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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2603.23809 |
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| _version_ | 1866914586693730304 |
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| author | Wojciechowski, Zbigniew |
| author_facet | Wojciechowski, Zbigniew |
| contents | 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 $μ$ 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 $(\mathbb{Q},<)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23809 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Infinite sequences via Lie algebra actions for oligomorphic groups Wojciechowski, Zbigniew Representation Theory Combinatorics Group Theory Logic 17B10, 20B27, 05A15, 20B30 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 $μ$ 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 $(\mathbb{Q},<)$. |
| title | Infinite sequences via Lie algebra actions for oligomorphic groups |
| topic | Representation Theory Combinatorics Group Theory Logic 17B10, 20B27, 05A15, 20B30 |
| url | https://arxiv.org/abs/2603.23809 |