Distance formulas in Bruhat-Tits building of \(\mathrm{SL}_d(\mathbb{Q}_p)\)

We study the distance on the Bruhat-Tits building of the group \(\mathrm{SL}_d(\mathbb{Q}_p)\) (and its other combinatorial properties). Coding its vertices by certain matrix representatives, we introduce a way how to build formulas with combinatorial meanings. In Theorem 1, we give an explicit formula for the graph distance \(\delta(\alpha,\beta)\) of two vertices \(\alpha\) and \(\beta\) (without having to specify their common apartment).Our main result, Theorem 2, then extends the distance formula to a formula for the smallest total distance of a vertex from a given finite set of vertices. In the appendix we consider the case of \(\mathrm{SL}_2(\mathbb{Q}_p)\) and give a formula for the number of edges shared by two given apartments.