Table Of ContentDEDEKIND’S η-FUNCTION AND
ROGERS–RAMANUJAN IDENTITIES
0
1 S. OLE WARNAAR AND WADIM ZUDILIN
0
2
Abstract. We provea q-seriesidentity thatgeneralisesMacdon-
n ald’s A(2) η-function identity and the Rogers–Ramanujan identi-
a 2n
J ties. We conjecture our result to generalise even further to also
2 include the Andrews–Gordon identities.
2
]
O
1. Introduction
C
h. In 1972 Macdonald published his seminal paper [26] in which he ex-
t tended Weyl’s denominator formula for classical reduced root systems
a
m to root systems of affine type. These identities, which include the Ja-
[ cobi triple product identity and the quintuple product identity as the
2 special cases A(1) and A(2), are now commonly known as the Macdon-
1 2
v ald identities. Through the procedure of specialisation the Macdonald
1
identities imply identities for powers of the Dedekind η-function
7
5 ∞
1 η(τ) = q1/24 (1−qj),
.
1
j=1
0 Y
0 where q = exp(2πiτ) and Im(τ) > 0. For example, the specialisa-
1
(2)
: tion [26, p. 138, (6)(c)] of the A Macdonald identity corresponds
v 2n
the following beautiful generalisation of the Euler pentagonal number
i
X
theorem:
r
a (1.1) η(τ)2n2−n = ξ(v/ρ)(−1)|v|−|ρ|q||v||2/(2(2n+1)).
Here v = (v ,...,v ), ρX= (1/2,3/2,...,n−1/2), |v| = v +···+v ,
1 n 1 n
||v||2 = v ·v = v2 +···+v2,
1 n
v2 −v2
ξ(v/w) = i j
w2 −w2
1≤i<j≤n i j
Y
and the sum on the right of (1.1) is over v ∈ (Z/2)n such that v ≡ ρ
i i
(mod 2n+1).
2000 Mathematics Subject Classification. 05A19, 05E05,11F20, 33D67.
Work supported by the Australian Research Council.
1
2 S.OLE WARNAARAND WADIMZUDILIN
Another famous family of combinatorial identities are the formulae
of Rogers and Ramanujan [2,31]
∞ qm2 (q2,q3,q5;q5)
∞
(1.2a) =
(q) (q)
m ∞
m=0
X
∞ qm2+m (q,q4,q5;q5)
∞
(1.2b) =
(q) (q)
m ∞
m=0
X
and their generalisations to arbitrary odd moduli due to Andrews and
Gordon [1,11]
(1.3)
qM12+···+Mk2−1+Mp+···+Mk−1 (qp,q2k−p+1,q2k+1;q2k+1)∞
= ,
(q) ···(q) (q)
m1,X...,mk−1 m1 mk−1 ∞
where 1 ≤ p ≤ k and M = m + ··· + m . In (1.2) and (1.3) we
i i k−1
employ the standard q-notation
m
(a) = (a;q) = (1−aqi−1)
m m
i=1
Y
and
(a ,...,a ) = (a ,...,a ;q) = (a ;q) ···(a ;q)
1 k m 1 k m 1 m k m
for m ∈ N∪{∞} (with the convention that N = {0,1,2,...}).
In this paper we link Macdonald’s identity (1.1) to the Rogers–
Ramanujan and Andrews–Gordon identities (1.2) and (1.3). More
specifically, we present a family of q-series identities depending on pos-
itive integers k, n and p ∈ {1,k} such that
(2)
(1) For k = 1 we recover Macdonald’s A identity (1.1) for the
2n
Dedekind η-function.
(2) For n = 1 and k = 2 we recover, modulo the Jacobi triple
product identity, the Rogers–Ramanujan identities (1.2).
(3) For n = 1 and general k we recover the p = k and p = 1
instances of the Andrews–Gordon identities (1.3).
(4) For general n and k → ∞ we recover the A case of an
2n−1
identity of Hua related to Kac’s conjectures for representations
of quivers.
