Toric principal bundles, piecewise linear maps and Tits buildings

Abstract

We define the notion of a piecewise linear map from a fan $\Sigma$ to $\tilde{\mathfrak{B}}(G)$, the cone over the Tits building of a linear algebraic group $G$. Let $X_\Sigma$ be a toric variety with fan $\Sigma$. We show that when $G$ is reductive the set of integral piecewise linear maps from $\Sigma$ to $\tilde{\mathfrak{B}}(G)$ classifies the isomorphism classes of (framed) toric principal $G$-bundles on $X_\Sigma$. This in particular recovers Klyachko's classification of toric vector bundles, and gives new classification results for the orthogonal and symplectic toric principal bundles.