PARTITION IDENTITIES FROM HIGHER LEVEL CRYSTALS OF A<SUP>(1)</SUP> ₁

The content of this paper lies in the intersection of combinatorics (in particular, the study of partition identities) and the representation theory of affine Kac-Moody algebras. Lepowsky, Milne and Wilson's foundational work [LM78, LW84, LW85] has led to a productive relationship between these two fields. We now recall some background and provide a brief description of the principal mechanism through which this interaction takes place. A partition of a natural number n is a non-increasing sequence of positive integers whose sum is n. For example, the partitions of 4 are (4), (3, 1), (2, 2), (2,1, 1) and (1,1, 1, 1). Among the most famous and ubiquitous partition and q-series identities are those of Rogers-Ramanujan [RR19]. In q-series form, they can be stated as follows: