Dedicated to Krishna Alladi on the occasion of his 60th birthday

Transcription

1 A classical q-hypergeometric approach to the A () standard modules Andrew V. Sills Dedicated to Krishna Alladi on the occasion of his 60th birthday Abstract This is a written expansion of the talk delivered by the author at the International Conference on Number Theory in Honor of Krishna Alladi for his 60th Birthday, held at the University of Florida, March 7, 06. Here we derive Bailey pairs that give rise to Rogers Ramanujan type identities, the product sides of which are known to be the principally specialized characters of the A () standard modules (l i + )Λ 0 + (i )Λ for any level l, and i,. Notation and Motivation. q-series notation and classical results Let q denote a formal variable. The standard notation for the infinite rising q- factorial is (a;q) : j0 ( aq j ). In order to allow for positive and negative values of n, we define the finite rising q-factorial as (a;q) n : (a;q) (aq n. ;q) We will also use the abbreviations (q) n and (q) for (q;q) n and (q;q) respectively. Additionally, Andrew V. Sills Georgia Southern University, Statesboro, Georgia, USA, ASills@georgiasouthern.edu

2 Andrew V. Sills and (a,a,...,a r ;q) n : (a ;q) n (a ;q) n (a r ;q) n (a,a,...,a r ;q) : (a ;q) (a ;q) (a r ;q). The bilateral basic hypergeometric series is given by [ ] a,a,...,a r (a,a,,a r ;q) n rψ r ;q,z : b,b,...,b r z n. n Z (b,b, b r ;q) n For m,n Z the q-binomial coefficients are defined as [ ] n (q) n if 0 m n : (q) m (q) n m m q 0 otherwise. We will require the following classical results: Jacobi s triple product identity [7, p. 5, Eq. (.6.)]. ( ) n z n q n (q/z,zq,q ;q ), () n Z and a result that follows from () and the quintuple product identity ([6], cf. [7, p. 47, ex. 5.6]) ( qz 3, q z 3,q 3 ;q 3 ) z( qx 3, q z 3,q 3 ;q 3 ) (q/z,z,q;q) (q/z,qz ;q ). () In fact, Eq. () is called The quintuple product identity; second version in [0, p. 9, Theorem.3.8]. Herein, we will only need to consider z of the form z ±q r for some r Z.. Certain affine Lie algebras and their connection to q-series Let g denote the affine Lie algebra A () or A () and let h 0,h denote the usual basis of a maximal toral subalgebra T of g. Let d denote the degree derivation of g and let T : T Cd. For all dominant integral λ T, there is a unique irreducible, integrable, highest weight module L(λ), assuming (without loss of generality) that λ(d) 0. Also, λ s 0 Λ 0 + s Λ where Λ 0 and Λ are the fundamental weights, given by Λ i (h j ) δ i j and Λ i (d) 0; s 0 and s are nonnegative integers. For A () the canonical central element is c h 0 + h, and for A (), the canonical central element is c h 0 + h. The level λ(c) λ g (c) of L(λ) is,

3 A classical q-hypergeometric approach to the A () standard modules 3 { s λ(c) 0 + s if g A (), s 0 + s if g A (), (cf. [0], [3]). For brevity, it is common to refer to L(λ) L(s 0 Λ 0 + s Λ ) as the (s 0,s )-module. Additionally ([3]), there is an infinite product F g associated with g, sometimes called the fudge factor, which needs to be divided out of the the principally specialized character χ(l(λ)) χ(s 0 Λ 0 + s Λ ), in order to obtain the quantities of interest here. The fudge factor F g is given by (q;q ) if g A () F g ( ), (q;q 6 ) (q 5 ;q 6 () ) if g A. Also, g has a certain infinite-dimensional Heisenberg subalgebra known as the principal Heisenberg vacuum subalgebra s (consult [4] for the construction of A () and [] for that of A () ). As demonstrated in [5], the principal character χ(ω(s 0 Λ 0 + s Λ )), where Ω(λ) is the vacuum space for s in L(λ), is χ(ω(s 0 Λ 0 + s Λ )) χ(l(s 0Λ 0 + s Λ )) F g, (3) where χ(l(λ)) is the principally specialized character of L(λ). By [3] applied to (3) in the case of A (), the standard modules of odd level correspond to Andrews analytic generalization of the Rogers Ramanujan identities [3], known as the Andrews Gordon identity, and the partition theoretic generalization of the Rogers Ramanujan identities due to B. Gordon [8]. Bressoud s even modulus counterpart to the Andrews Gordon identity [, p. 5, Eq. (3.4)] and its partition theoretic counterpart [, p. 64, Theorem, j 0 case]; was explained vertex-operator theoretically in [6] and [7] to correspond to the standard modules of even level in A (). The combined Andrews Gordon Bressoud identity (for both even and odd moduli) and its correspondence to the level l standard modules of A () can be stated compactly as χ(ω((l + i)λ 0 + (i )Λ )) n n n k 0 where k : k(l) + l/, i k, and q n +n + +n k +n i+n i+ + +n k (q) n n (q) n n 3 (q) nk n k (q) nk ( q) [ l]nk (qi,q l+ i,q l+ ;q l+ ) (q), (4)

