Algebraic Relations Between Partition Functions and the j -Function
aa r X i v : . [ m a t h . N T ] N ov ALGEBRAIC RELATIONS BETWEEN PARTITION FUNCTIONS ANDTHE j -FUNCTION ALICE LIN, ELEANOR MCSPIRIT, AND ADIT VISHNU
Abstract.
We obtain identities and relationships between the modular j -function, thegenerating functions for the classical partition function and the Andrews spt -function,and two functions related to unimodal sequences and a new partition statistic we call the“signed triangular weight” of a partition. These results follow from the closed formulawe obtain for the Hecke action on a distinguished harmonic Maass form M ( τ ) defined byBringmann in her work on the Andrews spt -function. This formula involves a sequence ofpolynomials in j ( τ ), through which we ultimately arrive at expressions for the coefficientsof the j -function purely in terms of these combinatorial quantities. Introduction and Statement of Results
Partitions, first and foremost combinatorial objects, permeate seemingly disparate areasof mathematics. The partition function p ( n ) gives the number of ways to write n as the sumof unordered positive integers. The generating function for p ( n ) is a weakly holomorphicmodular form of weight − /
2, namely P ( q ) := X n ≥ p ( n ) q n − = q − Y n ≥ − q n = 1 η (24 τ ) , (1.1)where η ( τ ) is Dedekind’s eta-function and we use the convention q = e πiτ . This is oneindication of partitions’ deep ties to number theory. Outside combinatorics and numbertheory, perhaps the most prominent role for partitions is in representation theory, wherethe theory of Young tableaux for partitions encodes the irreducible representations of allsymmetric groups [12, Theorem 2.1.11].Other modular forms and functions that were first studied in number theory have likewiseappeared in the representation theory of finite groups. In particular, the modular j -function,whose Fourier expansion is j ( τ ) = X n ≥− c ( n ) q n = q − + 744 + 196884 q + 21493760 q + · · · , (1.2)is well-known in number theory because the j -invariants, i.e. the values of j ( τ ) for τ ∈ H ,parametrize isomorphism classes of elliptic curves over C [17, Proposition 12.11].McKay famously observed that the first few coefficients of j ( τ ) satisfy striking relationssuch as(1.3) c (1) = 196884 = 1 + 196883 ,c (2) = 21493760 = 1 + 196883 + 21296876 , where the right-hand sides are linear combinations of dimensions of irreducible represen-tations of the monster group M . Such expressions inspired Thompson to conjecture [18] that there is a monstrous moonshine module, an infinite-dimensional graded M -module V ♮ = L n ≫− V n such that for n ≥ −
1, we have c ( n ) = dim( V n ) . Thompson further conjectured that, since the graded dimension is the graded trace of theidentity element of M , the traces of other elements g may likewise be related to naturally-occuring q -series. This was refined by Conway and Norton in [11], who conjectured that forevery element g ∈ M , the McKay-Thompson series T g ( τ ) := ∞ X n = − Tr( g | V n ) q n is the Hauptmodul which generates the function field for a genus 0 modular curve fora particular congruence subgroup Γ g ⊂ SL ( R ). Borcherds proved the Conway–Nortonconjecture for the Monster Moonshine Module in [6], an impactful result which, in part,solidifies the j -function’s connection to the representation theory of M .Since the j -function and partitions appear in both number theory and representationtheory, one can ask if there is a relation between c ( n ) and p ( n ). In this paper, we dis-cover that the coefficients of the Fourier expansion of both the