n + n ( q ; q ) n (cid:20) n n (cid:21) = 1( q , q , q , q , q , q ; q ) ∞ , ∞ X n =0 2 n X n =0 q n + n − n n + n ( q ; q ) n (cid:20) n n (cid:21) = 1( q, q , q , q , q , q ; q ) ∞ , ∞ X n =0 2 n +1 X n =0 q n + n − n n + n ( q ; q ) n (cid:20) n + 1 n (cid:21) = 1( q, q , q , q , q , q ; q ) ∞ , ∞ X n =0 2 n +1 X n =0 q n + n − n n + n ( q ; q ) n (cid:20) n + 1 n (cid:21) = 1( q, q , q , q , q , q ; q ) ∞ , ∞ X n =0 2 n X n =0 q n + n − n n ( q ; q ) n (cid:20) n n (cid:21) = 1( q, q, q , q , q , q ; q ) ∞ , where the Gaussian polynomial (cid:2) nk (cid:3) is defined by (cid:20) nk (cid:21) = ( q ; q ) n ( q ; q ) k ( q ; q ) n − k . All but the fourth of these identities were obtained in Theorem 5.2 of [3], whilethe fourth was conjectured in Section 2.4 of [8]. Note that the second and thirdexpressions are equal.In this note, we prove the following theorem, giving the generating functions F c,n ( q ) of cylindric partitions indexed by compositions c of 4 into 3 parts, withlargest entry at most n : HE A ROGERS-RAMANUJAN IDENTITIES REVISITED 3
Theorem 1.2. F (4 , , ,n ( q ) = n X n =0 2 n X n =0 q n + n − n n + n + n ( q ; q ) n − n ( q ; q ) n (cid:20) n n (cid:21) ,F (3 , , ,n ( q ) = n X n =0 2 n X n =0 q n + n − n n + n ( q ; q ) n − n ( q ; q ) n (cid:20) n n (cid:21) ,F (3 , , ,n ( q ) = n X n =0 2 n X n =0 q n + n − n n ( q ; q ) n − n q n ( q ; q ) n (cid:20) n n (cid:21) + n X n =1 2 n − X n =0 q n + n − n n ( q ; q ) n − n q n ( q ; q ) n − (cid:20) n − n (cid:21) ,F (2 , , ,n ( q ) = n X n =0 2 n X n =0 q n + n − n n ( q ; q ) n − n q n ( q ; q ) n (cid:20) n n (cid:21) + n X n =1 2 n − X n =0 q n + n − n n ( q ; q ) n − n q n (1 + q n n )( q ; q ) n − (cid:20) n − n (cid:21) ,F (2 , , ,n ( q ) = n X n =0 2 n X n =0 q n + n − n n ( q ; q ) n − n ( q ; q ) n (cid:20) n n (cid:21) . This result gives a finite version of the sum side of the A Rogers-Ramanujanidentities.In the n → ∞ limit, we recover the sum side of the identities of Theorem 1.1divided by ( q ; q ) ∞ . We also explain how to get the product side. It is a corollary ofa result of Borodin [4]. In Section 2 below, we start by defining cylindric partitionsand then obtain the product sides of particular cylindric partitions. These yieldthe right hand sides of the expressions in Theorem 1.1. The sum expressions onthe left hand sides are computed in Section 3. Acknowledgments.
SC was in residence at MSRI (NSF grant DMS-1440140) andwas visiting the Mathematics department at UC Berkeley during the completion ofthis work. TW acknowledges partial support from the Australian Research Council.The authors wish to thank Omar Foda for his interest in this work and usefuldiscussions. The authors also wish to thank the anonymous referee for her excellentsuggestions and careful reading.2.
Cylindric partitions and the product side
Cylindric partitions were introduced by Gessel and Krattenthaler [10] and ap-peared naturally in different contexts [4, 5, 7, 9, 12, 15]. Let ℓ and k be two positiveintegers. In this note, we choose to index cylindric partitions by compositions of ℓ into k non negative parts. Definition 2.1.
