Quasi-particles in the principal picture of $\widehat{\mathfrak{sl}}_{2}$ and Rogers-Ramanujan-type identities

Abstract

In their seminal work J. Lepowsky and R. L. Wilson gave a vertex-operator theoretic interpretation of Gordon-Andrews-Bressoud's generalization of Rogers-Ramanujan combinatorial identities, by constructing bases of vacuum spaces for the principal Heisenberg subalgebra of standard $\widehat{\mathfrak{sl}}_{2}$-modules, parametrized with partitions satisfying certain difference 2 conditions. In this paper we define quasi-particles in the principal picture of $\widehat{\mathfrak{sl}}_{2}$ and construct quasi-particle monomial bases of standard $\widehat{\mathfrak{sl}}_{2}$-modules for which principally specialized characters are given as products of sum sides of the corresponding analytic Rogers-Ramanujan-type identities with the character of the Fock space for the principal Heisenberg subalgebra.