19 March 2025 — Verona

Stable lattices in Bruhat-Tits buildings: algebra, geometry, and combinatorics

Mima Stanojkovski, Università di Trento

Abstract

Let \(K\) be a discretely valued field with ring of integers \(R\) and let \(d\) be a positive integer. Then the rank \(d\) free \(R\)-submodules of \(K^d\) (called \(R\)-lattices) are the \(0\)-simplices of an infinite simplicial complex called a Bruhat-Tits building. If \(O\) is an order in the ring of \(d\times d\) matrices over \(K\), then the collection of lattices that are also \(O\)-modules (called \(O\)-lattices) is a non-empty, bounded and convex subset of the building. Determining what these subsets are is in general a difficult question. I will report on joint work with Yassine El Maazouz, Gabriele Nebe, Marvin Hahn, and Bernd Sturmfels describing the geometric and combinatorial features of the set of \(O\)-lattices for some particular orders.