4 4 Andrew V. Sills { if P is true, [P] : 0 if P is false, is the Iverson bracket notation introduced by K.E. Iverson in [9] and advocated for by D.E. Knuth in []. A pair of sequences (α n (a,q),β n (a,q)) form a Bailey pair with respect to a if β n (a,q) n s0 α n (a,q). (q) n s (aq;q) n+s [5, p. 5 6]; cf. [9, pp., 5]. It is well known that identities of Rogers Ramanujan type may be derived by the insertion of Bailey pairs into limiting cases of Bailey s lemma [5, p. 5, Thm. 3.3; p. 7, Eq. (3.33)] such as a n q n β n (a,q) n0 (aq;q) a n q n α n (a,q), (5) n0 and setting a equal to a power of q. An efficient method for deriving (4) for odd l is via the Bailey lattice [], which is an extension of the Bailey chain ([4]; cf. [5, 3.5, pp. 7ff]) built upon the unit Bailey pair where δ i j is the Kronecker δ-function, β n (,q) δ n0, { if n 0 α n (,q) ( ) n q n(n )/ ( + q n ) if n > 0. Similarly, for even l, (4) follows from a Bailey lattice built upon the Bailey pair β n (,q) (q ;q, ) n { if n 0 α n (,q) ( ) n q n if n > 0. Thus the standard modules of A () correspond to two interlaced instances of the Bailey lattice. In contrast, the standard modules of A () are not as well understood, and a uniform q-series and partition correspondence analogous to what is known for A () has to date remained beyond our reach. As with A (), there are + l inequivalent level l standard modules associated with the Lie algebra A (), but the principal characters for the level l standard mod-

5 A classical q-hypergeometric approach to the A () standard modules 5 ules are given by instances of the quintuple product identity () (rather than the triple product identity) divided by (q) : χ(ω((l i + )Λ 0 + (i )Λ )) where i + l ; see [3]. (qi,q l+3 i,q l+3 ;q l+3 ) (q l+3 i,q l+i+3 ;q l+6 ) (q), (6) Bailey pairs for A () Let (α n (l,i),β n (l,i) ) denote the Bailey pair which, upon insertion into (5) with a, gives the principally specialized character of the A () standard module corresponding to the highest weight (l i + )Λ 0 + (i )Λ. The strategy is to choose α (l,i) n a standard transformation to find the corresponding β (l,i) n. that gives rise to the desired product side, and use. Bailey pairs for χ(ω(lλ 0 )) if n 0 α n α n (l,) q 3 (l 3)r (l 3)r + q 3 (l 3)r + (l 3)r if n 3r > 0 (,q) q 3 (l 3)r + (l 3)r if n 3r + q 3 (l 3)r (l 3)r if n 3r

