Uniform Semimodular Lattice & Valuated Matroid Insights

Dive into the intricate world of Uniform Semimodular Lattices and Valuated Matroids as discussed in the research by Hiroshi Hirai at the University of Tokyo. Explore the connections to Euclidean building and combinatorial geometries, offering a fresh perspective on geometric structures and matroid theory. Unveil the concepts of lattice theory, geometric lattices, and the extension of equivalences to valuated matroids. Delve into the applications of tree metrics and discrete convex analysis in this cutting-edge field of mathematics.

• Combinatorial Geometries
• Matroid Theory
• Lattice Theory
• Valuated Matroids
• Euclidean Building

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1. Uniform Semimodular Lattice, Valuated Matroid, & Euclidean building Hiroshi Hirai University of Tokyo hirai@mist.i.u-tokyo.ac.jp Combinatorial Geometries 2018: matroid, oriented matroids, and applications CIRM, Marseille-Luminy, France 9/27, 2018 1

2. Contents 1. Matroid geometric lattice 2. Valuated matroid 3. Uniform semimoduler lattice examples valuated matroid uniform semimodular lattice uniform semimodular lattice valuated matroid 4. Connection to Euclidean building (option) 5. Concluding remarks 2

3. Lattice Lattice := poset s.t. ?,? join ? ? = unique min { ? | ? ? ?}, meet ? ? = unique max { ? | ? ? ?} ?,? ? ? ? ?} ? :? ?,? = {?,?} 1 atom 0 0 3

4. Geometric Lattice := atomistic semimodular lattice semimodular: ? ? :? ? :? ? atomistic: ? = { atoms ?} 001 111 110 101 010 100 011 Ex: ? = {?1,?2, ,??} ? { subspaces spanned by subsets of ?} is a geometric lattice 4

5. Matroid Geometric Lattice Birkhoff 1940 M = ?, : matroid The family of flats 2?is a geometric lattice : geometric lattice ?: the set of atoms ? ? ? = 1,minimal} (?, ) is a simple matroid Goal: extend this equivalence to valuated matroids 5

6. Valuated Matroid Dress-Wenzel 1990 Valuated matroid ?: on matroid M = (?, ): [VEXC] ?,? , ? ? ? , ? ? ? ? ? + ?(? ) ? ? ? + ? + ?(? ? + ?) Greedy algorithm M-convexity, Discrete convex analysis (Murota 1996 ) Tropical Plucker vector (Speyer-Sturmfels 2004, Speyer 2008) 6

7. Representable Valuated Matroid ? (?)? {? ?| ? is a basis of (?)? } ?:? deg det ? is a valuated matroid [VEXC] = Tropicalization of Grassmann-Plucker relation det ? det ? = det (? ? + ?) det (? + ? ?) ? ( ,+) (+,max) deg det ? + deg det ? max? deg det (? ? + ?) + deg det (? + ? ?) 7

8. Tree Metric ? ? = ?,? : tree ? ? ? ? Four-point condition =VEXC ? ???,? + ???,? max {???,? + ???,? ,???,? + ???,? } ??:? is a valuated matroid 2 8

9. Result (H.18) Matroid Geometric Lattice integer-valued Uniform Valuated Matroid Semimodular Lattice 9

10. Definition Geometric lattice = atomistic semimodular lattice semimodular: ? ? :? ? :? ? atomistic: ? = { atoms ?} Uniform semimodular lattice semimodular: ? ? :? ? :? ? uniform: ? ?+ is automorphism Ascending operation: ?+ {? | ? ?} Lem: ?, ?+is a geometric lattice 10

11. Example 1: Integer Lattice? +? ?+ ? ? is uniform (semi)modular lattice = min, = max, ?+= ? + ? 11

12. Example 2. Tree ?+ ? Tree is uniform (semi)modular lattice corresponding to tree metric 12

13. Example 3. Lattice of Lattices ? = {?1,?2, ,??} ?? { ? -module generated by ??1?1,??2?2, ,????? }? ? ? := {?/? ? |deg ? deg?} is uniform semimodular lattice (with ?+= ??) corresponding to ? = deg det Recall ? = {?1,?2, ,??} ? { -vector space generated by ? }? ? is geometric lattice corresponding to M(?) 13

