Uniform Semimodular Lattice & Valuated Matroid Hiroshi Hirai University of Tokyo [email protected] JSIAM, Osaka University Osaka, March 15, 2018 1 Contents 1. Matroid v.s. geometric lattice 2. Valuated matroid 3. Result: uniform semimoduler lattice • examples • valuated matroid uniform semimodular lattice • uniform semimodular lattice valuated matroid 4. Concluding remarks 2 Matroid : base family [EXC] , 3 Representable Matroid is a basis } ): matroid 4 Lattice Lattice := partially ordered set s. t. join & meet exist atom 5 Geometric Lattice := atomistic semimodular lattice semimodular: atomistic: { atom } is geometric lattice ordered by 6 Matroid v.s. Geometric Lattice Birkoff 1940 : matroid The family of flats is a geometric lattice. geometric lattice of height , the set of atoms ) is a (simple) matroid 7 Valuated Matroid Dress-Wenzel 1990 Valuated matroid on matroid : • Greedy algorithm • M-convexity, Discrete convex analysis (Murota 1996～) • Tropical Plucker vector (Speyer-Sturmfels 2004, Speyer 2008) 8 Representable Valuated Matroid is a basis } is a valuated matroid [VEXC] = Tropicalization of Grassmann-Plucker identity 9 Tree Metric tree Four-point condition =VEXC is a valuated matroid 10 Result Matroid Geometric Lattice integer-valued Uniform Valuated Matroid Semimodular Lattice 11 Definition Geometric lattice = atomistic semimodular lattice semimodular: atomistic: { atom } Uniform semimodular lattice semimodular: uniform: is automorphism is geometric lattice 12 Example 1: Integer Lattice is uniform (semi)modular lattice = min, = max, 13 Example 2. is uniform (semi)modular lattice corresponding to tree metric 14 Example 3. Lattice of Lattice Lattice = rank free -submodule of ordered by is uniform semimodular lattice corresponding to deg det 15 Valuated matroid Uniform semimodular lattice valuated matroid on Tropical linear space (Dress-Terhalle 1993, Murota-Tamura 2001, Speyer 2008) Tropical convexity (Murota-Tamura 2001,Develin-Sturmfels 2004) automorphism semimodularity is uniform semimodular lattice 16 Uniform semimodular lattice Valuated matroid Geometric lattice matroid on atoms Uniform semimodular lattice valuated matroid on ends Ray: s.t. End: an equivalence class of rays by parallel relation 17 the set of all ends -rays representatives of geometric lattice atoms Matroid at : Matroid at infinity: Valuated matroid on 18 Fix arbitrarily is -sublattice is valuated matroid 19 Concluding remarks • Lattice-characterization to valuated matroid & tropical linear space • = Dress-Terhalle completion Dress-Terhalle 1993 • Modular matroid = Spherical building of type A Birkhoff 1940, Tits 1974 Modular valuated matroid = Euclidean building of type A 20 • L-convex function on uniform modular lattice References H. Hirai: Uniform semimodular lattice and valuated matroid, arXiv, 2018. H. Hirai: Uniform modular lattice and Euclidean building, arXiv, 2017. 21.