6 6 Andrew V. Sills β (l,) n (,q) n s0 α s (l,) (, q) (q) n s (q) n+s α 0 (q) + n r α 3r + (q) n 3r (q) n+3r (q) n 3r (q) n+3r+ + (q) n 3r+ (q) n+3r r α 3r r 0 α 3r+ q 3 (q) + (l 3)r (l 3)r + q 3 (l 3)r + (l 3)r n r (q) n 3r (q) n+3r + + (q) n 3r (q) n+3r+ (q) n 3r+ (q) n+3r r 0 q 3 (l 3)r + (l 3)r q 3 (l 3)r + (l 3)r r Z (q) n 3r (q) n+3r r Z r q 3 (l 3)r (l 3)r q 3 (l 3)r + (l 3)r (q) n 3r (q) n+3r+ q 3 (l 3)r + (l 3)r ( ) ( q n+3r+ ) ( q n 3r ) (q) n 3r (q) n+3r+ r Z (q n 3r q n+3r+ ) (q) n 3r (q) n+3r+ r Z q 3 (l 3)r + (l 3)r q 3 (l 3)r + (l 3)r+n 3r ( q 6r+ ) r Z (q) n 3r (q) n+3r+ q n q 3 (q) n (q) n+ (l 3)r + (l 9)r ( q 6r+ )(q n 3r+ ;q) 3r r Z (q n+ ;q) 3r q n ( q (q) n (q ;q) n 6r+ )(q n ;q) 3r r Z ( q)(q n+ ( ) r q 3 (l 6)r + (l 6)r. (7) ;q) 3r For each l,,3,..., the series expression in (7) is a limiting case of a verywell-poised bilateral basic hypergeometric series. For example, we have

7 A classical q-hypergeometric approach to the A () standard modules 7 q n (q) n (q) n+ β (l,) (,q) q [ 7 ] q, q 7,q n,q n,q n,e,e,e lim e 0 8 ψ 8 q, q,q n+4,q n+3,q n+, q4 e, ;q 3, q3n+6 q4 e e 3 if l 3 [ 7 ] q, q 7,q n,q n,q n,e,e, q lim e 0 8 ψ 8 q, q,q n+4,q n+3,q n+, q4 ;q3, q3n+4 e, q e if l 4 [ 7 ] q, q 7,q n,q n,q n,e lim e 0 6 ψ 6 q, q ;q 3, q3n+ if l 5,q n+4,q n+3,q n+, q4 e [ e 7 ] q, q 7,q n,q n,q n, q 6ψ 6 q, q,q n+4,q n+3,q n+, q ;q3, q 3n if l 6. [ 7 ] q, q 7,q n,q n,q n,e lim e 6 ψ 6 q, q ;q 3, q3n+ if l 7,q n+4,q n+3,q n+, q4 e [ e 7 ] q, q 7,q n,q n,q n,e,e, q lim e 8 ψ 8 q, q,q n+4,q n+3,q n+, q4 ;q3, q3n+4 e, q e if l 8 [ 7 ] q, q 7,q n,q n,q n,e,e,e lim e 8 ψ 8 q, q,q n+4,q n+3,q n+, q4 e, ;q 3, q3n+6 q4 e e 3 if l 9 Observe that the easiest cases are l 5,6,7, as these are instances of Bailey s summable bilateral very-well-poised 6 ψ 6 [8, Eq. (4.7)]; cf. [7, p. 357, Eq. (II.33)]. Indeed, Slater evaluated the cases l 5 and 7 [33, p. 464, Eqs. (3.4) and (3.3) resp.], while McLaughlin and Sills evaluated the case l 6 [8, p. 77, Table 3., line (P)]. We have β n (5,) (,q) qn, (8) (q) n β n (6,) (,q) qn ( ;q 3 ) n, (9) ( ;q) n (q) n β (7,) n (,q) qn (q) n. (0) To evaluate β n (l,) (,q) for levels l 3,4,8,9, we can use the following identity [7, p. 47, exercise 5.], analogous to Bailey s 6 ψ 6 sum: for nonnegative integer n, [ q a, q a,c,d,e, f,aq n,q n 8ψ 8 a, a,aq/c,aq/d,aq/e,aq/ f,q n+,aq n+ ;q, a q n+ ] (aq, q a, aq cd, aq e f ;q) n ( q c, q d, aq e, aq f ;q) n 4 ψ 4 cde f aqn+ e, f, cd,q n aq c, aq d,qn+, e f ;q,q. () aq n

