Affine buildings and tropical convexity.

*(English)*Zbl 1133.52003Let \(K= \mathbb C((z))\) be the field of Laurent series over the complex numbers, and denote by \(\mathcal B_d\) the Bruhat-Tits building of the special linear group \(SL_d\). The set of vertices of \(\mathcal B_d\), seen as a simplicial complex, is in bijection with the set of homothety classes of \(\mathbb C[[t]]\)-lattices in \(\mathbb C((t))^d\). There is a well-known geometric realization in terms of valuations (or “additive norms”) on \(K^d\), and hence a natural notion of convexity for \(\mathcal B_d\).

In the paper at hand, the focus is on two different notions of convexity. A subset \(\{ M_1, \dots, M_r \}\) of the set of vertices of \(\mathcal B_d\) is called max-convex, if the set of all lattices in the homothety classes \(M_i\) is closed under finite sums. It is called min-convex, if this set is closed under finite intersections. In a sense, these notions arise naturally from tropical geometry. Denote by \(\zeta \colon\mathbb R^N \rightarrow \mathbb R^N /\mathbb R(1,1,\dots, 1) := \mathbb T\mathbb P^{N-1}\) the projection from \(\mathbb R^N\) to the tropical projective space. The tropical projective space \(\mathbb T\mathbb P^{N-1}\) can naturally be identified with the standard apartment in \(\mathcal B_N\). A subset \(P \subset \mathbb R^N\) is called tropically convex, if it is closed under linear combinations in the min-plus-algebra. Then \(P=\zeta^{-1}(\zeta(P))\), so \(P\) is determined by \(\zeta(P)\). It is shown that any finite union of apartments in \(\mathcal B_d\) (a membrane) can be embedded in a suitable \(\mathbb T\mathbb P^{N-1}\) for \(N\) large (cf. also [S. Keel and J. Tevelev, Duke Math. J. 134, No. 2, 259–311 (2006; Zbl 1107.14026)], Thm. 4.11). Assume we want to compute the min-convex hull of a finite set of vertices in \(\mathcal B_d\), and we know that it is contained in a given membrane. It is proved that if we embed this membrane in a tropical projective space as above, then the min-convex hull in \(\mathcal B_d\) coincides with the tropical convex hull inside this tropical projective space. There remains the problem to find a membrane which contains the min-convex hull. This problem, as well as the “dual” problem of computing max-convex hulls, and the related question how to compute the intersection of a finite number of apartments in the building are discussed from an algorithmic point of view, and reduced to computational problems in tropical geometry (for which algorithms are known). The following theorem is obtained: The min-convex hull of a finite set of vertices in \(\mathcal B_d\) coincides with the standard triangulation of a tropical polytope in a suitable membrane. The max-convex hull coincides with the image of a max-tropical polytope under the nearest point map onto a min-tropical linear space.

In the paper at hand, the focus is on two different notions of convexity. A subset \(\{ M_1, \dots, M_r \}\) of the set of vertices of \(\mathcal B_d\) is called max-convex, if the set of all lattices in the homothety classes \(M_i\) is closed under finite sums. It is called min-convex, if this set is closed under finite intersections. In a sense, these notions arise naturally from tropical geometry. Denote by \(\zeta \colon\mathbb R^N \rightarrow \mathbb R^N /\mathbb R(1,1,\dots, 1) := \mathbb T\mathbb P^{N-1}\) the projection from \(\mathbb R^N\) to the tropical projective space. The tropical projective space \(\mathbb T\mathbb P^{N-1}\) can naturally be identified with the standard apartment in \(\mathcal B_N\). A subset \(P \subset \mathbb R^N\) is called tropically convex, if it is closed under linear combinations in the min-plus-algebra. Then \(P=\zeta^{-1}(\zeta(P))\), so \(P\) is determined by \(\zeta(P)\). It is shown that any finite union of apartments in \(\mathcal B_d\) (a membrane) can be embedded in a suitable \(\mathbb T\mathbb P^{N-1}\) for \(N\) large (cf. also [S. Keel and J. Tevelev, Duke Math. J. 134, No. 2, 259–311 (2006; Zbl 1107.14026)], Thm. 4.11). Assume we want to compute the min-convex hull of a finite set of vertices in \(\mathcal B_d\), and we know that it is contained in a given membrane. It is proved that if we embed this membrane in a tropical projective space as above, then the min-convex hull in \(\mathcal B_d\) coincides with the tropical convex hull inside this tropical projective space. There remains the problem to find a membrane which contains the min-convex hull. This problem, as well as the “dual” problem of computing max-convex hulls, and the related question how to compute the intersection of a finite number of apartments in the building are discussed from an algorithmic point of view, and reduced to computational problems in tropical geometry (for which algorithms are known). The following theorem is obtained: The min-convex hull of a finite set of vertices in \(\mathcal B_d\) coincides with the standard triangulation of a tropical polytope in a suitable membrane. The max-convex hull coincides with the image of a max-tropical polytope under the nearest point map onto a min-tropical linear space.

Reviewer: Ulrich Görtz (Bonn)