Spherical and Affine Buildings For SLn
The simplicial structure of the affine and spherical buildings associated to SLn(Qp), as well as their interconnections, are easy to explain directly using only concepts from elementary algebra and geometry (see Garrett 1997). In this case there are three different buildings, two spherical and one affine. Each is a union of apartments, themselves simplicial complexes. For the affine group, an apartment is just the simplicial complex obtained from the standard tessellation of Euclidean space En-1 by equilateral (n-1)-simplices; while for a spherical building it is the finite simplicial complex formed by all (n-1)! simplices with a given common vertex in the analogous tessellation in En-2.
Each building is a simplicial complex X which has to satisfy the following axioms:
-
- X is a union of apartments.
- Any two simplices in X are contained in a common apartment.
- If a simplex is contained in two apartments, there is a simplicial isomorphism of one onto the other fixing all common points.
Read more about this topic: Building (mathematics)
Famous quotes containing the word buildings:
“If the factory people outside the colleges live under the discipline of narrow means, the people inside live under almost every other kind of discipline except that of narrow means—from the fruity austerities of learning, through the iron rations of English gentlemanhood, down to the modest disadvantages of occupying cold stone buildings without central heating and having to cross two or three quadrangles to take a bath.”
—Margaret Halsey (b. 1910)