## Seminario

*Incontri di Algebra e Geometria allo SBAI*

Aula 2E, Pal. RM004

Ali Kemal Uncu (Research Institute for Symbolic Computation, Johannes Kepler University, Linz)

Polynomial identities that imply Capparelli's partition theorems, and more.

Abstract:

Capparelli's partition identities are a pair of combinatorial identities which has the implication that two sets of partitions with different constraints are equinumerous. More precisely, one set is counted with some difference conditions that the parts of the partitions satisfy and another set counted with some congruence conditions, and although the sets are seemingly unrelated, their total counts are the same for any number that is to be partitioned. Analytically, these identities tend to manifest themselves as an infinite sum being equal to an infinite product. In Capparelli's partition identities case this manifestation was not present. In 2018, using computer search, Kanade-Russell found an analytic representation for the generating function for the partitions counted with the difference conditions of the Capparelli's Partition Theorems. Kanade and Russell proved this identity relying on the the Capparelli's partition theorems. Kursungoz also provided generating function representations of these functions. His method on the other hand was combinatorial.

Joint with Alexander Berkovich, I have refined the above mentioned generating functions and this lead to polynomial identities, which directly imply the Capparelli's partition identities. This method can also be used to prove other partition theorems directly, such as Schur's partition theorem. Not only that, this study lead to many new identities such as two new Andrews-Gordon type infinite families.