Level three standard modules \(A^{(2)}_ 2\) and combinatorial identities
There is a close connection between the theory of representations of affine Lie algebras and combinatorial identities of the Rogers-Ramanujan type. Work of J. Lepowsky and R. L. Wilson gave a Lie-theoretic explanation and proof of the classic Rogers-Ramanujan identities, via the level 3 standard representations of the rank one affine algebra \(A_1^{(1)}\). They also laid the foundations for much of the work that was later done on this line of research by several researchers. Typically these works led to, or made use of, well-known partition identities. The first time that this method gave rise to partition identities that appeared to be unknown to specialists was in the paper by the reviewer [J. Algebra 154, 335-355 (1993; Zbl 0774.17029)]. This involved the study of the level 3 modules of the affine algebra \(A_2^{(2)}\). A proof of these identities was not included in that paper. The authors of the paper under review provide such a proof along precisely the lines of the \(\mathbb{Z}\)-algebra theory of Lepowsky and Wilson. They prove generalized anticommutation relations which, together with the Lepowsky-Wilson theory, allow them to prove the spanning of a suitable set of vectors in the level 3 representation space of \(A_2^{(2)}\). The proof of independence is carried out through a delicate analysis of relations among generating functions of operators acting on the vacuum space. As a consequence they obtain a proof of the conjectured identities. It must be noticed that a proof of one of the identities had been obtained by \textit{G. E. Andrews} [Contemp. Math. 166, 141-154 (1994; Zbl 0811.05001)], using combinatorial methods. A combinatorial proof of the other identity was presented in the reviewer's article [Commun. Algebra 23, 2959-2969 (1995; Zbl 0830.17012)]. A proof by the reviewer using Lie theory but a different approach has also appeared in Trans. Am. Math. Soc. 348, 481-501 (1996).
combinatorial identities of the Rogers-Ramanujan type
scientific article; zbMATH DE number 856979
scientific article