Given a composition c = ( c , . . . , c k ) , a cylindric partition ofprofile c is a sequence of k partitions Λ = ( λ (1) , . . . λ ( k ) ) such that : • λ ( i ) j ≥ λ ( i +1) j + c i +1 , • λ ( k ) j ≥ λ (1) j + c . SYLVIE CORTEEL AND TREVOR WELSH for all i and j . For example, the sequence Λ = ((3 , , , , (4 , , , , (4 , , , , j , λ (1) j ≥ λ (2) j +2 , λ (2) j ≥ λ (3) j and λ (3) j ≥ λ (1) j +2 for all j . Note that this definition implies that cylindric par-titions of profile ( c , . . . , c k ) are in bijection with cylindric partitions of profile( c k , c , . . . , c k − ).Our goal is to compute generating functions of cylindric partitions of a givenprofile c according to two statistics. Given a Λ = ( λ (1) , . . . , λ ( k ) ), let • | Λ | = P ki =1 P j ≥ λ ( i ) j , the sum of the entries of the cylindric plane partition,and • max(Λ) = max( λ (1)1 , . . . λ ( k )1 ), the largest entry of the cylindric plane parti-tion.Going back to our example, we have | Λ | = 24, and max(Λ) = 4.Let C c be the set of cylindric partitions of profile c and let C c,n be the set ofcylindric partitions of profile c and such that the largest entry is at most n . We areinterested in the following generating functions. F c ( q ) = X Λ ∈C c q | Λ | , (3) F c ( y, q ) = X Λ ∈C c q | Λ | y max(Λ) , (4) F c,n ( q ) = X Λ ∈C c,n q | Λ | . (5)A surprising and beautiful result is that for any c , the generating function F c ( q )can be written as a product. Namely, with t = k + ℓ , Theorem 2.2. [4]
The generating function F c ( q ) is equal to (6) 1( q t ; q t ) · k Y i =1 k Y j = i +1 c i Y m =1 q m + d i +1 ,j + j − i ; q t ) ∞ · k Y i =2 i − Y j =2 c i Y m =1 q t − ( m + d j,i − + i − j ) ; q t ) ∞ . where d i,j = c i + c i +1 + . . . + c j . The original result is written is a different but equivalent form.For what follows, we restrict attention to the case ℓ = 4 and k = 3. Ascylindric partitions of profile ( c , . . . , c k ) are in bijection with partitions of profile( c k , c , . . . , c k − ), we need only compute the generating functions for the composi-tions (4 , , , , , , , , , , HE A ROGERS-RAMANUJAN IDENTITIES REVISITED 5
Corollary 2.3. F (4 , , ( q ) = 1( q ; q ) ∞ ( q , q , q , q , q , q ; q ) ∞ ,F (3 , , ( q ) = 1( q ; q ) ∞ ( q, q , q , q , q , q ; q ) ∞ ,F (3 , , ( q ) = 1( q ; q ) ∞ ( q, q , q , q , q , q ; q ) ∞ ,F (2 , , ( q ) = 1( q ; q ) ∞ ( q, q , q , q , q , q ; q ) ∞ ,F (2 , , ( q ) = 1( q ; q ) ∞ ( q, q, q , q , q , q ; q ) ∞ . Note that these five products are precisely those in Theorem 1.1 divided by ( q ; q ) ∞ .3. The sum side
We first prove a general functional equation for F c ( y, q ) for any profile c . Supposethat k > c = ( c , . . . , c k ). Let I c be the subset of { , . . . , k } such that i ∈ I c if and only if c i >
0. For example if c = (2 , ,
0) then I c = { , } . Given a subset J of I c , we define the composition c ( J ) = ( c ( J ) , . . . , c k ( J )) by c i ( J ) = c i − i ∈ J and ( i − J,c i + 1 if i / ∈ J and ( i − ∈ J,c i otherwise . Here we set c = c k . Proposition 3.1.
For any composition c = ( c , . . . , c k ) , (7) F c ( y, q ) = X ∅⊂ J ⊆ I c ( − | J |− F c ( J ) ( yq | J | , q )1 − yq | J | . with the conditions F c (0 , q ) = 1 and F c ( y,
0) = 1 .Proof.
