Valuative invariants for polymatroids Harm Derksen 1 Alex Fink 2 1 University of Michigan 2 UC Berkeley → North Carolina State arXiv:0908.2988 FPSAC 2010 August 5, 2010 Derksen, Fink Valuative invariants for polymatroids 1 / 21
Outline ◮ Matroids and polymatroids ◮ The Tutte polynomial: a motivating example ◮ Valuations ◮ Canonical bases for (poly)matroids and valuations ◮ (Hopf) algebras of valuations Derksen, Fink Valuative invariants for polymatroids 2 / 21
Matroids Definition (Edmonds; Gelfand-Goresky-MacPherson-Serganova) A matroid M (on the ground set [ n ] ) is a polytope such that ◮ every vertex (basis) of M lies in { 0 , 1 } n ; ◮ every edge of M is parallel to e i − e j for some i , j ∈ [ n ] . 1100 001 101 011 0110 1010 1001 0101 1010 1001 0110 0101 1010 0110 0101 1001 100 010 110 0011 Derksen, Fink Valuative invariants for polymatroids 3 / 21
Polymatroids Definition (Edmonds) A polymatroid M (on [ n ] ) is a polytope such that ◮ every vertex of M lies in Z n ≥ 0 ; ◮ every edge of M is parallel to e i − e j for some i , j ∈ [ n ] . (4,1,2,3) (3,1,2,4) (4,2,1,3) (3,2,1,4) 202 022 (4,1,3,2) (2,1,3,4) (4,3,1,2) (2,3,1,4) (3,1,4,2) (4,2,3,1) (4,3,2,1) (2,1,4,3) 031 (1,2,3,4) (3,4,1,2) (1,3,2,4) (2,4,1,3) (3,2,4,1) (1,2,4,3) (3,4,2,1) (1,4,2,3) 400 (2,3,4,1) 130 (1,3,4,2) (2,4,3,1) this image (1,4,3,2) David Eppstein Polymatroids are Postnikov’s (lattice) generalised permutahedra. Derksen, Fink Valuative invariants for polymatroids 4 / 21
Ranks Let e X = � i ∈ X e i . The rank function of M is its support function on 0-1 vectors: y ∈ M � y , e X � . rk M ( X ) = max Fact 0-1 vectors are the only facet normals of 001 (poly)matroids. 101 011 M = { y ∈ R n : � y , e X � ≤ rk M ( X ) ∀ X ⊆ [ n ] , 100 010 � y , e [ n ] � = rk M ([ n ]) } . 110 r := rk M ([ n ]) is called the rank of M . Derksen, Fink Valuative invariants for polymatroids 5 / 21
A motivating example: the Tutte polynomial Matroids have two operations yielding minors : ◮ deletion, M \ i = { M ∩ x i = 0 } ◮ contraction, M / i = { M ∩ x i = 1 } Many invariants (e.g. # bases, independent sets, spanning sets; chromatic and flow polys of graphs; many hyperplane arr. properties; . . . ) can be evaluated by a deletion-contraction recurrence, f ( M ) = f ( M \ i ) + f ( M / i ) . (1) Theorem (Tutte ’54, Crapo ’69) The Tutte polynomial � ( x − 1 ) r − rk ( X ) ( y − 1 ) | X |− rk ( X ) T ( M ; x , y ) = X ⊆ [ n ] is universal for (1) . Z { matroids } / � M = M \ i + M / i � = Z [ x , y ] . Derksen, Fink Valuative invariants for polymatroids 6 / 21
A motivating example: the Tutte polynomial Matroids have two operations yielding minors : ◮ deletion, M \ i = { M ∩ x i = 0 } ◮ contraction, M / i = { M ∩ x i = 1 } Many invariants (e.g. # bases, independent sets, spanning sets; chromatic and flow polys of graphs; many hyperplane arr. properties; . . . ) can be evaluated by a deletion-contraction recurrence, f ( M ) = f ( M \ i ) + f ( M / i ) . (1) Theorem (Tutte ’54, Crapo ’69) The Tutte polynomial � ( x − 1 ) r − rk ( X ) ( y − 1 ) | X |− rk ( X ) T ( M ; x , y ) = X ⊆ [ n ] is universal for (1) . Z { matroids } / � M = M \ i + M / i � = Z [ x , y ] . Derksen, Fink Valuative invariants for polymatroids 6 / 21
Decompositions and valuations A decomposition Π = ( P ; P 1 , . . . , P k ) is a polyhedral complex. We write P I = � i ∈ I P i . Example P { 1 , 2 } P { 1 } P { 2 } P ∅ = P = + ! A valuation on a set M of polyhedra is an f : M → G such that any decomposition Π with all P I ∈ M satisfies � ( − 1 ) | I | f ( P I ) = 0 . I ⊆ [ k ] Derksen, Fink Valuative invariants for polymatroids 7 / 21
Examples of valuations I ⊆ [ k ] ( − 1 ) | I | f ( P I ) = 0 � General examples ◮ The map [ · ] sending P to its indicator function [ P ] : R n → Z . Many interesting evaluations, and sums and integrals of these: volume, Ehrhart polynomial, . . . ◮ Euler characteristic χ , χ ( P ) = 1 for P � = ∅ if P compact. From now on M = {matroids} or {polymatroids}. Matroidal examples ◮ the Tutte polynomial T ◮ Speyer’s invariant h , arising from K -theory of Grassmannians ◮ Billera-Jia-Reiner’s G , from combinatorial Hopf land Derksen, Fink Valuative invariants for polymatroids 8 / 21