8 8 Andrew V. Sills Notice that for level l and, q n (q) n (q ;q) n β n (l,) is a limiting case of a 0ψ 0, while for l > 6, it is a limiting case of a r ψ r with r l + ( + ( ) l )/. More precisely, if l is even and l 6, q n (q) n (q ;q) n+ β n (l,) (, q) lim e lψ l q 7, q 7,q n,q n,q n {}}{, e,e,...,e, q q, q,q n+4,q n+3,q n+, q4 q4,...,, q } e {{ e ;q3, q3n+l e l 6, } l 6 while if l is odd and l > 6, q n (q) n (q ;q) n+ β n (l,) (, q) lim e l ψ l l 6 l 6 q 7, q 7,q n,q n,q n {}}{, e,e,...,e q, q,q n+4,q n+3,q n+, q4 q4,..., e e } {{ } l 6 ;q 3, q3n+l e l 6 Then, to obtain ( the series and product ) expressions for χ(ω(lλ 0 )), one inserts the Bailey pair α n (l,) (,q),β n (l,) (, q) into (5) with a, and upon applying () and (), we find that q 9m( ) q 6m+ β (l,) (l,) 3m (,q) + β 3m (,q) + q6m+ β (l,) 3m+ (,q) m0. (q,ql+,q l+3 ;q l+3 ) (q l+,q l+5 ;q l+6 ) (q). () And thus in (), we have a uniform series-product identity for the principally specialized character of the (l,0) standard module of A () for any l. as a multisum for arbitrary l, one may employ the Andrews Baxter Forrester bilaterial very-well poised q-hypergeometric summation formula [5, p. 83, Eq. (8.56)]; cf. [7, Appendix B, pp. 6 65]. The level 3 case will be considered in detail in the next section. To express the β (l,) n. Bailey pairs for χ(ω((l )Λ 0 +Λ ))

9 A classical q-hypergeometric approach to the A () standard modules 9 if n 0 α n α n (l,) q 3 (l 3)r (9 l)r + q 3 (l 3)r + (9 l)r if n 3r > 0 (,q) q 3 (l 3)r + (l+3)r+. if n 3r + q 3 (l 3)r (l+3)r+ if n 3r The calculation of β n (l,) parallels that of β n (l,). The details of the l 7 case are given by Slater [33, p. 464]. β (l,) n (,q) Notice that n s0 α s (l,) (, q) (q) n s (q) n+s ( q (q) n (q ;q) n 6r+ )(q n ;q) 3r r Z ( q)(q n+ ( ) r q 3 (l 6)r + (l 6)r. (3) ;q) 3r q n β n (l,) (,q) β n (l,) (,q). (4) And so it follows that the series and product expressions for χ(ω((l )Λ 0 + Λ )) are q 9m( ) q 6m+ β (l,) (l,) 3m (,q) + β 3m (,q) + q6m+ β (l,) 3m+ (,q) m0 a general identity corresponding to the (l,) module. (q,q l+,q l+3 ;q l+3 ) (q l,q l+7 ;q l+6 ) (q), (5) 3 Level 3 Let us consider the case l 3 in detail. This level is of particular interest as it was the study of the the level 3 standard modules of A () that led S. Capparelli to discover two new Rogers Ramanujan type partition identities [, 6, 4, 5]. See [3, 3] for some historical notes. From the (3, 0)-module, Capparelli conjectured (and later proved [5], although the first proof was due to Andrews [6]) the following partition identity. A partition λ of an integer n is a finite weakly decreasing sequence (λ,λ,...,λ l ) of positive integers that sum to n; each λ i is called a part of the partition λ. Theorem (Capparelli s first partition identity). Let c (n) denote the number of partitions λ (λ,...,λ l ) of n wherein λ i for i,,...,l, λ i λ i+, for i,,...,l, λ i λ i+ only if λ i (mod 3),

10 0 Andrew V. Sills λ i λ i+ 3 only if λ i 0 (mod 3). Let c (n) denote the number of partitions of n into distinct parts ± (mod 6). Let c 3 (n) denote the number of partitions of n into parts congruent to ±,±3 (mod ). Then c (n) c (n) c 3 (n) for all n. From the (,)-module, Capparelli obtained the companion identity: Theorem (Capparelli s second partition identity). Let d (n) denote the number of partitions λ (λ,...,λ l ) of n wherein λ i for i,,...,l, λ i λ i+, for i,,...,l, λ i λ i+ only if λ i (mod 3), λ i λ i+ 3 only if λ i 0 (mod 3). Let d (n) denote the number of partitions of n into distinct parts ± (mod 6). Then d (n) d (n) for all n. 3. χ(ω(3λ 0 )) In order to use (), we need to consider three cases, n 3m, 3m +, and 3m. In (), replace q by q 3 ; then set a q, n m, f q 3m, and c d e, to obtain [ ] β (3,) 3m (,q) qn ( q) q 7/, q 7/,e,e,e,q 3m,q 3m,q 3m lim 8ψ 8 (q) n (q) n+ e 0 q /, q /, q4 e,q3m+,q 3m+3,q 3m+4 ;q3,q 3 q3m ( q) (q 4,q, q4, q3m+ e lim e ;q 3 ) m (q) 3m (q) 3m+ e 0 ( q3 e, q3 e,q3m+ ;q 3 ) m 4 ψ 4 e,q 3m, q3m+4,q 3m e q 4 e,q3m+3,eq 6m ;q3,q 3 m r m m r0 ( ) m+r q 3 m m r 3mr+ 3 r (q ;q 3 ) m (q 3 ;q 3 ) m+r (q;q 3 ) m r (q 3 ;q 3 ) m r ( ) r q 3 r + r (q ;q 3 ) r (q ;q 3 ) m (q 3 ;q 3. (6) ) m r (q) 3r In (), replace q by q 3 ; then set a q, n m, f q 3m, and c d e, to obtain