j -function and a certainsequence of polynomials in j have a combinatorial description in terms of partitions of inte-gers and unimodal sequences. This suggests the possibility of deeper connections betweenthe representation theory of the symmetric group and the monster Lie algebra.This research is inspired by recent work of Andrews [2] in which he defined spt ( n ) tocount the number of smallest parts among all integer partitions of n . For example, wecan determine that spt (4) = 10 by counting the following underlined parts across all fivepartitions of 4: 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 . Following the notation of [16], we define a renormalized generating function for spt ( n ) as S ( q ) := X n ≥ spt ( n ) q n − . (1.4)Paralleling Ramanujan’s notable congruences p (5 n + 4) ≡ ,p (7 n + 5) ≡ ,p (11 n + 6) ≡ , Andrews [2] showed that the spt function satisfies the congruences spt (5 n + 4) ≡ ,spt (7 n + 5) ≡ ,spt (13 n + 6) ≡ . Of further interest are the spt -function’s rich families of congruences modulo all primes ℓ ≥
5. As Ono proved in [16], if ℓ ≥ n ≥
1, and (cid:0) − nℓ (cid:1) = 1, then spt (cid:18) ℓ n − (cid:19) ≡ ℓ ) . (1.5) LGEBRAIC RELATIONS BETWEEN PARTITION FUNCTIONS AND THE j -FUNCTION 3 Subsequent work by Ahlgren et al. in [1] extended these congruences to arbitrary powersof ℓ . If m ≥
1, then spt (cid:18) ℓ m n + 124 (cid:19) ≡ ℓ m ) . (1.6)These congruences follow from studying a distinguished harmonic Maass form M ( τ )defined by Bringmann in [7] (see (2.1)). For background on harmonic Maass forms, werefer the reader to [8] and [15]. The function M ( τ ) is of particular interest because itsholomorphic part M + ( τ ) involves the generating functions for both p ( n ) and spt ( n ); namelywe have M + ( τ ) = S ( q ) + 112 q ddq P ( q ) . (1.7)For weight 3 / χ := (cid:0) · (cid:1) , we follow thenormalization given in [16] to define the Hecke operators T ( ℓ ) of index ℓ on a powerseries f ( τ ) = P n ≫−∞ a ( n ) q n by f ( τ ) | T ( ℓ ) := X n ≫−∞ (cid:20) a ( ℓ n ) + (cid:18) ℓ (cid:19) (cid:18) − nℓ (cid:19) a ( n ) + ℓa ( n/ℓ ) (cid:21) q n . (1.8)The congruences in (1.5) and (1.6) follow from the fact that M + ( τ ) | T ( ℓ ) ≡ (cid:18) ℓ (cid:19) M + ( τ ) (mod ℓ ) . (1.9)Ono asked whether there exist explicit identities which imply (1.9). We answer thisquestion. Using the standard notation ( q ; q ) ∞ := Q n ≥ (1 − q n ), we define a sequence ofmonic integer polynomials B m ( x ) of degree ( m −
1) by(1.10) B ( x, q ) = X m ≥ B m ( x ) q m := ( q ; q ) ∞ · j ( τ ) − x = q + ( x − q + ( x − x + 357395) q + · · · . In terms of the Eisenstein series E ( τ ) and E ( τ ), as well as Ramanujan’s Delta function∆( τ ), we offer the following solution to Ono’s problem. Theorem 1.1. If ℓ ≥ is a prime and δ ℓ := ℓ − , then M + ( τ ) | / T ( ℓ ) = (cid:18) ℓ (cid:19) (1 + ℓ ) M + ( τ ) − ℓ P ( q ) · B δ ℓ ( j (24 τ )) · E (24 τ ) E (24 τ )∆(24 τ ) . Remark.
We note that the identity in the theorem immediately reduces to (1.9) modulo ℓ .Moreover, this result gives an expression for the Hecke action in terms of only the originalmock modular form and the coefficient of q − ℓ produced by the Hecke operator. Therefore,the resulting mock modular form is determined by a single term.For notational clarity, we note that − q ddq j ( τ ) = E ( τ ) E ( τ )∆( τ ) = q − − X n ≥ nc ( n ) q n = q − − q − q + · · · . ALICE LIN, ELEANOR MCSPIRIT, AND ADIT VISHNU
Thus, B δ ℓ ( j (24 τ )) · E (24 τ ) E (24 τ )∆(24 τ ) is completely determined by the coefficients of j . Forconvenience, we write − q ddq j (24 τ ) = q − − q − q + · · · . Example.
