Stanley showed that finite sequences counting group orbits on subsets increase toward their midpoint by constructing an sl_2(C) Lie algebra action — the same algebraic structure that governs angular momentum in quantum mechanics. The raising and lowering operators enforce unimodality. But Stanley's construction works only for finite sets. For infinite sets, Cameron proved monotonicity by different methods, without the Lie algebra.
Wojciechowski (arXiv: 2603.23809) closes the gap. He constructs a full sl_2(C) action on the orbit algebra of oligomorphic groups — groups acting on infinite sets with finitely many orbits on finite subsets. The orbit algebra decomposes into a direct sum of Verma modules, the fundamental building blocks of sl_2 representation theory. The monotonicity of orbit-counting sequences follows immediately from the representation structure.
The applications are unexpected. The Fibonacci sequence arises as the orbit-counting sequence of a specific group action on ordered rational structures with a particular measure. The Tribonacci sequence arises similarly. These classical combinatorial sequences — usually defined by recurrences or generating functions — acquire a new identity as dimensions of weight spaces in Verma modules.
The through-claim: integer sequences are shadows of symmetry. The Fibonacci numbers aren't just defined by F(n) = F(n-1) + F(n-2). They count orbits of a group action on an infinite ordered structure, and their growth pattern is a consequence of the sl_2 representation theory governing that action. The recurrence is what you see. The Lie algebra is why it works. The sequence is a projection of a higher-dimensional algebraic structure onto one dimension of counting.
Wojciechowski, 2603.23809. Representation theory / Lie algebras / oligomorphic groups / Fibonacci sequence / Verma modules.