11 A classical q-hypergeometric approach to the A () standard modules [ ] β (3,) 3m+ (,q) qn ( q) q 7/, q 7/,e,e,e,q 3m,q 3m,q 3m lim 8ψ 8 (q) n (q) n+ e 0 q /, q /, q4 e,q3m+,q 3m+5,q 3m+4 ;q3,q 3 q3m+ ( q) (q 4,q, q4, q3m+5 e lim e ;q 3 ) m (q) 3m+ (q) 3m+ e 0 ( q3 e, q3 e,q3m+5 ;q 3 ) m 4 ψ 4 e,q 3m, q3m+4,q 3m e q 4 e,q3m+3,eq 6m ;q3,q 3 m r m m r0 ( ) m+r q 3 m + 7 m 7 r 3mr+ 3 r + (q ;q 3 ) m+ (q 3 ;q 3 ) m+r (q;q 3 ) m r+ (q 3 ;q 3 ) m r ( ) r q 3 r + 7 r+ (q ;q 3 ) r (q ;q 3 ) m+ (q 3 ;q 3. (7) ) m r (q) 3r+ For convenience, let us define the abbreviation σ(m,r) : so that we have immediately ( )r q 3 r + r (q ;q 3 ) r (q ;q 3 ) m (q 3 ;q 3 ) m r (q) 3r, (8) and with a bit of elementary algebra, β (3,) β (3,) 3m (,q) m r0σ(m,r), m ( ) 3m+ (,q) σ(m,r) r0 q 6m+ q 3r+, for m 0. The author could not find a direct substitution into (), analogous to the n 3m and n 3m+ cases, which yields the n 3m case. So we resort to an alternate method to obtain the n 3m case. From the Paule Riese qzeil.m Mathematica package available for download at and documented in [30], one can find that β n (3,) (,q) satisfies the recurrence β n q + q n + q n+ q ( q n )( q n ) β n ( q n )( q n ) β n (9) as certified by the rational function q n 6r ( q 3r q n)( q 3r+ q n)( q 3r+ q n) (q n )(q n + )(q n q)(q 6r+. )

12 Andrew V. Sills Setting n 3m+ in (9) and rearranging, we see how to express β 3m in terms of the two known expressions β 3m and β 3m+ : β 3m ( q 6m+ )( q 6m+ )β 3m+ ( q 6m q 6m+ )β 3m ( q 6m+ )( q 6m+ m ( ) σ(m,r) ) q 6m+ q 3r+ ( q 6m+ ) m r0 σ(m,r) r0 + ( q 6m+ )β 3m ( q 6m q 6m+ )β 3m m r0 σ(m,r) q 3r+ + q6m β 3m (q 6m q6m+ q 3r+ for m. ( ) Inserting α n (3,) (,q),β n (3,) (, q) we find that ), (0) into (5) with a, and applying () and (), q 9m( ) q 6m+ β (3,) (3,) 3m (,q) + β 3m (,q) + q6m+ β (3,) 3m+ (,q) m0 m (q (q 6m 6m q6m+ m0 r0 q 9m σ(m,r) ( + q6m+ q 6m+ (q,q5,q 6 ;q 6 ) (q 4,q 8 ;q ) (q) q 3r+ q 3r+ )) ) + (q,q 3,q 9,q 0 ;q ) ( q ;q ) ( q 3 ;q 6 ). () The series expansion of (q,q 3,q 9,q 0 ;q ) in () is quite different than others that have appeared in the literature, due to Alladi, Andrews, and Gordon [, pp , Lemma (b)] (cf. [3, p. 399, Eq. (.3)]), the author [3, p. 399, Eq. (.4) and Eq. (.5)], and Bringmann and Mahlburg [3]. 3. χ(ω(λ 0 +Λ )) ( In light of (4), it is trivial ) to obtain β n (3,) (,q) from β n (3,) (, q), and upon inserting α n (3,) (,q),β n (3,) (, q) into (5) with a, and applying () and (), we find that