Here we illustrate Theorem 1.1 for the primes 5, 7, and 11. In the notationof [16], we define M ℓ ( τ ) := M + ( τ ) | / T ( ℓ ) − (cid:18) ℓ (cid:19) (1 + ℓ ) M + ( τ ) . (1.11)For ℓ = 5, note that δ = 1 and B ( x ) = 1. Therefore, we find that M ( τ ) = 512 P ( q ) · q ddq j (24 τ ) = − q − − q − + 4922056 q + · · · . For ℓ = 7, δ = 2 and B ( x ) = x − M ( τ ) = 712 P ( q ) · ( j (24 τ ) − · q ddq j (24 τ ) = − q − − q − + 14907812512 q + · · · . For ℓ = 11, δ = 5 and B ( x ) = x − x + 2732795 x − x + 4947668669 . Therefore, we have M ( τ ) = 1112 P ( q ) · B ( j (24 τ )) · q ddq j (24 τ ) = − q − + 1112 q − + · · · . In view of (1.11), the case ℓ = 5 gives an expression for M + ( τ ) | T (25) in terms of thecoefficients c ( n ) of the j -function, thus deriving an unexpected relationship between thesecoefficients and the values of p ( n ) and spt ( n ). Namely, we offer the following partition-theoretic counterparts to (1.3): c (1) = 196884 = 2 + 49 + 15708 + 181125 ,c (2) = 21493760 = 12 (cid:16) −
49 + 182 − − (cid:17) . The two identities above are examples of a more general theorem. To make this precise,it is important to illustrate how the summands above correspond to p ( n ) and spt ( n ). Werequire the following notation. For n ≥
1, we define(1.12) h (24 n −
1) := 125 spt (25 n −
1) + 5(24 n − p (25 n − µ n · (cid:16) spt ( n ) + 5(24 n − p ( n ) (cid:17) ,h (25(24 n − spt ( n ) + (24 n − p ( n ) , where µ n := 6 − (cid:0) − n (cid:1) . We define h ( m ) = 0 if m
23 mod 24 and h ( m ) = 0 if m
23 mod 24 or if m n = 1,we set s ( n ) = 2, and for n >
1, let s ( n ) := ( ( − k +1 if 24 n = (6 k + 1) −
25 or 24 n = (6 k + 1) − k ∈ Z , . Remark.
It is an easy exercise to confirm s ( n ) is well-defined. LGEBRAIC RELATIONS BETWEEN PARTITION FUNCTIONS AND THE j -FUNCTION 5 Then we have the following result.
Theorem 1.2. If n ≥ , then c ( n ) = s ( n ) n + 1 n X k ∈ Z h ( − k h (24 n − (6 k + 1) ) + ( − k h (24 n − (6 k + 1) ) i . Remark.
The formula in Theorem 1.2 bears a strong resemblance to another well-knownexpression for the coefficients of j . Work of Kaneko [13] shows for n ≥ c ( n ) = 1 n X r ∈ Z (cid:20) t ( n − r ) − ( − n + r t (4 n − r ) + ( − r t (16 n − r ) (cid:21) , (1.13)where t are traces of singular moduli, i.e. the sums of the j -invariants of elliptic curveswith complex multiplication. In view of the similarity of these expressions, it is natural towonder whether Theorem 1.2 suggests a deep connection between partitions and traces ofsingular moduli.In [3], Andrews related spt ( n ) to a number of other combinatorial and number-theoreticfunctions. One connection of particular interest is the relationship of spt to strongly uni-modal sequences. We ask whether this relationship reveals deeper connections to the j -function and representation theory.A sequence of integers { a k } sk =1 is a strongly unimodal sequence of size n if P sk =1 a k = n and for some r it satisfies 0 < a < a < · · · < a r > a r +1 > a r +2 > · · · > a s > { a k } sk =1 is s − r + 1, the number of terms after the maximal term minusthe number of terms preceding it. The function U ( t ; q ) counts specific types of stronglyunimodal sequences [10]. For t = − U ( − q ) = X n ≥ u ∗ ( n ) q n = q + q − q − q + 2 q + · · · , where u ∗ ( n ) is the difference of the number of even-rank strongly unimodal sequences ofsize n and the number of odd-rank strongly unimodal sequences of size n . Andrews provedin [3] that U ( − q ) = − X n ≥ spt ( n ) q n + 2 A ( q ) , (1.14)where A ( q ) = X n ≥ a ( n ) q n := 1( q ; q ) ∞ ∞ X n =1 ( − n − nq n n − q n = q + q − q + q − q + 4 q + · · · . It is natural to ask what A ( q ) is counting. We find that A ( q ) is the generating functionfor a partition statistic that we call the “signed triangular weight” of a partition, a resultwhich is of independent interest. Given a partition λ ⊢ N , where we write the size of thepartition as | λ | := N , let n λ be the maximal number such that λ contains parts of size1 , , . . . , n λ . Letting m k denote the number of times that the part k appears in λ , we definethe signed triangular weight of λ to be t s ( λ ) := P n λ k =1 ( − k − km k . If λ does not contain apart of size 1, then let t s ( λ ) = 0. Example.
