Jehanne Dousse: Partition identities of Capparelli and Primc

Abstract

A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. A Rogers-Ramanujan type identity is a theorem stating that for all n, the number of partitions of n satisfying some difference conditions equals the number of partitions of n satisfying some congruence conditions. In the 1980's, Lepowsky and Wilson established a connection between the Rogers-Ramanujan identities and representation theory. Other representation theorists have then extended their method and obtained new identities yet unknown to combinatorialists, such as Capparelli's and Primc's identities. Though these two identities did not seem related from the representation theoretic point of view, we show combinatorially that Capparelli's identity can actually be deduced from Primc's identity, through q-difference equations or a bijection.