Siladi\'c's theorem: weighted words, refinement and companion

TL;DR

This paper refines and extends Siladić's theorem, a partition identity linked to Lie algebras, using recurrences and q-difference equations, and introduces a new companion to Schur's theorem with distinct difference conditions.

Contribution

It provides a non-dilated version, further refinement, and a companion of Siladić's theorem using recurrences and q-difference equations, diverging from previous transformation methods.

Findings

Derived a non-dilated version of Siladić's theorem.

Established a new companion to Schur's theorem with different difference conditions.

Connected the identities to Lie algebra studies and partition theory.

Abstract

In a previous paper, the author gave a combinatorial proof and refinement of Siladi\'c's theorem, a Rogers-Ramanujan type partition identity arising from the study of Lie algebras. Here we use the basic idea of the method of weighted words introduced by Alladi and Gordon to give a non-dilated version, further refinement and companion of Siladi\'c's theorem. However, while in the work of Alladi and Gordon, identities were proved by doing transformations on generating functions, we use recurrences and $q$-difference equations as the original method seems difficult to apply in our case. As the non-dilated version features the same infinite product as Schur's theorem, another dilation allows us to find a new interesting companion of Schur's theorem, with difference conditions very different from the original ones.