Consider λ = { , , , , , , , } . Then λ ⊢ n λ = 5, and t s ( λ ) = 1 · − · · − · · . ALICE LIN, ELEANOR MCSPIRIT, AND ADIT VISHNU
We prove the following result relating t s ( λ ) for all partitions λ of all positive integers to theseries A ( q ). Theorem 1.3.
The following q -series identity is true: A ( q ) = X λ t s ( λ ) q | λ | . From this, we may conclude that a ( n ) = P | λ | = n t s ( λ ). Given this relationship, the spt congruence given in (1.6) immediately implies the following result. Corollary 1.4. If ℓ ≥ is prime, (cid:0) − nℓ (cid:1) = 1 , and m ≥ , then u ∗ (cid:18) ℓ m n − (cid:19) ≡ a (cid:18) ℓ m n − (cid:19) (mod ℓ m ) . Combining our explicit expression for the action of the Hecke operator T (25) in Theo-rem 1.1 and our combinatorial expressions for c ( n ), we arrive at new expressions for thecoefficients of j ( τ ) in terms of p ( n ) and the coefficients of a ( n ) and u ∗ ( n ). For ease ofnotation, we define the functions(1.15) g (24 n −
1) := − u ∗ (25 n −
1) + 245 a (25 n −
1) + 5(24 n − p (25 n − µ n · (cid:18) − u ∗ (25 n −
1) + 245 a (25 n −
1) + 5(24 n − p ( n ) (cid:19) ,g (25(24 n − − u ∗ ( n ) + 24 a ( n ) + (24 n − p ( n ) , where as in (1.12), g ( m ) = 0 if m
23 mod 24 and g ( m ) = 0 if m
23 mod 24 and m Corollary 1.5. If n ≥ , then c ( n ) = s ( n ) n + 1 n X k ∈ Z h ( − k g (24 n − (6 k + 1) ) + ( − k g (24 n − (6 k + 1) ) i . Example.
Using our result, we find the following identities: c (1) = 196884 = s (24) + 168 a (1) − u ∗ (1) + 161 p (1)5 + 245 a (24) − u ∗ (24) + 115 p (24) c (2) = 21493760 = 12 (cid:16) s (48) − a (1) − u ∗ (1) + 161 p (1)5 + 14 a (2) − u ∗ (2) + 329 p (2)5 − a (24) + 125 u ∗ (24) − p (24) + 245 a (49) − u ∗ (49) + 235 p (49) (cid:17) . Question.
Are the combinatorial interpretations of the coefficients of the j -function inTheorem 1.2 and Corollary 1.5 glimpses of hidden structure of the monster module? Inparticular, do spt ( n ), u ∗ ( n ), and a ( n ) play roles in representation theory? Remark.
After this paper was submitted, Toshiki Matsusaka informed the authors [14] thathe has obtained further similar results along these lines which frame the spt function interms of a weakly holomorphic Jacobi form. This structure also provides a connectionto the formulation by Kaneko [13] of the j -function’s coefficients using traces of singularmoduli. LGEBRAIC RELATIONS BETWEEN PARTITION FUNCTIONS AND THE j -FUNCTION 7 This paper is organized as follows. In Section 2, we investigate the specific harmonicMaass form M ( τ ) and derive an expression for the action of the Hecke operator on itsholomorphic part. To do this, we study canonical families of polynomials in j ( τ ) andexplore the relationship of modular forms to modular functions on SL ( Z ). In Section 3,we prove Theorem 1.3. In Section 4, we prove Theorem 1.2 and Corollary 1.5.2. Harmonic Maass Forms
Preliminaries.
To motivate our study and to ground the methods used here, webegin by introducing the harmonic Maass form of interest for this paper. Recall that aweakly holomorphic modular form for a congruence subgroup Γ of SL ( Z ) is a function thatis holomorphic on H , whose poles, if any, are supported on the cusps of Γ \H , and whichsatisfies the corresponding modularity properties for its weight. If f is a weakly holomorphicmodular form of weight k for Γ and Nebentypus χ , we write f ∈ M ! k (Γ , χ ).Likewise, a smooth function f : H → C is a harmonic Maass form of weight k for Γ and χ if it satisfies the standard modular transformation laws, is annihilated by the harmonicLaplacian ∆ k , and has at most growth-order 1 exponential growth at each cusp on Γ \H .We denote the vector space of harmonic Maass forms of weight k for Γ and χ as H k (Γ , χ ).Recalling the definitions of P ( q ) and S ( q ) in (1.1) and (1.4), we define M ( τ ) following[16] as M ( τ ) := S ( q ) + 112 q ddq P ( q ) − i π √ · Z i ∞− ¯ τ η (24 z )[ − i ( z + τ )] / dz. (2.1)By Theorem 2.1 of [16], M ( τ ) ∈ H / (Γ (576) , χ ), where χ := (cid:0) · (cid:1) . By M + ( q ) wedenote the holomorphic part of M ( τ ). This may be expressed as M + ( q ) := S ( q ) + 112 q ddq P ( q ) = − q − + 3512 q + 656 q + · · · . The Hecke Action.
