Infinite sequences via Lie algebra actions for oligomorphic groupsOberseminar Darstellungstheorie

by Zbigniew Wojciechowski (Technische Universität Dresden)

Friday Apr 24, 2026, 2:00 PM → 4:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Many integer sequences arise by counting G-orbits on the set of n-element subsets of a set X, for a group G acting on X. For finite X, Stanley proved that these finite sequences increase towards the middle using an action of the Lie algebra sl2. For infinite sets X, and hence infinite sequences, Cameron provided an argument for monotonicity. He first identifies orbits with a vector space basis of a certain commutative k-algebra H*, called the orbit algebra. He then considers the operator which forms the product with the constant 1-function on X, and proves its injectivity. In this paper, we generalize Stanley's approach to oligomorphic groups, and in particular extend Cameron's operator to a full sl2-action on H*.

Most crucially, we define for every oligomorphic permutation group G the X-th tensor power of the standard representation of the general linear Lie algebra gl_d, generalizing work of Entova-Aizenbud. This space carries natural commuting actions of G and gl_d, the latter depending on a Harman–Snowden measure on G. In the case d = 2, Cameron's result implies that H* decomposes into a direct sum of Verma modules, which provides a representation-theoretic explanation for monotonicity. A highlight of the talk is an sl2-representation whose weight space dimensions are given by the Fibonacci numbers.