Lie Algebras And Classical Partition Identities

Lie Algebras and Classical Partition Identities J. Lepowsky, and S. Milne PNAS 1978;75;578-579 doi:10.1073/pnas.75.2.578 This information is current as of March 2007. This article has been cited by other articles: www.pnas.org#otherarticles E-mail Alerts

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Proc. Nati. Acad. Sci. USA Vol. 75, No. 2, pp. 578-579, February 1978

Mathematics

Lie algebras and classical partition identities (Macdonald identities/Rogers-Ramanujan identities/Weyl-Kac character formula/generalized Cartan matrix Lie algebras)

J. LEPOWSKY* AND S. MILNE Department of Mathematics, Yale University, New Haven, Connecticut 06520

Communicated by Walter Feit, November 7,1977

ABSTRACT In this paper we interpret Macdonald's unspecialized identities as multivariable vector partition theorems and we relate the well-known Rogers-Ramanujan partition identities to the Weyl-Kac character formula for an infinitedimensional Euclidean generalized Cartan matrix Lie algebra. In this paper we announce relationships between Lie algebra theory and certain partition formulas which are important in combinatorial analysis. Specifically, we interpret Macdonald's unspecialized identities (1) as multivariable vector partition theorems and we relate the well-known Rogers-Ramanujan partition identities to the Weyl-Kac character formula for an infinite-dimensional Euclidean generalized Cartan matrix (GCM) Lie algebra. The details will appear elsewhere. [For background material, including the definitions of concepts used here on GCM (or Kac-Moody) Lie algebras, see refs. 2-5.] When Macdonald's unspecialized identities are rewritten with the formal exponentials e(-simple roots) used as new variables, they translate easily into a large family of multivariable vector partition theorems to the effect that the excess of the number of suitably restricted partitions of an integral n-vector into an even number of allowable parts over those into an odd number of such parts is 1, -1, or 0. In case it is i1, the vector in question has only one suitably restricted partition into allowable parts. The types of allowable parts are determined by the root system of an appropriate GCM Lie algebra. Any number of variables can be achieved. This observation suggests that the "reason" why certain combinatorial identities always have quadratic exponents for the variable q is that consecutive integral multiples of a fixed imaginary root are added together in the computation of p wp in the denominator formula. Recall Kac's generalization (3) of Weyl's character formu-

There is in general no such expansion for the unspecialized numerator in Eq. 1. However, we have found a product expansion for the numerator when all the variables e(-simple roots) are set equal. Such a numerator is said to be principally specialized. The reason for this definition is given elsewhere (J. Lepowsky, unpublished). To be more precise, when ao and al are the simple roots of the infinite-dimensional GCM Lie algebra A l'), which is related to 9f(2, C), we have THEOREM 1. (Numerator Formula): Let X be a dominant integral linear form and let V be the standard module for A1(l) with highest weight X. Then we have:

x(V) WeW E (-1)l(w)e(wp p)/e()e(-ao)=e(-aj)=q -

co

=

E

X(V)/e(X) weW

(-l)1(W)e(w(X

+

-

=

Ei

weW

(-1)'(w)e(wp- P) I e(-aj)=qni,

where ni = (X + p)(h1)(i = 0, 1).

(The hi are among the standard generators of AlMl) and p(hi) = 1.) By Theorem I it is not hard to see that:

f1 (1 q2nl)x(V)/e(\) e(-ao)=e(-al)=q -

n.1

=

rln (i q)-

n=l

[2]

n w 0, +(X + p)(ho) [mod(X + p)(ho + hi)].

From Eq. 2 we obtain: THEOREM 2. After multiplication by lln>(1 - q2n-l)-l, the product sides of the pair of Rogers-Ramanujan identities become the principally specialized characters for the standard modules for Al1l)corresponding to the dominant integral linearforms X such that X(ho) = i - 1, and X(h1) = 4-i, where i = 1, 2. Furthermore, the expression IInI(1 -q2"-1)-1 is itself the principally specialized character for the standard module for Al l) corresponding to A such that X(ho) = 0 and

la: =

Ij (1 - q(no+nI)n)(j -q(no+nl)n-no)(j q(nO+nl)n-nl) n=1

p) - (A + p)) [1]

WE (-1W )We(wp-p) where X is the highest weight of the standard module V, x(V) is the character of V, e(-) is a formal exponential, W is the Weyl group, I(w) is the length of w e W, and p is the analogue defined by Kac (3) of the half-sum of positive roots of a semisimple Lie algebra (cf. ref. 5). Both the numerator and denominator in Eq. 1 are formal power series in e(-simple roots). The Macdonald-Kac denominator formula enables us to factor the formal power series in the denominator of Eq. 1 into an infinite product indexed by the set A+ of positive roots.

X(h1) = 1. We do not have a Lie algebraic proof of the Rogers-Ramanujan identities. There are many generalizations and analogues of the Rogers-Ramanujan identities due to Gordon, G6llnitz-Gordon, and Andrews (see ref. 6) which can be "explained", just as in Theorem 2, by using different standard modules for Al1'). Slater (7) has a list of identities among which there are 21 of

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Abbreviation: GCM, generalized Cartan matrix. * Current address: Department of Mathematics, Rutgers University, New Brunswick, NJ 08903.

578

Mathematics: Lepowsky and Milne the Rogers-Ramanujan type which are not included in Andrews' theory. All 21 are "explained", as in Theorem 2,.byA2(2), an infinite-dimensional GCM Lie algebra related to 61(3, C). An abstract argument (. Lepowsky, unpublished) generalizes for our numerator formula for A10') and A2(2) to all GCM Lie algebras. Kac has pointed out that this abstract argument is in fact classical. Theorem 2 suggests studying the standard modules further. It has been found (A. Feingold and J. Lepowsky, unpublished) that the weight multiplicities of the AIM')-module with-X(ho) = 0, X(hi) = 1 are precisely given by the classical partition function. This provides a new Lie algebraic interpretation of a classical formula of Gauss. In addition, analogous results are obtained for other modules. These facts lead to ideas that illuminate the structure of certain of these Lie algebras (J. Lepowsky and R. L. Wilson, unpublished). During this work, J.L. was partially supported by a Sloan Foundation Fellowship and National Science Foundation Grant MCS76-10435,

Proc. Natl. Acad. Sci. USA 75 (1978)

579

and S.M. was partially supported by National Science Foundation Grant MCS74-24249. 1. Macdonald, I. G. (1972) "Affine root systems and Dedekind's 7-function," Inventiones Math. 15,91-143. 2. Kac, V. G. (1968) "Simple irreducible graded Lie algebras of finite growth," (in Russian), Izv. Akad. Nauk SSSR 32, 1323-1367 (English translation: Math. USSR-Izvestija 2, 1271-1311). 3. Kac, V. G. (1974) "Infinite-dimensional Lie algebras and Dedekind's 7-function," (in Russian), Funkt. Anal. i Ego Przlozheniya 8, 77-78 (English translation: Functional Analysis and its Applications 8,68-70). 4. Moody, R. V. (1968) "A new class of Lie algebras," J. Algebra 10, 211-230. 5. Garland, H. & Lepowsky, J. (1976) "Lie algebra homology and the Macdonald-Kac formulas," Inventiones Math. 34, 37-76. 6. Andrews, G. E. (1976) "The Theory of Partitions," in Encyclopedia of Mathematics and Its Applications, ed. Rota, G.-C. (AddisonWesley, Reading, MA), Vol. 2. 7. Slater, L. J. (1952) "Further identities of the Rogers-Ramanujan type," Proc. London Math. Soc. (2) 54, 147-167.