To understand the action of the Hecke operator on M + , wewill need the following result that produces a weakly holomorphic modular form involving M + ( τ ) | T ( ℓ ). We produce this modular form via the following result. Lemma 2.1. If M ℓ ( τ ) := M + ( τ ) | T ( ℓ ) − (cid:18) ℓ (cid:19) (1 + ℓ ) M + ( τ ) , then M ℓ ( τ ) ∈ M !3 / (Γ (576) , χ ) . Proof.
Up to a constant, the nonholomorphic part of M ( τ ) is the period integral for η (24 τ ).Write τ = x + iy for x, y ∈ R . Under the action of the differential operator ξ k := 2 iy k ∂∂τ ,we have ξ / ( M ) = − π η (24 τ ). Note that η (24 τ ) is an eigenform for Hecke operators ofweight 1 / χ ( ℓ )(1 + ℓ − ) = ( ℓ )(1 + ℓ − ). If we define M ℓ ( τ ) := M ( τ ) | T ( ℓ ) − (cid:18) ℓ (cid:19) (1 + ℓ ) M ( τ ) , we observe that ξ / (cid:20) M ( τ ) | T ( ℓ ) − (cid:18) ℓ (cid:19) (1 + ℓ ) M ( τ ) (cid:21) = 0 . (2.2) ALICE LIN, ELEANOR MCSPIRIT, AND ADIT VISHNU
Here we have used the commutativity relation ξ k ( f | k T ( ℓ )) = ℓ k − ( ξ k f ) | − k T ( ℓ )for half-integral weight harmonic Maass forms given in Proposition 7.1 of [9]. Since theHecke algebra preserves modularity, M ( τ ) | T ( ℓ ) ∈ H / (Γ (576) , χ ). By (2.2), M ℓ ( τ )is in the kernel of ξ / and is therefore holomorphic on the upper half plane. Since theaction of the Hecke and ξ operators are linear and thus split over the holomorphic andnonholomorphic parts of M ( τ ), the same result holds for M + ( τ ). In particular, M ℓ ( τ ) ∈ M !3 / (Γ (576) , χ ) . (cid:3) Canonical polynomials in j ( τ ) . We show that the set of all B m ( j ( τ )) form a conve-nient C -basis for the ring of weakly holomorphic modular functions on SL ( Z ) as a C -vectorspace. Recall that the ring of weakly holomorphic modular functions on SL ( Z ) is preciselythe ring of complex polynomials in j ( τ ), i.e. M !0 (SL ( Z )) = C [ j ( τ )] [4, Theorem 2.8]. Asdefined in (1.10), we have B ( x ) = 1 ,B ( x ) = x − ,B ( x ) = x − x + 357395 . From these first few examples, the set of B m ( x ) appears to form a C -basis for the polynomialring C [ x ] as a C -vector space, and hence the set of B m ( j ( τ )) would form a C -basis for M !0 (SL ( Z )). In the following lemma, we show that this is indeed the case. To do so, wedefine the function α ( q ) := ( q ; q ) ∞ − q ddq j ( τ ) = q + O ( q ) . (2.3) Lemma 2.2. If f ( τ ) is a weakly holomorphic modular function on SL ( Z ) and is of theform f ( τ ) = α ( q ) − X n ≫−∞ t ( n ) q n ! + O ( q ) , (2.4) then f ( τ ) = − X n ≫−∞ t ( n ) B − n ( j ( τ )) . Remark.
The above lemma gives a clean formulation for modular functions f of the formgiven in (2.4) when the principal part of f /α is known. Proof of Lemma 2.2.
