An interesting construction of a Tits building?

The notion of Tits building was introduced by Jacques Tits to study certain questions in group theory. The wikipedia entry gives a way to construct a Tits building from a vector space, but I would be interested in a simpler construction defined as follows.

Let $\mathbb{V}$ be a vector space over a field $\mathbb{F}$. Construct a simplicial complex $SCI(\mathbb{V})$ with ground set $\mathbb{V}$, where the simplices are the linearly independent subsets of $\mathbb{V}$ - note that this essentially defines a representable matroid.

Consider two basis $B^1 = \{u^1_1,\ldots,u^1_n\}$ and $B^2 = \{u^2_1,\ldots,u^2_n\}$ of $\mathbb{V}$, such that the basis-change matrix $M_{B^1,B^2}$ is lower-triangular. We can then define an apartment $O_{B^1,B^2}$ which is the subcomplex of $SCI(\mathbb{V})$ formed by the subsets $\{u^{i_1}_1,\ldots,u^{i_p}_p\}$ for $1 \leq p \leq n, i_j \in \{1,2\}$.

Question: does this construction satisfy the four axioms of a Tits building (given in the above link)? If so, what are the possible applications to matroid theory?

• Wouldn't this have a better chance of getting an answer at Math Overflow? – David Richerby Apr 12 '14 at 23:27