(Poly)matroid valuations Matroid polytope decompositions come up in ◮ labelling fine Schubert cells in the Grassmannian (Lafforgue); connections to realisability. ◮ describing linear spaces via tropical geometry (Speyer, Ardila-Klivans). ◮ compactifying moduli of hyperplane arrangements (Hacking-Keel-Tevelev). Problem Describe all (poly)matroid valuations. Find a universal one. Prove [Derksen ’08]’s conjectured universal invariant G . Notation Let P M be the Z -module generated by indicators [ M ] for M ∈ M . Grading: P M ( r , n ) is gen. by rank r matroids on [ n ] . M := � Hom ( P M ( r , n ) , G ) is the group of valuations. Prop’n: P ∨ Derksen, Fink Valuative invariants for polymatroids 9 / 21
(Poly)matroid valuations Matroid polytope decompositions come up in ◮ labelling fine Schubert cells in the Grassmannian (Lafforgue); connections to realisability. ◮ describing linear spaces via tropical geometry (Speyer, Ardila-Klivans). ◮ compactifying moduli of hyperplane arrangements (Hacking-Keel-Tevelev). Problem Describe all (poly)matroid valuations. Find a universal one. Prove [Derksen ’08]’s conjectured universal invariant G . Notation Let P M be the Z -module generated by indicators [ M ] for M ∈ M . Grading: P M ( r , n ) is gen. by rank r matroids on [ n ] . M := � Hom ( P M ( r , n ) , G ) is the group of valuations. Prop’n: P ∨ Derksen, Fink Valuative invariants for polymatroids 9 / 21
(Poly)matroid valuations Matroid polytope decompositions come up in ◮ labelling fine Schubert cells in the Grassmannian (Lafforgue); connections to realisability. ◮ describing linear spaces via tropical geometry (Speyer, Ardila-Klivans). ◮ compactifying moduli of hyperplane arrangements (Hacking-Keel-Tevelev). Problem Describe all (poly)matroid valuations. Find a universal one. Prove [Derksen ’08]’s conjectured universal invariant G . Notation Let P M be the Z -module generated by indicators [ M ] for M ∈ M . Grading: P M ( r , n ) is gen. by rank r matroids on [ n ] . M := � Hom ( P M ( r , n ) , G ) is the group of valuations. Prop’n: P ∨ Derksen, Fink Valuative invariants for polymatroids 9 / 21
(Poly)matroid valuations Matroid polytope decompositions come up in ◮ labelling fine Schubert cells in the Grassmannian (Lafforgue); connections to realisability. ◮ describing linear spaces via tropical geometry (Speyer, Ardila-Klivans). ◮ compactifying moduli of hyperplane arrangements (Hacking-Keel-Tevelev). Problem Describe all (poly)matroid valuations. Find a universal one. Prove [Derksen ’08]’s conjectured universal invariant G . Notation Let P M be the Z -module generated by indicators [ M ] for M ∈ M . Grading: P M ( r , n ) is gen. by rank r matroids on [ n ] . M := � Hom ( P M ( r , n ) , G ) is the group of valuations. Prop’n: P ∨ Derksen, Fink Valuative invariants for polymatroids 9 / 21
Bases Define the polyhedra (full-dimensional cones) R ( X , r ) = { y ∈ R n : � y , e X i � ≤ r i for i = 1 , . . . , ℓ − 1 , � y , e [ n ] � = r } and the (almost dual) valuations � 1 if rk M ( X i ) = r i for i = 1 , . . . , ℓ , s X , r ( M ) = 0 otherwise and r = ( r 1 , . . . , r ℓ = r ) ∈ Z ℓ . for ∅ � X 1 � · · · � X ℓ − 1 � X ℓ = [ n ] Let ∆ M ( r , n ) be the largest polyhedron in M ( r , n ) . Theorem (Derksen-F) ◮ The distinct nonzero [ R ( X , r ) ∩ ∆ M ( r , n )] form a basis for (poly)matroids mod subdivisions P M ( r , n ) . ◮ The distinct nonzero s X , r | M form a basis for valuations P ∨ M ( r , n ) . Derksen, Fink Valuative invariants for polymatroids 10 / 21
Why these cones? Theorem (Brianchon, Gram) If the polyhedron P does not contain a line, then � ( − 1 ) dim F [ cone F ( P )] [ P ] = F where F runs over all the bounded faces of P. Proposition (Derksen-F) � ( − 1 ) n − ℓ ( X ) [ R ( X , rk M ( X ))] , [ M ] = X where X ranges over all chains. Derksen, Fink Valuative invariants for polymatroids 11 / 21
Example of the Brianchon-Gram Theorem Example This polytope has the combinatorial type of the permutahedron. = + + + + + ! ! ! ! ! ! + Derksen, Fink Valuative invariants for polymatroids 12 / 21
Why these cones? Theorem (Brianchon, Gram) If the polyhedron P does not contain a line, then � ( − 1 ) dim F [ cone F ( P )] [ P ] = F where F runs over all the bounded faces of P. Proposition (Derksen-F) � ( − 1 ) n − ℓ ( X ) [ R ( X , rk M ( X ))] , [ M ] = X where X ranges over all chains. Derksen, Fink Valuative invariants for polymatroids 13 / 21
Example of the proposition Example We decompose this polymatroid polytope in R s by inflating it to the previous one: = + + + + + ! ! ! ! ! ! + Derksen, Fink Valuative invariants for polymatroids 14 / 21
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