For each m ≥
0, note that there exists a unique weakly holomorphicmodular function j m ( τ ) on SL ( Z ) such that j m ( τ ) = q − m + O ( q ). The Faber polynomials J n ( x ) are the coefficients of the generating function ∞ X n =0 J n ( x ) q n := E ( τ ) E ( τ )∆( τ ) · j ( τ ) − x = 1 + ( x − q + · · · . LGEBRAIC RELATIONS BETWEEN PARTITION FUNCTIONS AND THE j -FUNCTION 9 By Corollary 4 in [5], J n ( j ( τ )) = j n ( τ ) for all n ≥
0. By comparing the generating functionsfor J n ( x ) and B n ( x ) and using the identity (1.10), we see that α ( q ) X n ≥ J n ( x ) q n = α ( q ) · E ( τ ) E ( τ )∆( τ ) · j ( τ ) − x = ( q ; q ) ∞ · j ( τ ) − x = X n ≥ B n ( x ) q n . Since α ( q ) = q + O ( q ), we compare coefficients and deduce that for each n ≥ α ( q ) J n ( j ( τ )) = B n ( j ( τ )) = α ( q ) q − n + O ( q ) . And hence we can conclude that f ( τ ) = α ( q ) − X n ≫−∞ t ( n ) q n ! + O ( q ) = − X n ≫−∞ t ( n ) B − n ( j ( τ )) . (cid:3) Proof of Theorem 1.1.
Note that we may write M ℓ ( z ) = − ℓ q − ℓ + (cid:18) ℓ (cid:19) ℓ q − + X n ≥ n ≡
23 mod 24 a ℓ ( n ) q n , (2.5)where we observe that, since ℓ ≡ M ℓ are supportedon integral exponents that are 23 mod 24. Following this, we define F ℓ (24 τ ) := η ℓ (24 τ ) M ℓ ( τ ) . (2.6)By Lemma 2.1, it is immediate that F ℓ (24 τ ) is a weakly holomorphic modular form ofweight ℓ +32 over Γ (576) with trivial Nebentypus. In fact, by Theorem 2.2 in [16], F ℓ ( τ ) isa weight ℓ +32 holomorphic modular form on SL ( Z ). We recall that the proof makes use ofthe observation that F ℓ ∈ Z [[ q ]] by construction, and that the behavior of F ℓ under thematrix S = (cid:0) −
11 0 (cid:1) can be determined using a result of Bringmann in [7] which gives that M ( τ ) is an eigenform of the Fricke involution.2.4.1. Getting to Weight 0.
Now that we have a holomorphic modular form of weight ℓ +32 onall of SL ( Z ), we will leverage this information, along with some properties of the Eisensteinseries E and the j -function, to produce a closed formula for the Hecke action. We first notethat ℓ +32 ≡ E ( τ ) E ( τ ). To make use of this seeminglyinnocuous fact, define δ ℓ := ℓ − and note that G ℓ ( τ ) := E ( τ ) E ( τ )∆ δ ℓ − ( τ ) = q δ ℓ − + . . . ∈ M ℓ (SL ( Z )) . (2.7)Since we now have another modular form of the same weight on SL ( Z ), we would liketo prove that their quotient, F ℓ ( τ ) /G ℓ ( τ ), is a weakly holomorphic modular function onSL ( Z ), which, coupled with our preceding characterization of the Faber polynomials, willallow for a unique expression of the quotient as a polynomial in j ( τ ). Lemma 2.3.
