In this talk we review construction of toric varieties and classification of (torus equivariant) line bundles and vector bundles on them (after Klyachko). We interpret Klyachko's data of a vector bundle as a "piecewise linear map" into the Tits building of the general linear group GL(r). This "building" perspective helps to approach many questions about vector bundles on toric varieties in a new light. As an application of this idea, we obtain a classification of (torus equivariant) vector bundles on toric schemes in terms of "piecewise affine maps" to the Bruhat-Tits building of GL(r). This is work in progress with Chris Manon and Boris Tsvelikhovsky. I try to cover most of the background material.

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