Open problems session 1

Gleb Koshevoy

Good Exercise:

For any set of vectors A in \mathbb{Z}^n, let H(A) be the family of integral polytopes whose facet normals are in A directions and let Ed(A) be the family of integral polytopes whose edge directions are in A directions  (H for Hoffman or Hyperplanes, and Ed for Edmond or Edges).

Let A_n=\{e_i - e_j : i \neq j \text{ in } \{1,\dots,n\}\} \cup \{\pm e_i : i \in \{1,\dots,n\}\}, which are edge directions of the simplex with vertices 0,e_1,\dots,e_n.  It is a maximal unimodular system.  It is easy to see that H(A_n) (also called L^\# convex polytopes or alcoved polytopes) is closed under intersection.  The family Ed(A_n) (also called M^\# convex polytopes or generalized polymatroids or generalized permutohedra) is not closed under intersection (e.g. take two pyramid in an octahedron, which is the matroid polytope of U_{4,2} that do not share a square face).

But Ed(A_n) has the property that the intersection of any two (but not three!) polytopes in the the family is an integral polytope.  We will call this property (*).

Problem (proven by Isedora Heller): Suppose A is a maximal cardinality unimodular system with property (*).  Is A = A_n after a change of lattice basis?

Remark: Let G be a graph and A the matrix whose columns are e_i - e_j for all edges (i,j) in the graph.  We can then construct a graphical matrix [I | A]and cographical matrix [I | A^T] of GSeymour’s decomposition theorem says that every unimodular system is obtained from graphical matrices, cographical matrices, and an exceptional system R_{10} via two operation \oplus, which is putting the matrices (whose columns are the vectors) in a block diagonal form, and \oplus' is like \oplus' with an extra row containing a one and the rest zero.

Question:

Let \mathcal{L} be a finite collection of some linear subspaces of \mathbb{Q}^n.  We say that \mathcal{L} is unimodular if for any subcollection \{L_1,\dots,L_r\} \subset \mathcal{L}, the group \mathbb{Z}^n / (\mathbb{Z}^n \cap L_1+\cdots L_r) is torsion-free.

Conjecture: Suppose A is a unimodular set of vectors such that the lines spanned by vectors in A form a maximal unimodular system of subspaces, i.e. we cannot add any other subspaces to the collection while keeping the unimodularity property.  Then A = A_n for some n after a change of lattice basis.

John Weymark

Intersect the boundary of the sets Q_a‘s (see the slides) with the set of valuations, which can be assumed to be convex. Call this intersection S

Question: Write $late S$ as the intersection of a tropical hyperplane with the set of valuations.  Find conditions under which this is possible.  How do you find the tropical hyperplane?

 

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