The function F ℓ ( τ ) /G ℓ ( τ ) is a polynomial in j ( τ ) .Proof. By construction, G ℓ has a zero of degree 2 at e πi/ , a simple zero at i , and no otherzeros in the fundamental domain F of SL ( Z ).Since the weight of F ℓ is k = ( ℓ + 3) / ≡ S = (cid:0) −
11 0 (cid:1) to get that F ℓ ( − /i ) = i k F ℓ ( i ) = − F ℓ ( i ), hence F ℓ ( i ) = 0. Similarly, applying the transformation law under γ = (cid:0) −
11 1 (cid:1) yields F ℓ ( e πi/ ) = 0. Differentiatingboth sides of F ℓ ( γτ ) = ( τ + 1) k F ℓ ( τ ) and letting τ = e πi/ gives that ddτ F ℓ ( τ ) | τ = e πi/ = 0.Hence F ℓ vanishes at e πi/ with order at least 2. Therefore the quotient F ℓ /G ℓ has no polesin F , and we may deduce that F ℓ /G ℓ is a weakly holomorphic modular form of weight 0 onSL ( Z ). Since the modular functions on SL ( Z ) are precisely the polynomials C [ j ( τ )], wemay conclude that F ℓ /G ℓ is a polynomial in j ( τ ). (cid:3) It remains to construct this polynomial in j ( τ ). Using the modular functions B δ ℓ ( j (24 τ )),we arrive at the following conclusion: F ℓ ( τ ) G ℓ ( τ ) = η ( τ ) ℓ M ℓ ( τ / E ( τ ) E ( τ )∆ δ ℓ − ( τ )= ( q ; q ) ∞ − q ddq j ( τ ) q / M ℓ ( τ / q ; q ) ∞ − q ddq j ( τ ) (cid:20) − ℓ q − δ ℓ + (cid:18) ℓ (cid:19) ℓ
12 + O ( q ) (cid:21) = α ( q ) (cid:20) − ℓ q − δ ℓ + (cid:18) ℓ (cid:19) ℓ
12 + O ( q ) (cid:21) = − ℓ B δ ℓ ( j ( τ )) , where the last equality follows from Lemma 2.2. Hence we may rearrange to get the ex-pression M ℓ ( τ /
24) = F ℓ ( τ ) η ℓ ( τ ) = − ℓ E ( τ ) E ( τ )∆( τ ) η − ( τ ) B δ ℓ ( j ( τ )) . (2.8)Sending τ τ and using the fact that P ( q ) = η − (24 τ ), M ℓ ( τ ) = P ( q ) (cid:18) E (24 τ ) E (24 τ )∆(24 τ ) (cid:19) (cid:20) − ℓ B δ ℓ ( j (24 τ )) (cid:21) . We can finally conclude that the action of the Hecke operator T ( ℓ ) is M + ( τ ) | / T ( ℓ ) = (cid:18) ℓ (cid:19) (1 + ℓ ) M + ( τ ) − ℓ P ( q ) · B δ ℓ ( j (24 τ )) · E (24 τ ) E (24 τ )∆(24 τ ) , concluding the proof of Theorem 1.1. (cid:3) The Signed Triangular Weight
In light of the connection between the generating functions of the Andrews spt -functionand a particular class of unimodal sequences given in (1.14) mediated by the series A ( q ),we present the proof of Theorem 1.3.3.1. Proof of Theorem 1.3.
We begin by examining the summation X n ≥ q n ( n +1)2 − q n . (3.1)By considering each summand to be of the form q n ( n +1)2 (1 + q n + q n + q n + · · · ), notethat if we formally expand the above power series as P m ≥ α ( m ) q m , then α ( m ) counts the LGEBRAIC RELATIONS BETWEEN PARTITION FUNCTIONS AND THE j -FUNCTION 11 number of ways to choose integers ( n, k ) with n ≥ k ≥ m = T n + kn , where T n = n ( n + 1) / n th triangular number. Similarly, the coefficient β ( m ) of q m in the formal expansion of X n ≥ ( − n − nq n ( n +1)2 − q n := X m ≥ β ( m ) q m (3.2)denotes a sum over all such pairs ( n, k ), weighted by the parity and size of n .Multiplying the above series by the generating function 1 / ( q ; q ) ∞ for partitions then givesa formal power series 1( q ; q ) ∞ X n ≥ ( − n − nq n ( n +1)2 − q n := X m ≥ γ ( m ) q m (3.3)where γ ( m ) runs over all partitions λ ⊢ m such that λ contains a subpartition consisting ofthe parts { , , . . . , n } and also possibly k more parts of size n , for n ≥ k ≥
0, butweighting this count by the parity and size of n . (cid:3) Combinatorial interpretations of the coefficients of j ( τ )As we have now developed a variety of both combinatorial and number-theoretic objects,all of which are tied together by a class of polynomials in j (24 τ ), it is natural to ask if wemay formalize and explicate this connection. To do this, we make use of both the standarddefinition of the Hecke operator on q -series expansions as well as the result of Theorem 1.1in order to pull the functions spt ( n ) and p ( n ) through to the j -function. We restrict ourattention to the case where ℓ = 5 since δ ℓ = 1 and B ( j (24 τ )) = 1 . While at first glance itmay seem as though we have removed j from our expressions by looking at this case, werecall that − q ddq j (24 τ ) = E (24 τ ) E (24 τ )∆(24 τ ) = q − − X n ≥ nc ( n ) q n , (4.1)where c ( n ) is the n th coefficient of the j -function. Thus, we need only solve for the c ( n ) ′ s interms of the combinatorial information given by M + ( τ ) to arrive at our final conclusions.4.1. Proof of Theorem 1.2.
