Linear representations of twin cities

TL;DR

This paper extends the explicit flag complex descriptions from classical spherical and affine buildings to twin cities, providing linear representations as flag complexes of subspaces in Hilbert spaces, relevant for Kac-Moody groups.

Contribution

It generalizes the flag complex description to twin cities and constructs their linear representations in Hilbert spaces, advancing understanding of Kac-Moody geometric structures.

Findings

Explicit construction of twin city representations as flag complexes

Extension of classical building descriptions to twin cities

Framework for analyzing Kac-Moody group completions

Abstract

For spherical Tits buildings of the classical types there are well-known explicit descriptions as flag complexes. Similarly for affine buildings of the classical types there are explicit constructions in terms of lattices. In this article we generalize the flag complex description to twin cities, a generalization of twin buildings adapted to analytic completions of Kac-Moody groups as they appear for example in Kac-Moody geometry. We construct linear representations of twin cities as flag complexes of certain subspaces in Hilbert spaces.