The fact that the Rogers–Ramanujan identities have a close con-
nection with affine root systems or, more generally, affine Kac–Moody
algebras is not new, and well-known are the interpretations of (1.2)
(1)
and (1.3) in terms of standard modules of A , see e.g., [22–25,29]
1
DEDEKIND’S η-FUNCTION AND ROGERS–RAMANUJAN IDENTITIES 3
and as characters corresponding to certain non-unitary Virasoro mod-
ules, see e.g., [8,30]. For our generalisation of the Rogers–Ramanujan
and Andrews–Gordon identities, however, it is crucial to interpret the
(2) (1)
right-hand sides of (1.2) and (1.3) as of type type A , not A .
2 1
Before stating our main results we observe that, by an appeal to
Jacobi’striple product identity [10, (II.28)], theright-handside of (1.3)
may be rewritten as
∞
1
(−1)jq(2k+1)(2j)+pj
(q)
∞
j=−∞
X
q−(2k−2p+1)2/(8(2k+1))
= (−1)v−k+p−1/2qv2/(2(2k+1)),
(q)
∞
where in the second expression the suXm is over v ∈ Z/2 such that
v ≡ k − p + 1/2 (mod 2k + 1). Comparing the sum on the right
with the sum in (1.1) it takes little imagination to make the following
conjecture.
For 1 ≤ a,b ≤ N − 1, let C be the Cartan integers of the Lie
ab
algebra A , i.e., C = 2, C = −1 and C = 0 otherwise. By
N−1 aa a,a±1 ab
abuse of notation, for w = (w ,...,w ) and a a scalar, set w + a =
1 n
(w +a,...,w +a) so that, in particular,
1 n
||w+a||2 = ||w||2 +2a|w|+na2 and |w+a| = |w|+na.
Conjecture 1.1. For k,n positive integers, N = 2n and p ∈ {1,k},
q21PNa,b−=11Pki=−11CabMi(a)Mi(b)+PNa=−11Pki=−p1(−1)aMi(a)
(1.4)
N−1 k−1(q)
a=1 i=1 m(a)
X i
1Q Q ||v||2−||ρ+k−p||2
= ξ(v/ρ)(−1)|v|−|ρ+k−p|q 2(2k+2n−1) ,
(q)2n2−n
∞
X
where the sum on the left is over m(a) ∈ N (for all 1 ≤ a ≤ N −1 and
i
1 ≤ i ≤ k −1) and the sum on the right is over v ∈ (Z/2)n such that
v ≡ ρ + k −p (mod 2k +2n−1). The integers M(a) are defined as
i i i
M(a) = m(a) +···+m(a) , i.e., m(a) = M(a) −M(a) for 1 ≤ i ≤ k −2
i i k−1 i i i+1
and m(a) = M(a) .
k−1 k−1
Theorem 1.2 (Generalised Rogers–Ramanujan identities). Conjec-
ture 1.1 is true for k = 2. That is, for n a positive integer and N = 2n,
q1mCmt 1 ||v||2−||ρ||2
(1.5) 2 = ξ(v/ρ)(−1)|v|−|ρ|q 2(2n+3) ,
(q) (q)2n2−n
m∈NN−1 m ∞
X X
4 S.OLE WARNAARAND WADIMZUDILIN
where the sum on the right is over v ∈ (Z/2)n such that v ≡ ρ
i i
(mod 2n+3), and
q1mCmt+|m|− 1 ||v||2−||ρ+1||2
2 = ξ(v/ρ)(−1)|v|−|ρ+1|q 2(2n+3) ,
(q) (q)2n2−n
m∈NN−1 m ∞
X X
where the sum on the right is over v ∈ (Z/2)n such that v ≡ ρ + 1
i i
(mod 2n+3). In the above (q) = (q) ...(q) , C is the Cartan
m m1 mN−1
matrix of A , i.e.,
N−1
N−1 N−2
1
mCmt = m2 − m m ,
2 i i i+1
i=1 i=1
X X
and, for m ∈ NN−1,
N−1
|m| = (−1)i−1m .