Writing the q -series expansion for M + ( τ ) out in terms of spt ( n ) and p ( n ), we arrive at M + ( τ ) = − q − + X n ≥ (cid:20) spt ( n ) + (24 n − p ( n ) (cid:21) q n − . For n ≥
1, we then write for ease m (24 n −
1) := spt ( n ) + 24 n − p ( n ) . (4.2)Now we can describe the action of the Hecke operator T (25) as follows: M + ( τ ) | T (25) = − q − + 112 q − + X n ≥ m (24 n − q n − + X n ≥ (cid:20) m (25(24 n − − (cid:18) − n + 15 (cid:19) m (24 n − (cid:21) q n − . Since M ( τ ) = M + ( τ ) | T (25) + 6 M + ( τ ), we have M ( τ ) = − q − − q − + X n ≥ m (24 n − q n − + X n ≥ (cid:20) m (25(24 n − (cid:20) − (cid:18) − n + 15 (cid:19)(cid:21) m (24 n − (cid:21) q n − . Thus, when ℓ = 5, the statement of Theorem 1.1 reduces to M ( τ ) = − η − (24 τ ) q − − X n ≥ nc ( n ) q n and we are able to rearrange as follows: − q − + X n ≥ nc ( n ) q n = η (24 τ ) h − q − − q − + 125 X n ≥ h m (25(24 n − h − (cid:16) − n (cid:17)i m (24 n − i q n − + X n ≥ m (24 n − q n − i . Recall the definitions of h ( m ) and h ( m ) in (1.12). Using these, we define δ ( n ) := X k ∈ Z ( − k h ( n − (6 k + 1) ) ,δ ( n ) := X k ∈ Z ( − k h ( n − (6 k + 1) ) . Then we may write − q − + X n ≥ nc ( n ) q n = X n ≥ δ (24 n ) q n + X n ≥ δ (24(25 n − q n − − ∞ X n = −∞ ( − n q (6 n +1) − − ∞ X n = −∞ ( − n q (6 n +1) − . We note that for n ≥ X n ≥ s ( n ) q n = − ∞ X n = −∞ ( − n q (6 n +1) − − ∞ X n = −∞ ( − n q (6 n +1) − . Thus, Theorem 1.2 follows by solving for c ( n ). (cid:3) Proof of Corollary 1.5.
This result follows immediately from Theorem 1.2 and the relation spt ( n ) = − u ∗ ( n ) + 2 a ( n ). Remark.
While the results above use only the action of the specific Hecke operator T (25),one should note that the entire sequence of operators T ( ℓ ) generate similar results for thepolynomials B δ ℓ ( j (24 τ ). We outline this process below. We define q ddq j (24 τ ) · B δ ℓ ( j (24 τ )) := X n ≫−∞ r ℓ ( n ) q n . LGEBRAIC RELATIONS BETWEEN PARTITION FUNCTIONS AND THE j -FUNCTION 13 Then likewise if m ℓ (24 n −
1) := m ( ℓ (24 n − (cid:18) ℓ (cid:19) (cid:20)(cid:18) − (24 n − ℓ (cid:19) − (1 + ℓ ) (cid:21) m (24 n − ℓm ((24 n − /ℓ ) , we may write M ℓ ( τ ) = − ℓ q − ℓ + (cid:18) ℓ (cid:19) ℓ q − + X n ≥ m ℓ (24 n − q n − . Rewriting the result of Theorem 1.1, we have X n ≫−∞ r ℓ ( n ) q n = 12 ℓ η (24 τ ) M ℓ ( τ ) . (4.3)Thus, expanding the right-hand side using the pentagonal number theorem allows one tosolve for r ℓ ( n ). By Theorem 1.2 and Corollary 1.5, the coefficients of q ddq j (24 τ ) are knownin terms of combinatorial quantities, and so the coefficients of B δ ℓ ( j ( τ )) themselves canwritten as a sequence of combinatorial expressions as well. Acknowledgments
We thank Professor Ken Ono for his invaluable mentorship and guidance in producingthis paper. We also thank the reviewer for their careful reading and helpful suggestions.We additionally give thanks to the support of the Asa Griggs Candler Fund and the grantsfrom the National Security Agency (H98230-19-1-0013), the National Science Foundation(1557960, 1849959), and the The Spirit of Ramanujan Talent Initiative.
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