Presentations of the principal subspaces of the higher-level standard $\widehat{\mathfrak{sl}(3)}$-modules

Using the theory of vertex operator algebras and intertwining operators, we obtain presentations for the principal subspaces of all the standard $\widehat{\goth{sl}(3)}$-modules. Certain of these presentations had been conjectured and used in work of Calinescu to construct exact sequences leading to the graded dimensions of certain principal subspaces. We prove the conjecture in its full generality for all standard $\widehat{\goth{sl}(3)}$-modules. We then provide a conjecture for the case of $\widehat{\goth{sl}(n+1)}$.