In recent joint work with Kiumars Kaveh, we have shown how Klyachko data of a toric vector bundle can be repackaged in a way that is at once more combinatorial and geometric: a piecewise-linear map from the fan of the toric variety to a spherical building of type A. We have since used this perspective to give a classification of toric G-principal bundles (G a connected algebraic group) using more general spherical buildings, bundles over a toric scheme using the affine building (joint with Boris

Tsevlikhovskiy) and introduce a notion of tropical vector bundles. The Chern-Weil map, reduction of structure group, and various notions of positivity can be phrased naturally in this language.

After introducing some of this background, I will describe a recent result along these lines classifying torus equivariant vector bundles on normal varieties of equipped with the action of a codimension 1 torus. Such varieties are called complexity 1 T-varieties.

I'll also give some tentative remarks on an ongoing project to construct moduli, and connections to vector bundles over stacky curves. This is joint work with Jyoti Dasgupta, Chandranandan Gangopadhyay, and Kiumars Kaveh.