Infinite sequences via Lie algebra actions for oligomorphic groups

Abstract

Many integer sequences arise as numbers of $G$-orbits on $\left(\genfrac{}{}{0ex}{}{X}{n}\right)$ as $n$ varies, for a permutation group $G\subseteq \mathrm{Sym}(X)$. 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 $G\subseteq \mathrm{Sym}(X)$ 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 $(\mathbb{Q},<)$.

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