The proof make use of an inclusion-exclusion argument.First, for fixed J such that ∅ ⊂ J ⊆ I c , we require the generating function ofcylindric partitions Λ of profile c such that λ ( j )1 = max(Λ) for all j ∈ J .Let M = ( µ (1) , . . . , µ ( k ) ) be a cylindric partition of profile c ( J ), and set n =max( M ). Then, for a fixed integer m ≥
0, create a cylindric partition Λ =( λ (1) , . . . , λ ( k ) ) using the following recipe: λ ( j ) = ( ( m + n, µ ( j )1 , µ ( j )2 , . . . ) if j ∈ J , µ ( j ) if j / ∈ J .It is easily checked that Λ is a cylindric partition of profile c and that max(Λ) = m + n . Moreover, λ ( j )1 = max(Λ) for all j ∈ J . The generating function for allcylindric partitions Λ obtained from M in this way is(8) ∞ X m =0 y m + n q | J | ( m + n ) q | M | = y n q | J | n + | M | ∞ X m =0 ( yq | J | ) m = y n q | J | n + | M | − yq | J | . SYLVIE CORTEEL AND TREVOR WELSH
Then the generating function for all cylindric partitions Λ obtained in this wayfrom any cylindric partition M of profile c ( J ) is(9) X M ∈C c ( J ) y max( M ) q | J | max( M )+ | M | − yq | J | = F c ( J ) ( yq | J | , q )1 − yq | J | , making use of the definition (4).Let Λ = ( λ (1) , . . . , λ ( k ) ) be an arbitrary cylindric partition of profile c , and let p = max(Λ). Because λ ( i − ≥ λ ( i )1 whenever i / ∈ I c , it must be the case that p = λ ( j )1 for some j ∈ I c . Then, if J = ∅ is such that p = λ ( j )1 for each j ∈ J (this J might not be unique), we see that Λ is one of the cylindric partitions enumerated by(9). However, because Λ can arise from various different J , the generating functionfor cylindric partitions of profile c is obtained via the inclusion-exclusion process.This immediately gives (7). (cid:3) Now, for each composition c , define(10) G c ( y, q ) = ( yq ; q ) ∞ F c ( y, q ) . In terms of this, the previous result translates to(11) G c ( y, q ) = X ∅⊂ J ⊆ I ( − | J |− ( yq ; q ) | J |− G c ( J ) ( yq | J | , q )with G c (0 , q ) = G c ( y,
0) = 1.
Theorem 3.2. G (4 , , ( y, q ) = ∞ X n =0 2 n X n =0 y n q n + n − n n + n + n ( q ; q ) n (cid:20) n n (cid:21) ,G (3 , , ( y, q ) = ∞ X n =0 2 n X n =0 y n q n + n − n n + n ( q ; q ) n (cid:20) n n (cid:21) ,G (3 , , ( y, q ) = ∞ X n =0 2 n X n =0 y n q n + n − n n q n ( q ; q ) n (cid:20) n n (cid:21) + ∞ X n =1 2 n − X n =0 y n q n + n − n n q n ( q ; q ) n − (cid:20) n − n (cid:21) ,G (2 , , ( y, q ) = ∞ X n =0 2 n X n =0 y n q n + n − n n q n ( q ; q ) n (cid:20) n n (cid:21) + ∞ X n =1 2 n − X n =0 y n q n + n − n n q n (1 + q n n )( q ; q ) n − (cid:20) n − n (cid:21) ,G (2 , , ( y, q ) = ∞ X n =0 2 n X n =0 y n q n + n − n n ( q ; q ) n (cid:20) n n (cid:21) . HE A ROGERS-RAMANUJAN IDENTITIES REVISITED 7
Proof.
