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I am very new to this topic, and my background is mostly in graph theory and basic algebra. What I want for now is to understand the structure of the dimension $2$ complex $\mathcal{B}(PGL_3(K))$ where $K$ is a $p$-adic field. In this case, we have vertices, edges and triangles, just one step above graphs and so easier to approach in elementary terms, I hope.

So what does this building look like? I heard somewhere that the link of every vertex is a bipartite graph which is the Levy graph of the incidence structure of the projective plane over $\mathbb{Z}/p\mathbb{Z}$.

But I am unable to find any accessible references on this. All references I find start from and go deep into representation theory, while all I am seeking, at present, is an explicit combinatorial description of the building of $PGL_3$ as a hypergraph: what are the vertices, which vertices are connected by edges, what are the triangles, and what is the structure of the star or link of a vertex. I am not looking for the most general definitions or abstractions.

Is there a simple combinatorial description here, like in the case of graphs? While I will approach it using representation theory in time, it could help to have some feel for this structure just as a hypergraph at present. I want to play around with the combinatorial structure for now.

Any explanations, or even accessible references, would help greatly. Thanks!

BharatRam
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In the first case of $PGL_2(K)$, a good reference is the Serre's book "Trees" (Ch. II $\S 2$ page 69). The construction of the building begins with lattices of rank two as a vertices and the incidence relation (an edge from L to L') between $L\subset L'$ given by $L/L' \cong k$ (residue field). This generate a tree (a graph without cycles), when $|k| =2$ the tree is bipartite. In general, the number of neighbors of a vertex is $|k| +1$.

For the case of $PGL_n(K)$ ($n\geq 2$), a simpler construction of the building is by flag complexes. Consider $V=k^n$, for $0\leq r \leq n-1$ an $r$-flag is a chain $$\{0\}\subset V_0\subset V_1\subset \cdots \subset V_r\subset k^n,$$ where each $V_i$ is a $k$-subspace of $k^n$. An $r$-simplex is an $r$-flag. The $(n-1)$-simplex corresponds to complete (or maximal) chain of subspaces, they called $\textit{chambers}$. In your case, $n=3$, the chambers correspond to $2$-simplex, in particular they are equilateral triangles. Fixing a basis (e.g canonical basis) the chambers forms a subcomplex called $\textit{apartment}$ which is a tilling of the plane. Thee building is the gluing by certain axiom of these apartments.

The main reference for me, is the Kenneth Brown's book "Buildings" that is a short reference. For more complete theory, the book of Brown & Abramenko "Building, theory and applications" it will be very useful for you, i think.

Regards.

P. Gudu
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