New homogeneous ideals for current algebras: filtrations, fusion products and Pieri rules

New graded modules for the current algebra of ${\mathfrak{s}\mathfrak{l}}_n$ are introduced. Relating these modules to the fusion product of simple ${\mathfrak{s}\mathfrak{l}}_n$-modules and local Weyl modules of truncated current algebras shows their expected impact on several outstanding conjectures. We further generalize results on PBW filtrations of simple ${\mathfrak{s}\mathfrak{l}}_n$-modules and use them to provide decomposition formulas for these new modules in important cases.