For the general linear group, the Bruhat-Tits building can be realized explicitly in terms of periodic lattice chains in the standard representation. Further, each parahoric group scheme can be described as an automorphism group of a particular chain. I will explain a Tannakian formalism which establishes analogous descriptions for arbitrary connected reductive groups over complete discretely valued fields. This complements previously known results for classical groups, and fits in with Mumford's Geometric Invariant Theory, where spherical buildings are similarly described. This is joint work with Kevin Wilson.