Towards plethystic \(\mathfrak{sl}_2\) crystals

To find crystals of \(\mathfrak{sl}_2\) representations of the form \(\Lambda^n\operatorname{Sym}^r\mathbb{C}^2\) it suffices to solve the combinatorial problem of decomposing the Young lattice into symmetric, saturated chains. We review the literature on this latter problem, and present a strategy to solve it. For \(n \le 4\), the strategy recovers recently discovered solutions. We obtain (i) counting formulas for plethystic coefficients, (ii) new recursive formulas for plethysms of Schur functions, and (iii) formulas for the number of constituents of \(\Lambda^n\operatorname{Sym}^r\mathbb{C}^2\). Mathematics Subject Classifications: 05E10, 17B10, 05A30 Keywords: Plethysm, crystals, symmetric chain decompositions, Young lattice, Gaussian coefficients