The $\boldsymbol{SO(2r)_2}$ string functions as $\boldsymbol{q}$-diagrams

TL;DR

This paper introduces q-diagrams to interpret string functions of simply laced Lie algebras at level two, proving a conjecture for SO(2r) and linking classical identities through a diagrammatic approach.

Contribution

It proposes a novel diagrammatic method to analyze string functions and proves a conjecture for SO(2r), extending classical identities to a broader algebraic context.

Findings

Proof of the conjecture for SO(2r) string functions.

Introduction of q-diagrams as a diagrammatic interpretation tool.

Establishment of infinite series of diagrammatic identities.

Abstract

We discuss our conjecture for simply laced Lie algebras level two string functions of mark one fundamental weights and prove it for the $SO(2r)$ algebra. To prove our conjecture we introduce $q$-diagrams and examine the diagrammatic interpretations of known identities by Euler, Cauchy, Heine, Jacobi and Ramanujan. Interestingly, the diagrammatic approach implies these identities are related in the sense that they represent the first few terms in an infinite series of diagrammatic identities. Furthermore, these diagrammatic identities entail all the identities needed to prove our conjecture as well as generalise it to all $SO(2r)$ level two string functions. As such, our main objective is proving these series of diagrammatic identities thus extending the works mentioned and establishing our conjecture for the $SO(2r)$ level two string functions.