Abstract: The standard monomial basis for $\mathfrak{sl}_r$-invariant polynomials is indexed by rectangular standard Young tableaux, but it lacks rotational invariance. Although promotion induces a cyclic action on tableaux, a more symmetric basis is desirable. For $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, such bases are given by non-crossing matchings and non-elliptic $\mathfrak{sl}_3$ webs, respectively. In the case of $\mathfrak{sl}_4$, a web basis has recently been constructed using hourglass plabic graphs, but this approach relies on intricate growth rules and does not readily generalize to higher rank.

In this thesis, we introduce a simpler and more direct construction of $\mathfrak{sl}_4$ webs. Starting from a rectangular four-row standard Young tableau, we construct the associated web by stacking the $\mathfrak{sl}_3$ webs corresponding to the top three rows and the bottom three rows, identifying along the non-crossing matching determined by the middle two rows. We prove that the resulting web is fully reduced and lies in the same equivalence class as the $\mathfrak{sl}_4$ web obtained via growth rules.