− i
i=1
X
Analogous to the above theorem, the left-hand side of (1.4) can be
expressed without the use of indices by introducing the square matrix
B of dimension d := (N − 1)(k − 1) given by the Kronecker product
of the Cartan matrix C of A and the (k−1)×(k−1) matrix T−1
N−1
with entries (T−1) = min{i,j}:
ij
B = (C ⊗T−1) = C min{i,j}.
ai,bj ai,bj ab
For example, the k = p instance of (1.4) generalises (1.5) to
q1mBmt 1 ||v||2−||ρ||2
(1.6) 2 = ξ(v/ρ)(−1)|v|−|ρ|q2(2k+2n−1).
(q) (q)2n2−n
mX∈Nd m ∞ X
We should remark that the expression on the left-hand side of (1.6) has
arepresentation theoreticinterpretationduetoFeiginandStoyanovsky
[8]. Their interpretation in fact holds for B = C ⊗T−1 where C is a
Cartan matrix of any semi-simple simply laced Lie algebra g. Let gˆ
be the (nontwisted) affine counterpart of g and V the level-l vacuum
l
integrable highest weight module of g with vacuum vector v. Then W
l
is the space W = U(nˆ ) · v ⊂ V , with U the universal enveloping
l + 0 l
algebra and n ⊕h⊕n the Cartan decomposition of g. The Feigin–
− +
Stoyanovsky formula states that
q1mBmt
Tr(qL0)| = 2 ,
Wk−1 (q)
m
mX∈Nd
where d = (k −1)rank(g) and L the energy operator.
0
DEDEKIND’S η-FUNCTION AND ROGERS–RAMANUJAN IDENTITIES 5
There is another elegant expression for the left-hand side of (1.4)
using notation from the theory of Hall–Littlewood polynomials, see
[27,33]. Let λ = (λ ,λ ,...) be a partition and λ′ its conjugate. Then
1 2
the q-function
bλ(q) = (q)λ′−λ′
i i+1
i≥1
Y
features in the Cauchy identity for the Hall–Littlewood polynomials
P (x) = P (x;q) and in the principal specialisation formula on an
λ λ
infinite alphabet:
qn(λ)
P (1,q,q2,...) = ,
λ
b (q)
λ
where
λ′
(1.7) n(λ) = (i−1)λ = i .
i
2
i≥1 i≥1 (cid:18) (cid:19)
X X
If we further denote (λ|µ) = λ′µ′ then the left-hand side of (1.4)
i≥1 i i
for p = k or p = 1 corresponds to
P
N−1 |λ(a)|
(1.8) q21PNa,b−=11Cab(λ(a)|λ(b)) za
b (q)
a=1 λ(a)
X Y
summed over partitions λ(1),...,λ(N−1) such that λ(1),...,λ(N−1) ≤
1 1
k −1, i.e., such that the largest parts of the λ(i) do not exceed k −1.
In (1.8) z = 1 for all i if p = k and z = q and z = q−1 for all i if
i 2i−1 2i
p = 1.
The rewriting (1.8) of the left-hand side of (1.4) shows that not only
the k = 1 case of (1.4) is known, but also the limiting case k → ∞
(with p = k). Indeed, in [12] Hua derived a combinatorial identity
relatedtoKac’sconjectures [14]concerning thenumber ofisomorphism
classes of absolutely indecomposable representations of quivers over
finite fields. For finite and tame quivers (i.e., classical and affine ADE)
Kac’s conjectures are known to be true, ensuring that in these cases
Hua’s combinatorial identities are true as well. For the finite quiver
A (N not necessarily even) Hua’s identity (corrected in [9]) is
N−1
N−1 |λ(a)|
q12PNa,b−=11Cab(λ(a)|λ(b)) za = 1 ,
b (q) (zαq)
λ(1),.X..,λ(N−1) aY=1 λ(a) αY∈R+ ∞
where R is the set of positive roots of A :
+ N−1
R = {α +α +···+α : 1 ≤ i ≤ j ≤ N −1},
+ i i+1 j
and zαi+αi+1+···+αj = z z ···z .