In this proof, we abbreviate G c ( y, q ) to G c ( y ) for convenience. Applying theform (11) of Propostion 3.1 to the case ℓ = 4 and k = 3 yields G (4 , , ( y ) = G (3 , , ( yq ) ,G (3 , , ( y ) = G (3 , , ( yq ) + G (2 , , ( yq ) − (1 − yq ) G (2 , , ( yq ) ,G (3 , , ( y ) = G (4 , , ( yq ) + G (2 , , ( yq ) − (1 − yq ) G (3 , , ( yq ) ,G (2 , , ( y ) = G (3 , , ( yq ) + G (2 , , ( yq ) − (1 − yq ) G (2 , , ( yq ) ,G (2 , , ( y ) = G (2 , , ( yq ) + G (2 , , ( yq ) + G (3 , , ( yq ) − (1 − yq )( G (2 , , ( yq ) + G (2 , , ( yq ) + G (3 , , ( yq ))+ (1 − yq )(1 − yq ) G (2 , , ( yq ) . By manipulating these equations, we obtain G (4 , , ( y ) = G (3 , , ( yq ) ,G (3 , , ( y ) = G (2 , , ( yq ) + yq G (3 , , ( yq ) + yqG (2 , , ( yq ) ,G (3 , , ( y ) = G (2 , , ( yq ) + yqG (3 , , ( yq ) ,G (2 , , ( y ) = G (2 , , ( yq ) + yqG (2 , , ( yq ) + yq G (3 , , ( yq ) ,G (2 , , ( y ) = G (2 , , ( yq ) + yqG (2 , , ( yq ) + yqG (2 , , ( yq )+ yq G (3 , , ( yq ) + yq G (2 , , ( yq ) . (12)We claim that this system of equations (12) together with the boundary conditions G c (0 , q ) = G c ( y,
0) = 1 for each composition c , is uniquely solved by the expressionsstated in the theorem. This is proved using an induction argument involving allfive expressions.We use induction on the exponents of y . For each composition c , let g c ( n )denote the coefficient of y n in the solution G c ( y ) of (12). The boundary conditions G c (0 , q ) = 1 imply that each g c (0) = 1. This holds for the expressions of thetheorem. So now, for n >
0, assume that g c ( n ) agrees with the coefficient of y n in the statement of the theorem for each n < n and each composition c . We mustcheck that g c ( n ), as determined by the expressions (12), is equal to the coefficientof y n in the statement of the theorem for each c .The fifth expression in (12) implies that(1 − q n ) g (2 , , ( n ) = ( q n + q n − ) g (2 , , ( n − q n − g (3 , , ( n −
1) + q n − g (2 , , ( n − . Using the expressions for g c ( n ) implied by the induction hypothsis then yields g (2 , , ( n ) = n X n =0 q n + n − nn ( q ; q ) n (cid:20) nn (cid:21) . This expression, along with the other expressions for g c ( n ) implied by (12), enablesus to compute, in turn, g (2 , , ( n ), g (3 , , ( n ), g (3 , , ( n ) and finally g (4 , , ( n ). Be-cause the expressions that result agree with the corresponding coefficients of y n inthe statement of the theorem, the induction argument is complete. (cid:3) SYLVIE CORTEEL AND TREVOR WELSH
Proof of Theorem 1.2.
Comparing (4) and (5) leads to F c,n ( q ) = [ y ] F c ( y, q ) + [ y ] F c ( y, q ) + · · · + [ y n ] F c ( y, q )= [ y n ] F c ( y, q )1 − y = [ y n ] G c ( y, q )( y ; q ) n having used (10). By the q -binomial theorem [1], we have1( y, q ) n = n X i =0 y i ( q ; q ) i , and therefore it follows that F c,n ( q ) = n X n =0 q ; q ) n − n [ y n ] G c ( y, q ) . Applying this to the expressions of Theorem 3.2 then yields those of Theorem1.2. (cid:3)
Proof of Theorem 1.1.
