Dipartimento di Scienze di Base e Applicate per l'Ingegneria

Jehanne Dousse (Université Lyon 1)

 

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. Lepowsky and Wilson were the first to exhibit a connection between Rogers-Ramanujan type partition identities and representation theory in the 1980s. Shortly after, studying representations of a different Lie algebra, Capparelli discovered a partition identity yet unknown to combinatorialists. Since then, interactions between representation theory and partition identities have been very fruitful, giving rise to many new identities, such as Primc's identity from crystal base theory. 

After an introduction to the above-mentioned partition identities, we will show that Capparelli's identity can be deduced combinatorially from Primc's identity, even though they don't seem related from the representation theoretic point of view.