Abstract: Klyachko famously classified ($T$-linearized) vector bundles on a toric variety by certain data of filtrations associated to the rays in the fan. We give an interpretation of Klyachko’s data as a ``piecewise linear map'' and make a connection with the notion of ``building''. This leads us to a generalization of Klyachko’s classification to principal $G$-bundles on toric varieties for semisimple groups $G$ other than $\operatorname{GL}(n).$
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