Infinite sequences via Lie algebra actions for oligomorphic groups

Many integer sequences arise as numbers of $G$-orbits on $\binom{X}{n}$ as $n$ varies, for a permutation group $G\subseteq \operatorname{Sym}(X)$. For finite $X$, Stanley proved that these finite sequences increase towards the middle using an action of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. For infinite sets $X$, and hence infinite sequences, Cameron provided an argument for monotonicity by identifying orbits with a vector space basis of the orbit algebra $\mathsf{H}_{G,X}^{\star}$, and proving injectivity of a certain operator $\mathsf{H}_{G,X}^{\star}\to \mathsf{H}_{G,X}^{\star+1}$. In this paper we generalize Stanley's approach to oligomorphic groups, and in particular extend Cameron's operator to a full $\mathfrak{sl}_2(\mathbb{C})$-action on $\mathsf{H}_{G,X}^{\star}$. As intermediate step, we define for every oligomorphic permutation group $G\subseteq \operatorname{Sym}(X)$ the $X$-th tensor power $(k^r)^{\otimes X}$, generalizing work of Entova-Aizenbud. We show that this space carries natural commuting actions of $G$ and the Lie algebra $\mathfrak{gl}_r(k)$, the latter depending on a Harman-Snowden measure $\mu$ on $G$. We then show that $\mathsf{H}_{G,X}^{\star}\subseteq (\mathbb{C}^2)^{\otimes X}$ has an ascending filtration by $\mathfrak{sl}_2(\mathbb{C})$-Verma modules. We explain how our approach applies to Fibonacci numbers, Tribonacci numbers, etc. by constructing measures on products with $(\mathbb{Q},<)$.