Solitary and edge-minimal bases for representations of the simple lie algebra $G_2$

Abstract

We consider two families of weight bases for “one-rowed” irreducible representations of the simple Lie algebra $G_2$ constructed in Donnelly et al [Constructions of representations of $\mathfrak{o}(2n+1,\mathbb{C})$ that imply Molev and Reiner–Stanton lattices are strongly Sperner, Discrete Math. 263 (2003) 61–79] using two corresponding families of distributive lattices (called “supporting graphs”), here denoted $L_{G_2}^{\mathrm{LM}}(k)$ and $L_{G_2}^{\mathrm{RS}}(k)$. We exploit the relationship between these bases and their supporting graphs to give combinatorial proofs that the bases enjoy certain uniqueness and extremal properties (the “solitary” and “edge-minimal” properties, respectively). Our application of the combinatorial technique we develop in this paper to obtain these results relies on special total orderings of the elements and edges of the lattices. We also apply this technique to another family of lattice supporting graphs to re-derive results obtained in Donnely et al. $\mathrm{using}$ different, more algebraic methods.