Let us now comment on how to get the end of the proofof Theorem 1.1. The right hand side of the five identities correspond to Corollary2.3. To get the sum side; we first let n tend to infinity in Theorem 1.2 and multiplyby ( q ; q ) ∞ . We get directly the left hand side of the first, second and fifth identities.Let us show how to get the left hand side third identity. Theorem 1.2 states that thegenerating function of the cylindric partitions of profile (3 , ,
1) and largest entryat most n is:(13) F (3 , , ,n ( q ) = X n ,n q n + n − n n ( q ; q ) n − n ( q ; q ) n (cid:18) q n (cid:20) n n (cid:21) + q n (1 − q n ) (cid:20) n − n (cid:21)(cid:19) We let n → ∞ and multiply by ( q ; q ) ∞ in (13), we get X n ,n q n + n − n n + n ( q ; q ) n (cid:20) n n (cid:21) + X n ,n q n + n − n n +2 n ( q ; q ) n − (cid:20) n − n (cid:21) After the change ( n , n ) → ( n + 1 , n −
1) in the second sum, we get= X n ,n q n + n − n n + n ( q ; q ) n (cid:20) n n (cid:21) + X n ,n q ( n − +( n +1) − ( n − n +1)+( n − n − − ( n +1)+1) ( q ; q ) n − (cid:20) n − n (cid:21) = X n ,n q n + n − n n + n ( q ; q ) n (cid:20) n n (cid:21) + X n ,n q n + n − n n + n +(2 n − n +1) ( q ; q ) n (cid:20) n n − (cid:21) = X n ,n q n + n − n n + n ( q ; q ) n (cid:20) n + 1 n (cid:21) , using the well known recurrence relation (cid:20) nk (cid:21) = (cid:20) n − k (cid:21) + q n − k (cid:20) n − k − (cid:21) . This is the left hand side of the third identity of Theorem 1.1. HE A ROGERS-RAMANUJAN IDENTITIES REVISITED 9
We leave the computation for the left hand side of the fourth identity to thereader. One needs to show that X n ,n q n + n − n n + n ( q ; q ) n (cid:20) n + 1 n (cid:21) equals X n ,n q n + n − n n ( q ; q ) n (cid:18) q n (cid:20) n n (cid:21) + q n (1 + q n n )(1 − q n ) (cid:20) n − n (cid:21)(cid:19) . (cid:3) References [1] G.E. Andrews, The Theory of Partitions. Cambridge University Press. ISBN 0-521-63766-X,1976.[2] G.E. Andrews, On the General Rogers-Ramanujan Theorem. Providence, RI: Amer. Math.Soc., 1974.[3] G.E. Andrews, A. Schilling, S.O. Warnaar, An A Bailey lemma and Rogers-Ramanujan-typeidentities, J. Amer. Math. Soc., 12 (3), 677-702, 1999.[4] A. Borodin, Periodic Schur process and cylindric partitions. Duke Math. J. 140 (2007), no.3, 391468.[5] J. Bouttier, G. Chapuy and S. Corteel, From Aztec diamonds to pyramids: steep tilings,Trans. Amer. Math. Soc. 369 (2017), no. 8, 5921–5959.[6] S. Corteel, Rogers-Ramanujan identities and the Robinson-Schensted-Knuth Correspondence,Proc. Amer. Math. Soc. 145 (2017), no. 5, 2011–2022.[7] S. Corteel, C. Savelief and M. Vuletic, Plane overpartitions and cylindric partitions, Jour. ofComb. Theory A, Vol.118, Issue 4, (2011), 1239-1269.[8] B. Feigin, O. Foda, and T. Welsh, Andrews-Gordon type identities from combinations ofVirasoro characters. Ramanujan J. 17 (2008), no. 1, 33-52.[9] T. Gerber, Crystal isomorphisms in Fock spaces and Schensted correspondence in affine typeA. Algebras and Representation Theory 18 (2015), 1009-1046.[10] Ira Gessel, and Christian Krattenthaler, Cylindric partitions. Trans. Amer. Math. Soc. 349(1997), no. 2, 429-479.[11] A. Garsia, and S.C. Milne, A Rogers-Ramanujan Bijection. J. Combin. Th. Ser. A 31, 289-339,(1981).[12] O. Foda and T. Welsh, Cylindric partitions, W r characters and the Andrews-Gordon-Bressoud identities, Journal of Physics. A, Mathematical and theoretical, 49(16), [164004].arXiv:1510.02213.[13] I. Pak, Partition bijections, a survey. Ramanujan J. 12 (2006), no. 1, 5-75.[14] L.J. Rogers and Srinivasa Ramanujan, Proof of certain identities in combinatorial analysis,Camb. Phil. Soc. Proc., Vol 19, 1919, p. 211-216.[15] P. Tingley, Three combinatorial models for b sl n crystals, with applications to cylindric parti-tions. Int. Math. Res. Not. IMRN 2008, no. 2, Art. ID rnm143, 40 pp. IRIF, CNRS et Universit´e Paris Diderot, France
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