13 A classical q-hypergeometric approach to the A () standard modules 3 m (q (q 6m 6m q6m+ m0 r0 q 9m 3m σ(m,r) q 3r+ ) + ( )) + q6m q 6m+ q 3r+ (q,q 4,q 6 ;q 6 ) (q,q 0 ;q ) (q) ( q;q ) ( q 6 ;q 6 ). () 3.3 Nandi s recent work on level 4 It should be noted that recently D. Nandi, in his Ph.D. thesis [9] conjectured the partition identities corresponding to the three inequivalent level 4 standard modules (4,0), (,) and (0,). These identities, while still in the spirit of the Rogers Ramanujan and Capparelli identities, involve difference conditions that are much more complicated than anything that has been considered previously in the theory of partitions. It is no wonder that after Capperelli s discoveries for level 3, it took a quarter century to successfully perform the analogous feat for level 4. 4 Bailey pairs for levels 3 through 9 summarized 4. Level 3 β (3,) 3m (,q) m r m m r0 β (3,) 3m+ (,q) m r m m r0 ( ) m+r q 3 m m+3mr+ 3 r r (q ;q 3 [ ] ) m m (q) 6m (q ;q 3 ) m+r m + r q 3 ( ) r q 3 r r (q;q 3 ) m (q ;q 3 ) r (q 3 ;q 3 ) m r (q 3 ;q 3 ) r ( ) m+r q 3 m + 5 m+3mr+ 3 r + 5 r+ (q ;q 3 [ ] ) m m (q) 6m+ (q ;q 3 ) m+r m + r q 3 ( ) r q 3 r + 5 r (q;q 3 ) m+ (q ;q 3 ) r+ (q 3 ;q 3 ) m r (q 3 ;q 3 ) r β 3m ( q 6m+ )( q 6m+ )β 3m+ ( q 6m q 6m+ )β 3m

14 4 Andrew V. Sills 4. Level 4 β (4,) 3m (,q) m r m m r0 β (4,) 3m+ (,q) m r m m r0 ( ) m+r q 3m +3mr+ 3 r r ( q ;q 3 ) r (q ;q 3 [ ] ) m m (q) 6m ( q ;q 3 ) m (q ;q 3 ) m+r m + r q 3 ( ) r q 3m 3mr+3r ( q;q 3 ) m r (q;q 3 ) m ( q ;q 3 ) m (q ;q 3 ) r (q 3 ;q 3 ) m r (q 3 ;q 3 ) r ( ) m+r q 3m +3m+3mr+ 3 r + 5 r+ ( q ;q 3 ) r (q ;q 3 [ ] ) m m (q) 6m+ ( q ;q 3 ) m (q ;q 3 ) m+r+ m + r q 3 ( ) r q 3m 3mr+3r +3r+ ( q;q 3 ) m r (q;q 3 ) m+ ( q ;q 3 ) m (q ;q 3 ) r+ (q 3 ;q 3 ) m r (q 3 ;q 3 ) r q 3m β 3m ( q 6m+ )( q 6m+ )( + q 3m )β 3m+ q 3m ( + q 3m )( q 3m q 3m+ )β 3m 4.3 Level 5 β (5,) 3m q9m (,q), (q) 6m and β (5,) 3m+ (,q) q9m +6m+ (q) 6m+, which, together, simplifies to Eq. (8). q 6m β 3m ( q 6m )( q 6m )β 3m, 4.4 Level 6 β (6,) 3m (,q) q3m ( ;q) 3m ( ;q 3 ) 3m (q) 6m, β (6,) 3m+ (,q) q3m+ ( ;q 3 ) 3m+ (q) 6m+ ( ;q) 3m+,

