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

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