Uniform Semimodular & Valuated

Hiroshi Hirai University of Tokyo [email protected]

JSIAM, Osaka University Osaka, March 15, 2018

1 Contents

1. Matroid v.s. 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) = 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.

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