Distance in the Affine Buildings of SL_n and Sp_n

Details Distance in the Affine Buildings of SL_n and Sp_n Distance in the Affine Buildings of SL_n and Sp_n

2005-11-22

arxiv.org/abs/math/0511556v6

INTEGERS: Electronic Journal of Combinatorial Number Theory 8 (2008), #A48

Mathematics

Number Theory

16 pages, 3 figures; minor corrections; accepted for publication in INTEGERS: The Electronic Journal of Combinatorial Number T

Scientific paper

For a local field $K$ and $n \geq 2$, let $\Xi_n$ and $\Delta_n$ denote the affine buildings naturally associated to the special linear and symplectic groups $\SL_n(K)$ and $\Sp_n(K)$, respectively. We relate the number of vertices in $\Xi_n$ ($n \geq 3$) close (i.e., gallery distance 1) to a given vertex in $\Xi_n$ to the number of chambers in $\Xi_n$ containing the given vertex, proving a conjecture of Schwartz and Shemanske. We then consider the special vertices in $\Delta_n$ ($n \geq 2$) close to a given special vertex in $\Delta_n$ (all the vertices in $\Xi_n$ are special) and establish analogues of our results for $\Delta_n$.

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