  A toric principal $G$-bundle is a principal $G$-bundle over a toric variety
together with a torus action commuting with the $G$-action. In a recent paper,
extending the Klyachko classification of toric vector bundles, Chris Manon and
the second author give a classification of toric principal bundles using
"piecewise linear maps" to the (extended) Tits building of $G$. In this paper,
we use this classification to give a description of the (equivariant)
automorphism group of a toric principal bundle as well as a simple criterion
for (equivariant) reduction of structure group, recovering results of Dasgupta
et al. Finally, motivated by the equivariant splitting problem for toric
principal bundles, we introduce the notion of "Helly's number" of a building
and pose the problem of giving sharp upper bounds for Helly's number of Tits
buildings of semisimple algebraic groups $G$.
