The Rogers-Ramanujan Identities: Lie Theoretic Interpretation and Proof

• Published in: Proceedings of the National Academy of Sciences - PNAS 1981-02, Vol.78 (2), p.699-701
• Main Authors: Lepowsky, James, Wilson, Robert Lee
• Format: Article
• Language: English
• Subjects: Algebra; Integers; Mathematical vectors; Physical Sciences: Mathematics; Polynomials; Power series; Subalgebras
• ISSN: 0027-8424; 1091-6490
• DOI: 10.1073/pnas.78.2.699

The two Rogers-Ramanujan identities, which equate certain infinite products with infinite sums, are among the most intriguing of the classical formal power series identities. It has been found by Lepowsky and Milne that the product side of each of them differs by a certain factor from the principally specialized character of a certain standard module for the Euclidean Kac-Moody Lie algebra A1 (1). On the other hand, the present authors have introduced an infinite-dimensional Heisenberg subalgebra s of A1 (1)which leads to a construction of A1 (1)in terms of differential operators given by the homogeneous components of an ``exponential generating function.'' In the present announcement, we use s to formulate a natural ``abstract Rogers-Ramanujan identity'' for an arbitrary standard A1 (1)-module which turns out to coincide with the classical identities in the cases of the two corresponding standard modules. The abstract identity equates two expressions, one a product and the other a sum, for the principally specialized character of the space Ω of highest weight vectors or ``vacuum states'' for s in the module. The construction of A1 (1)leads to a concrete realization of Ω as the span of certain spaces of symmetric polynomials occurring as the homogeneous components of exponential generating functions. The summands in the Rogers-Ramanujan identities turn out to ``count'' the dimensions of these spaces. For general standard A1 (1)-modules, we conjecture that the abstract identities agree with generalizations of the Rogers-Ramanujan identities due to Gordon, Andrews, and Bressoud.