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Lie Algebras and Classical Partition Identities (Macdonald Identities/Rogers-Ramanujan Identities/Weyl-Kac Character Formula/Generalized Cartan Matrix Lie Algebras) J

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Lie Algebras and Classical Partition Identities (Macdonald Identities/Rogers-Ramanujan Identities/Weyl-Kac Character Formula/Generalized Cartan Matrix Lie Algebras) J 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 un- There is in general no such expansion for the unspecialized specialized identities as multivariable vector partition theorems numerator in Eq. 1. However, we have found a product ex- and we relate the well-known Rogers-Ramanujan partition identities to the Weyl-Kac character formula for an infinite- pansion for the numerator when all the variables e(-simple dimensional Euclidean generalized Cartan matrix Lie alge- roots) are set equal. Such a numerator is said to be principally bra. specialized. The reason for this definition is given elsewhere (J. Lepowsky, unpublished). To be more precise, when ao and In this paper we announce relationships between Lie algebra al are the simple roots of the infinite-dimensional GCM Lie theory and certain partition formulas which are important in algebra Al'), which is related to 9f(2, C), we have combinatorial analysis. Specifically, we interpret Macdonald's THEOREM 1. (Numerator Formula): Let X be a dominant unspecialized identities (1) as multivariable vector partition integral linearform and let V be the standard module for A1(l) theorems and we relate the well-known Rogers-Ramanujan with highest weight X. Then we have: partition identities to the Weyl-Kac character formula for an infinite-dimensional Euclidean generalized Cartan matrix x(V) E (-1)l(w)e(wp - p)/e()e(-ao)=e(-aj)=q (GCM) Lie algebra. The details will appear elsewhere. [For WeW background material, including the definitions of concepts used co = Ij (1 - q(no+nI)n)(j -q(no+nl)n-no)(j - q(nO+nl)n-nl) here on GCM (or Kac-Moody) Lie algebras, see refs. 2-5.] n=1 When Macdonald's unspecialized identities are rewritten with the formal exponentials e(-simple roots) used as new = Ei (-1)'(w)e(wp- P)I e(-aj)=qni, variables, they translate easily into a large family of multi- weW variable vector partition theorems to the effect that the excess where of the number of suitably restricted partitions of an integral n-vector into an even number of allowable parts over those into ni = (X + p)(h1)(i = 0, 1). an odd number of such parts is 1, -1, or 0. In case it is i1, the (The hi are among the standard generators of AlMl) and p(hi) vector in question has only one suitably restricted partition into = 1.) allowable parts. The types of allowable parts are determined By Theorem I it is not hard to see that: by the root system of an appropriate GCM Lie algebra. Any number of variables can be achieved. f1 (1 - q2nl)x(V)/e(\) e(-ao)=e(-al)=q This observation suggests that the "reason" why certain n.1 combinatorial identities always have quadratic exponents for the variable q is that consecutive integral multiples of a fixed = rln (i - q)- [2] imaginary root are added together in the computation of p - n=l wp in the denominator formula. n w 0, +(X + p)(ho) [mod(X + p)(ho + hi)]. Recall Kac's generalization (3) of Weyl's character formu- From Eq. 2 we obtain: la: THEOREM 2. After multiplication by lln>(1 - q2n-l)-l, the product sides ofthe pair ofRogers-Ramanujan identities E (-l)1(W)e(w(X + p) - (A + p)) become the principally specialized characters for the standard X(V)/e(X) = weW [1] modules for Al1l)corresponding to the dominant integral )We(wp-p) linearforms X such that X(ho) = i - 1, and X(h1) = 4-i, where WE (-1W i = 1, 2. Furthermore, the expression IInI(1 -q2"-1)-1 is where X is the highest weight of the standard module V, x(V) itself the principally specialized character for the standard is the character of V, e(-) is a formal exponential, W is the Weyl modulefor Al l) corresponding to A such that X(ho) = 0 and group, I(w) is the length of w e W, and p is the analogue de- X(h1) = 1. fined by Kac (3) of the half-sum of positive roots of a semisimple We do not have a Lie algebraic proof of the Rogers-Ram- Lie algebra (cf. ref. 5). Both the numerator and denominator anujan identities. in Eq. 1 are formal power series in e(-simple roots). There are many generalizations and analogues of the Rog- The Macdonald-Kac denominator formula enables us to ers-Ramanujan identities due to Gordon, G6llnitz-Gordon, and factor the formal power series in the denominator of Eq. 1 into Andrews (see ref. 6) which can be "explained", just as in The- an infinite product indexed by the set A+ of positive roots. orem 2, by using different standard modules for Al1'). Slater (7) has a list of identities among which there are 21 of The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked Abbreviation: GCM, generalized Cartan matrix. "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate * Current address: Department of Mathematics, Rutgers University, this fact. New Brunswick, NJ 08903. 578 Downloaded by guest on September 29, 2021 Mathematics: Lepowsky and Milne Proc. Natl. Acad. Sci. USA 75 (1978) 579 the Rogers-Ramanujan type which are not included in An- and S.M. was partially supported by National Science Foundation drews' theory. All 21 are "explained", as in Theorem 2,.byA2(2), Grant MCS74-24249. an infinite-dimensional GCM Lie algebra related to 61(3, C). 1. Macdonald, I. G. (1972) "Affine root systems and Dedekind's An abstract argument (. Lepowsky, unpublished) generalizes 7-function," Inventiones Math. 15,91-143. for our numerator formula for A10') and A2(2) to all GCM Lie 2. Kac, V. G. (1968) "Simple irreducible graded Lie algebras of finite algebras. Kac has pointed out that this abstract argument is in growth," (in Russian), Izv. Akad. Nauk SSSR 32, 1323-1367 fact classical. (English translation: Math. USSR-Izvestija 2, 1271-1311). Theorem 2 suggests studying the standard modules further. 3. Kac, V. G. (1974) "Infinite-dimensional Lie algebras and Dede- (A. Feingold and Lepowsky, unpublished) kind's 7-function," (in Russian), Funkt. Anal. i Ego Przlozheniya It has been found J. 8, 77-78 (English translation: Functional Analysis and its Ap- that the weight multiplicities of the AIM')-module with-X(ho) plications 8,68-70). = 0, X(hi) = 1 are precisely given by the classical partition 4. Moody, R. V. (1968) "A new class of Lie algebras," J. Algebra 10, function. This provides a new Lie algebraic interpretation of 211-230. a classical formula of Gauss. In addition, analogous results are 5. Garland, H. & Lepowsky, J. (1976) "Lie algebra homology and obtained for other modules. These facts lead to ideas that illu- the Macdonald-Kac formulas," Inventiones Math. 34, 37-76. minate the structure of certain of these Lie algebras (J. Le- 6. Andrews, G. E. (1976) "The Theory of Partitions," in Encyclopedia powsky and R. L. Wilson, unpublished). of Mathematics and Its Applications, ed. Rota, G.-C. (Addison- Wesley, Reading, MA), Vol. 2. During this work, J.L. was partially supported by a Sloan Foundation 7. Slater, L. J. (1952) "Further identities of the Rogers-Ramanujan Fellowship and National Science Foundation Grant MCS76-10435, type," Proc. London Math. Soc. (2) 54, 147-167. Downloaded by guest on September 29, 2021.
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