Towards plethystic $\mathfrak{sl}_2$ crystals

TL;DR

This paper explores the combinatorial problem of decomposing Young's lattice to find $rak{sl}_2$ crystals, providing new formulas and strategies that recover recent solutions for small cases.

Contribution

It introduces a new strategy for decomposing Young's lattice into symmetric chains, leading to counting formulas and recursive methods for plethystic coefficients.

Findings

Recovered solutions for $n \,\leq\, 4$

Derived new recursive formulas for plethysms

Provided formulas for constituents of $\\Lambda^n\text{Sym}^r\mathbb{C}^2$

Abstract

To find crystals of $\mathfrak{sl}_2$ representations of the form $\Lambda^n\text{Sym}^r\mathbb{C}^2$ it suffices to solve the combinatorial problem of decomposing Young's 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 $\Lambda^n\text{Sym}^r\mathbb{C}^2$.