Toric principal bundles, Tits buildings and reduction of structure group (Q6572075)

The groundwork for the classification of toric vector bundles was laid by \textit{T. Kaneyama} [Nagoya Math. J. 57, 65--86 (1975; Zbl 0283.14008)] and \textit{A. A. Klyachko} [Izv. Akad. Nauk SSSR Ser. Mat. 53, no. 5, 1001--1039 (1989; Zbl 0706.14010)]. Building on Kaneyama's work, the authors in [\textit{J. Dasgupta} et al., ``Classification, reduction and stability of toric principal bundles'', Preprint, \url{arXiv:2012.13540}] extended the classification results for toric principal bundles and established several key results regarding the description of automorphism group and reduction of structure group of a toric principal bundle. In line with Klyachko's classification results, \textit{K. Kaveh} and \textit{C. Manon} [Math. Z. 302, no. 3, 1367--1392 (2022; Zbl 1510.14036)] provided a classification of toric principal bundles in terms of piecewise linear functions to the Tits building of the structure group.\N\NIn this paper, the authors adopt Kaveh and Manon's framework to offer a concise proof of a result by Dasgupta et al. regarding the equivariant automorphism group of a toric principal bundle. Moreover, they present a simple criterion for the reduction of the structure group of a toric principal bundle, formulated in terms of the associated piecewise linear map. Motivated by the equivariant splitting problem for toric principal bundles, the authors introduce the notion of Helly's number of a building and pose the problem of determining sharp upper bounds for it.