14. Proof outline tropical geometry tropical linear space, tropical convexity valuated matroid uniform semimodular lattice ray, end, matroid at infinity inspired by building theory: spherical building at infinity in Euclidean building 14

15. Valuated matroid Uniform semimodular lattice ?: valuated matroid on (?, ) ? Argmax? ? ? + ? ??(?) (? ?) Tropical linear space (Dress-Terhalle 1993, Murota-Tamura 2001, Speyer 2008) ? ? ? ? M?= ?, ? is loop-free} Tropical convexity (Murota-Tamura 2001,Develin-Sturmfels 2004) ? ? ? ? + ?? ? ? ?,? ? ? min(?,?) ? ? +automorphism ----> semimodularity ? ? ?is uniform semimodular lattice 15

16. Uniform semimodular lattice Valuated matroid Geometric lattice matroid on atoms Uniform semimodular lattice valuated matroid on ???? ends Ray: ?0 ?1 ?2 s.t. ?0,??= ?0,?1,?2, ,?? ( ?) End: an equivalence class of rays by parallel relation ?? ?? ?? ?0= (??) ?0 16

17. ?: the set of all ends ?-rays representatives of ? ?+ geometric lattice atoms ? Matroid at ?: M?= ?, ? Matroid at infinity: M = ?, ? ? Valuated matroid ? on M 17

18. ? ? ? ? ?} ? (? (M)) ~ apartement ? {?, } ? ?{?, } ? ? {?,?} 2 ? ? Fix ?, define ?: (M ) by valuated matroid ? ? ? ??,? ?? ? ? ? ? ?} 18

19. (?,?) ? ? ? ?,? ? ? ? (? ,? ) ?,? Completion of valuated matroid Dress-Terhalle 1993 c.f. p-adic completion 19

20. Connection to Euclidean building Modular matroid := matroid whose lattice of flats is a modular lattice Modular valuated matroid := valuated matroid (?,?) s.t. ? ? ? is a modular lattice Birkhoff, Tits H.17 Spherical building of type A Euclidean building of type A Modular lattice := lattice with ? ? ? ? ? = ? (? ?) Cor (H.18) ?: modular valuated matroid ?(?)/ ?is a geometric realization of Euclidean building For ? = ??, ? = deg det: A.Dress and W. Terhalle: The tree of life and other affine buildings, ICM, Berlin 1998 20

21. Concluding remarks Lattice-theoretic characterization to valuated matroid & tropical linear space More application to tropical geometry ? Invariants for uniform semimodular lattices e.g., characteristic polynomial ? 21

22. References H. Hirai: Uniform modular lattice and Euclidean building, arXiv, 2017. H. Hirai: Uniform semimodular lattice and valuated matroid, arXiv, 2018. H. Hirai: Computing degree of determinant via discrete convex optimization over Euclidean building, arXiv, 2018. 22

23. Spherical/Euclidean building of type A := simplicial complex = apartements : B0: apartment Coxeter complex of type A. B1: ?,? , apartment ?,?. B2: apartments , ?,? isomorphism fixing ?,?. spherical Euclidean Coxeter complex of type A Decomposition of ?? 1 by ??? {?|??= ??} Decomposition of ?/ ? by ???,? {?|??= ??+ ?} 23

24. Spherical building of type A = modular matroid Thm (Tits) Spherical building of type A = chain complex of the subspace lattice of generalized projective geometry Thm (Birkhoff) Modular geometric lattice = the subspace lattice of generalized projective geometry Spherical building of type A = chain complex of modular geometric lattice 24

25. Euclidean building of type A = modular valuated matroid Thm (H.17) Euclidean building of type A = complex of short chains/~ of a uniform modular lattice short chain ? ?1 ?? ?+ ? ? ? = ?+?or ? = ?+? 25