i i+1 j
6 S.OLE WARNAARAND WADIMZUDILIN
2. Further conjectures
At first sight it may seem surprising that the left-hand side of (1.4)
features the root system A instead of, more simply, A . However,
2n−1 n
this is not that unexpected in view of the following theorem due to
Feigin and Stoyanovsky [8] (n = 1) and Stoyanovsky [32] (n > 1).
Theorem 2.1. For k,n positive integers, N = 2n+1 and p = k,
q21PNa,b−=11Pki=−11CabMi(a)Mi(b)+PNa=−11Pki=−p1(−1)aMi(a)
(2.1)
N−1 k−1(q)
a=1 i=1 m(a)
X i
Q Q 1 ||v||2−||ρ∗+k−p||2
= χ(v/ρ∗)q 4(k+n) ,
(q)2n2+n
∞
X
where ρ∗ = (1,2,...,n),
n v v2 −v2
(2.2) χ(v/w) = i i j
w w2 −w2
i=1 i 1≤i<j≤n i j
Y Y
and the sum on the right is over v ∈ Zn such that v ≡ ρ∗ + k − p
i i
(mod 2k +2n).
(1)
The Feigin–Stoyanovsky theorem generalises Macdonald’s C η-
n
function identity [26, p. 136, (6)]:
η(τ)2n2+n = χ(v/ρ∗)q||v||2/(4(n+1)),
which, for n = 1, is equivalenXt to Jacobi’s well-known
∞
(q)3 = (−1)m(2m+1)q(m2+1).
∞
m=0
X
Conjecture 2.2. Equation (2.1) is true for p = 1.
Theorem 2.3. Conjecture 2.2 is true for k = 2, i.e., for n a positive
integer and N = 2n+1,
q1mCmt+|m|− 1 ||v||2−||ρ∗+1||2
2 = χ(v/ρ∗)q 4(n+2) ,
(q) (q)2n2+n
m∈NN−1 m ∞
X X
where the sum on the right is over v ∈ Zn such that v ≡ ρ∗ + 1
i i
(mod 2n+4).
Of course, by the Feigin–Stoyanovsky theorem we also have
q1mCmt 1 ||v||2−||ρ∗||2
2 = χ(v/ρ∗)q 4(n+2) ,
(q) (q)2n2+n
m∈NN−1 m ∞
X X
DEDEKIND’S η-FUNCTION AND ROGERS–RAMANUJAN IDENTITIES 7
where the sum on the right is over v ∈ Zn such that v ≡ ρ∗ (mod 2n+
i i
4).
There is a well-known even-modulus counterpart of the Andrews–
Gordon identities (1.3) due to Bressoud [3]:
qM12+···+Mk2−1+Mp+···+Mk−1 (qp,q2k−p,q2k;q2k)∞
(2.3) = ,
(q) ···(q) (q2;q2) (q)
m1,X...,mk−1 m1 mk−2 mk−1 ∞
where k > 1 and, again, 1 ≤ p ≤ k and M = m +···+m . It will
i i k−1
be convenient to interpret 1/(q2;q2) as (q) /(q2;q2) so that (2.3)
m0 ∞ ∞
is true for all positive integers k.