15 A classical q-hypergeometric approach to the A () standard modules 5 and and thus (9) holds. (q + q 6m q 3m )β 3m ( q 6m )( q 6m )β 3m, 4.5 Level 7 β (7,) 3m q3m (,q), (q) 6m and and thus (0) holds. β (7,) q3m+ 3m+ (,q), (q) 6m+ qβ 3m ( q 6m )( q 6m )β 3m, 4.6 Level 8 β (8,) 3m (,q) m r m m r0 β (8,) 3m+ (,q) m r m m r0 q 3 r + r+3m ( q ;q 3 ) r (q ;q 3 [ ] ) m m (q) 6m ( q ;q 3 ) m (q ;q 3 ) m+r m + r q 3 q 3m +m+3r +r 6mr ( q;q 3 ) m r (q;q 3 ) m ( q ;q 3 ) m (q ;q 3 ) r (q 3 ;q 3 ) m r (q 3 ;q 3 ) r, q 3 r + r+3m+ ( q ;q 3 ) r (q ;q 3 [ ] ) m m (q) 6m+ ( q ;q 3 ) m (q ;q 3 ) m+r+ m + r q 3 q 3m +m+3r +r 6mr+ ( q;q 3 ) m r (q;q 3 ) m+ ( q ;q 3 ) m (q ;q 3 ) r+ (q 3 ;q 3 ) m r (q 3 ;q 3 ) r, q β 3m ( q 6m+ )( q 6m+ )( + q 3m )β 3m+ + q( + q + q 3m q 6m+ )β 3m. 4.7 Level 9

16 6 Andrew V. Sills β (9,) 3m+ (,q) m β (9,) 3m (,q) m r m r m m r0 m r0 q 3r +r+3m (q ;q 3 [ ] ) m m (q) 6m (q ;q 3 ) m+r m + r q 3 q 3m +m+3r +r 6mr (q;q 3 ) m (q ;q 3 ) r (q 3 ;q 3 ) m r (q 3 ;q 3 ) r, q 3r +r+3m+ (q ;q 3 ) m ( q 3m+ )( q 3m+ )(q) 6m (q ;q 3 ) m+r q 3m +m+3r +r 6mr+ [ ] m m + r q 3 ( q 3m+ )( q 3m+ )(q;q 3 ) m (q ;q 3 ) r (q 3 ;q 3 ) m r (q 3 ;q 3 ) r, q 3 β 3m ( q 6m+ )( q 6m+ )β 3m+ + q( + q q 6m+ )β 3m. 5 Conclusion and Open Questions It is the hope of the author that the results presented here will help to provide some insight into the structure of A () that can be exploited by vertex operator algebraists. Questions of course remain. For instance, as pointed out by Ole Warnaar during the question-and-answer period following my talk at the Alladi 60 conference, it is not at all clear how the series expressions in () and () enumerate the partition functions c (n) and d (n) respectively. It would be very nice indeed if this connection could be established. The referee echoed this same concern, pointing out that the double sums in () and () involve complicated, highly non-hypergeometric and non-positive terms, and this is only at level 3. Christian Krattenthaler pointed out that it was conceivable that there are other families of Bailey pairs that could give rise to identities with the same product sides. Indeed, the approach employed here has limitations, and it may be that this is not the best way to approach this topic. The referee suggested that in the increasingly unwieldy multisum expressions, using only q-notation masks what is really going on, and that it is conceivable that the sum sides could be more meaningfully expressed in terms of some specialized orthogonal polynomals. While the above is admittedly true, the choice of the α n employed here is motivated by classical work; in particular the level 5 and 7 identities and the corresponding Bailey pairs coincide with the work of Slater [33, 34]. Further the availability of the Andrews Baxter Forrester transformation to express the β n as a multisum for any level l is an encouraging sign that this may point to a fruitful direction in the effort to better understand A () as a whole.