Our next conjecture unifies Bressoud’s identity for p = k with Mac-
(2)
donald’s η-function identity for A [26, p. 136, (6)(b)]:
2n−1
η(τ)2n2+n−1
(2.4) = ξ(v/ρ⋆)(−1)(|v|−|ρ⋆|)/(2n)q||v||2/(4n),
η(2τ)2n−1
X
where ρ⋆ = (0,1,...,n−1) and v is summed over Zn such that v ≡ ρ⋆
i i
(2)
(mod 2n), and a second η-function identity for A [26, p. 138, (6)(a)]:
2n
η(τ)2n2+3n
(2.5) = χ(v/ρ)q||v||2/(2(2n+1)),
η(2τ)2n
X
where the sum is over v ∈ Zn such that v ≡ ρ (mod 2n+1).
i i
Conjecture 2.4. For k,N positive integers and n = ⌊N/2⌋,
q12PNa,b−=11Pki=−11CabMi(a)Mi(b)
N−1 k−2(q) (q2;q2)
X a=1 i=1 m(ia) m(ka−)1
Q (cid:0)Q1 (cid:1) |v|−|ρ⋆| ||v||2−||ρ⋆||2
(2.6a) = ξ(v/ρ⋆)(−1)2k+N−2q 2(2k+N−2)
N(N−1)/2
(q)
∞
X
1 ||v||2−||ρ||2
(2.6b) = χ(v/ρ)q2(2k+N−2),
N(N−1)/2
(q)
∞
X
where (2.6a) applies for even N, in which case the sum is over v ∈ Zn
such that v ≡ ρ⋆ (mod 2k +N −2) and (2.6b) applies for odd N, in
i i
which case the sum is over v ∈ Zn such that v ≡ ρ (mod 2k+N−2).
i i
As before, to recover (2.4) and (2.5) as the k = 1 case of (2.6) we
have to interpret 1/(q2;q2) as (q) /(q2;q2) .
m(a) ∞ ∞
0
8 S.OLE WARNAARAND WADIMZUDILIN
3. Dilogarithm identities
To provide further support for Conjecture 1.1, we show below that
a standard asymptotic analysis applied to (1.4) implies an identity for
the Rogers dilogarithm due to Kirillov.
We begin by recalling the definition of the Rogers dilogarithm func-
tion
1 x log(1−t) logt
L(x) = − + dt, x ∈ [0,1].
2 t 1−t
Z0 (cid:18) (cid:19)
Note in particular that L(1) = π2/6.
In [17] Kirillov proved the following A type dilogarithm identity
N−1
(3.1)
1 N−1K−1 sin aπ sin (N−a)π (N2 −1)(K −1)
L K+N−1 K+N−1 = .
L(1) sin (cid:0)(i+a)π (cid:1)sin (cid:0)(i+N−a)π(cid:1) ! K +N −1
a=1 i=1 K+N−1 K+N−1
X X
If we denote the summ(cid:0)and on(cid:1)the(cid:0)left by S((cid:1)K,N;a,i) then
S(2k,N;a,i) = S(2k,N;a,2k−i−1) for 1 ≤ i ≤ k −1
and S(2k,N;a,2k −1) = L(1). Hence
(3.2)
1 N−1k−1 sin aπ sin (N−a)π N(N −1)(k −1)
L 2k+N−1 2k+N−1 = .
L(1) sin (cid:0)(i+a)π (cid:1)sin (cid:0)(i+N−a)π(cid:1) ! 2k +N −1
a=1 i=1 2k+N−1 2k+N−1
XX
We now recall the f(cid:0)ollowing(cid:1)resu(cid:0)lt from [1(cid:1)8].
Lemma 3.1. Let B be a d × d symmetric, positive definite, rational
matrix and let
∞ q1mBmt
a qi = 2 .
i
(q)
m
i=0 m∈Nd
X X
Then
log2a d
m
lim = L(x ),
i
m→∞ 4m
i=1
X
where the x for 1 ≤ i ≤ d are the solutions of
i
d
x = (1−x )Bij
i j
j=1
Y
such that x ∈ (0,1) for all i.
i
DEDEKIND’S η-FUNCTION AND ROGERS–RAMANUJAN IDENTITIES 9
If we apply the above lemma to the expression on the left-hand side
of (1.6) we are led to the system of equations
N−1k−1
f(a) = (1−f(b))Cabmin{i,j},
i i
b=1 j=1
Y Y
for 1 ≤ a ≤ N −1 and 1 ≤ i ≤ k −1. It is readily verified that this is
solved by
sin aπ sin (N−a)π
f(a) = 2k+N−1 2k+N−1 .
i sin (cid:0)(i+a)π (cid:1)sin (cid:0)(i+N−a)π(cid:1)
2k+N−1 2k+N−1
Hence, denoting the q-series(cid:0)on eith(cid:1)er s(cid:0)ide of (1.(cid:1)6) by a qi, we
i≥0 i
find that
P
1 log2a
m
lim = LHS(3.2).