17 A classical q-hypergeometric approach to the A () standard modules 7 Acknolwedgments The author thanks Jim Lepowsky for assistance with the exposition and the referee for carefully reading the manuscript and providing a number of useful suggestions for improvement. References. Alladi, K., Andrews, G.E., Gordon, B.: Refinements and generalizations of Capparelli s conjecture on partitions. J. Algebra 74, (995). Agarwal, A.K., Andrews, G.E., Bressoud, D.M.: The Bailey lattice. J. Indian Math. Soc. (N. S.) 5, (987) 3. Andrews, G.E.: An analytic generalization of the Rogers Ramanujan identities for odd moduli. Proc. Nat. Acad. Sci. USA, 7, (974) 4. Andrews, G.E.: Multiple series Rogers Ramanujan type identities. Pacific J. Math (984) 5. Andrews, G.E.: q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra. CBMS Regional Conferences Series in Mathematics, no. 66, American Mathematical Society, Providence (986) 6. Andrews, G.E.: Schur s theorem, Capparelli s conjecture and q-trinomial coefficients. The Rademacher legacy to mathematics (University Park, PA, 99) 4 54, Contemp. Math., 66, Amer. Math. Soc., Providence, RI (994) 7. Andrews, G.E., R. J. Baxter, P. J. Forrester, Eight-vertex SOS model and generalized Rogers Ramanujan type identities. J. Stat. Phys. 35,93 66 (984) 8. Bailey, W.N.: Series of hypergeometric type which are infinite in both directions. Quart. J. Math. (Oxford) 7, 05 5 (936) 9. Bailey, W.N.: Identities of the Rogers-Ramanujan type. Proc. London Math. Soc. () 50, 0 (949) 0. Berndt, B.C.: Number Theory in the Spirit of Ramanujan, American Mathematical Society (006). Bressoud, D.M.: A generalization of the Rogers-Ramanujan identities for all moduli. J. Combin. Theory Ser. A (979). Bressoud, D.M.: Analytic and combinatorial generalizations of the Rogers-Ramanujan identities. Mem. Amer. Math. Soc. 4, no. 7, 54 (980) 3. Bringmann, K., Mahlburg, K.: False theta functions and companions to Capparelli s identities. Adv. Math. 78, 36 (05) 4. Capparelli, S.: Vertex operator relations for affine algebras and combinatorial identities. Ph.D thesis, Rutgers University (988) 5. Capparelli, S.: A construction of the level 3 modules for the affine Lie algebra A () and a new combinatorial identity of the Rogers-Ramanujan type. Trans. Amer. Math. Soc. 348, (996) 6. Fricke, R.: Die Elliptischen Funktionen und ihre Anwendungen. Ers Teil, Teubner, Leipzig (96) 7. Gasper, G., Rahman, M.: Basic Hypergeometric Series, nd ed. Cambridge University Press (004) 8. Gordon, B.: A combinatorial generalization of the Rogers-Ramanujan identities. Amer. J. Math. 83, (96) 9. Iverson, K.E.: A programming language, New York: Wiley (96) 0. Kac, V.G.: Infinite Dimensional Lie Algebras, 3rd ed. Cambridge Univ. Press (990)

18 8 Andrew V. Sills. Kac, V.G.: Kazhdan, D.A., Lepowsky, J., Wilson, R.L.: Realization of the basic representations of the Euclidean Lie algebras. Adv. Math. 4, 83 (98). Knuth, D.E.: Two notes on notation, Amer. Math. Monthly (99) 3. Lepowsky, J., Milne, S.: Lie algebraic approaches to classical partition identities. Adv. Math. 9, 5 59 (978) 4. Lepowsky, J., Wilson, R.L.: Construction of the affine Lie algebra A (). Comm. Math. Phys. 6, (978) 5. Lepowsky, J., Wilson, R.L.: A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. Math. 45, 7 (98) 6. Lepowsky, J., Wilson, R.L.: The structure of standard modules I: Universal algebras and the Rogers-Ramanujan identities. Invent. Math. 77, (984) 7. Lepowsky, J., Wilson, R.L.: The structure of standard modules II: the case A (), principal gradation. Invent. Math. 79, (985) 8. McLaughlin, J., Sills, A.V.: Ramanujan Slater type identities related to the moduli 8 and 4. J. Math. Anal. Appl. 344, (008) 9. Nandi, D.: Partition identities arising from the standard A () modules of level 4. Ph.D. thesis, Rutgers University (04) 30. Paule, P., Riese, A.: A Mathematica q-analogue of Zeilberger s algorithm based on an algebraically motivated approach to q-hypergeometric telescoping, in Special Functions, q-series and Related Topics. Fields Inst. Commun., vol. 4, 79 0 (997) 3. Sills, A.V.: On series expansions of Capparelli s infinite product. Adv. Appl. Math (004) 3. Sills, A.V.: Rademacher-type formulas for restricted partition and overpartition functions. Ramanujan J (00). 33. Slater, L.J.: A new proof of Rogers transformation of infinite series. Proc. London Math Soc. () 53, (95) 34. Slater, L.J.: Further identities of the Rogers-Ramanujan type. Proc. London Math Soc. () 54, (95)