L(1) m→∞ 4m
(2)
The right-hand side of (1.6) is a specialised standard module of A
2n
[13,15]. Exploiting its modular properties [16] we obtain
1 log2a
m
lim = RHS(3.2)
L(1) m→∞ 4m
(recall that N = 2n), leading to (3.2).
In much the same way an asymptotic analysis of Theorem 2.1 gives
(3.2) for odd N. The asymptotics of Conjecture 2.4, onthe other hand,
can be related to the odd K case of (3.1).
4. Proof of Theorems 1.2 and 2.3
The proof given below uses the theory of Hall–Littlewood polynomi-
als, and for notation and definitions pertaining to these functions we
refer the reader to [27,33].
For x = (x ,...,x ) and µ a partition of length l(µ) ≤ n, let
1 n
Q′ (x) = Q′ (x;q) be the modified Hall–Littlewood polynomial [21]
µ µ
Q′ (x) = K (q)s (x),
µ λµ λ
λ
X
where the K (q) are the Kostka–Foulkes polynomials and s (x) the
λµ λ
Schur functions. In the ring of symmetric functions, the polynomials
Q′ form the adjoint basis of P with respect to the Hall inner prod-
µ λ
uct. They may also be viewed in λ-ring notation [19] as Q′ (x) =
µ
b (q)P (x/(1−q)).
µ µ
10 S.OLE WARNAARAND WADIMZUDILIN
Theorem 4.1. Let C be the A Cartan matrix. Then
n
∞ qm q1rCrt
(4.1a) Q′ (1n) = 2
(q) (2m) (q)
m=0 m r∈Nn r
X X
and
∞ q2m q1rCrt+|r|−
(4.1b) Q′ (1,q−1,1,q−1,...) = 2 .
(q) (2m) (q)
mX=0 m n terms rX∈Nn r
Our proof requires a gener|alisatio{zn of [27}]
(4.2) Q′ (1) = qn(λ)
λ
due to Lascoux. Extend (1.7) to skew shapes by
λ′ −µ′
n(λ/µ) = i i .
2
i≥1 (cid:18) (cid:19)
X
Theorem 4.2 ([20, Theorem 3.1]). For µ ⊆ λ
l(µ)
Q′ (1) = qn(λ/µ) (1−qλ′µi−i+1),
λ/µ b (q)
µ
i=1
Y
and Q′ (1) = 0 otherwise.
λ/µ
Before we show how this implies Theorem 4.1 we note that the above
may be rewritten as
λ′ −µ′
(4.3) Q′ (1) = qn(λ/µ) i i+1 ,
λ/µ λ′ −µ′
i≥1(cid:20) i i (cid:21)
Y
where m is a q-binomial coefficient. A classical result in the theory
k
of abelian p-groups states that if α (µ;p) is the number of subgroups
λ
(cid:2) (cid:3)
of type µ in a finite abelian p-group of type λ, then [4–6,34]
λ′ −µ′
qn(λ)−n(µ)α (µ;q−1) = qn(λ/µ) i i+1 .
λ λ′ −µ′
i≥1(cid:20) i i (cid:21)
Y
This obviously implies that
Q′ (1) = qn(λ)−n(µ)α (µ;q−1),
λ/µ λ
a result we failed to find in the literature.
Proof of Theorems 4.1. For n = 1 (4.1) is trivial since, by (4.2),
Q′ (1) = qm2−m.
(2m)
