Divisors on matroids and their volumes Christopher Eur Department - PowerPoint PPT Presentation

Divisors on matroids and their volumes Christopher Eur Department of Mathematics University of California, Berkeley FPSAC 2019

0. Goal today Today: the volume polynomial of the Chow ring of a matroid new invariants of matroids, “Hopf-y structures,” ◮ (Comb) volumes of generalized permutohedra ◮ (Alg Geom) degrees of certain varieties, E.g. M 0 , n and L n ◮ (Trop Geom) first step in tropical Newton-Okounkov bodies ◮ (Rep) “Taking the Chow ring of a matroid respects its Type A structure.”

1. Graphs G : a finite simple graph chromatic polynomial of G : χ G ( t ) := # of ways to color vertices of G with at most t many colors with no adjacent vertices same color Example χ G ( t ) = t ( t − 1)( t − 1)( t − 2) = t ( t 3 − 4 t 2 + 5 t − 2) Conjecture [Rota ’71] The unsigned coefficients of χ G are unimodal ( ր ց ) .

2. Matroids A matroid M = ( E , I ): ◮ a finite set E , the ground set ◮ a collection I of subsets of E , the indepedent subsets Examples E = { v 0 , . . . , v n } vectors, ◮ realizable matroids: independent = linearly independent E = edges of a graph G , ◮ graphical matroids: independent = no cycles characteristic polynomial of M : χ M ( t ) Conjecture [Rota ’71, Heron ’72, Welsh ’74] The unsigned coefficients of χ M ( t ) are unimodal.

3. History Resolution of the Rota-Heron-Welsh conjecture: KEY: coefficients of χ M = intersection numbers of (nef) divisors ◮ graphs [Huh ’12], realizable matroids [Huh-Katz ’12] → volumes of convex bodies (Newton-Okounkov bodies) ◮ general matroids [Adiprasito-Huh-Katz ’18] → no explicit use of convex bodies & their volumes → Hodge theory on the Chow ring of a matroid

4. More matroids M of rank r : nonzero vectors E = { v 0 , . . . , v n } spanning V ≃ C r . ◮ For S ⊆ E , set rk M ( S ) := dim C span( S ). ◮ F ⊆ E is a flat of M if rk( F ∪ { x } ) > rk( F ) ∀ x / ∈ F . ◮ hyperplane arrangement A M = { H i } in P V ∗ , where H i := { f ∈ P V ∗ | v i ( f ) = 0 } . Example A M drawn on P 2 M as 4 vectors in 3-space

5. Chow rings of matroids ◮ M a matroid of rank r = d + 1 with ground set E , ◮ L M := the set of nonempty proper flats of M . Definition [Feichtner-Yuzvinsky ’04, de Concini-Procesi ’95] Chow ring A • ( M ): a graded R -algebra A • ( M ) = � d i =0 A i ( M ) R [ x F : F ∈ L M ] A • ( M ) := � � � x F x F ′ | F , F ′ incomparable � + � x F − x G | i , j ∈ E � F ∋ i G ∋ j Elements of A 1 ( M ) called divisors on M . A • ( M ) = cohomology ring of the wonderful compactification X M : ◮ built via blow-ups from P V ∗ ; compactifies P V ∗ \ � A M ◮ E.g. M 0 , n , L n (moduli of stable rational curves with marked points)

6. Poincar´ e duality & the volume polynomial Theorem [6.19, Adiprasito-Huh-Katz ’18] The ring A • ( M ) satisfies Poincar´ e duality: 1. the degree map deg M : A d ( M ) ∼ → R (where deg M ( x F 1 x F 2 · · · x F d ) = 1 for every maximal chain F 1 � · · · � F d of nonempty proper flats) 2. non-degenerate pairings A i ( M ) × A d − i ( M ) → A d ( M ) ≃ R . Macaulay inverse system: Poincar´ e duality algebras ↔ volume polynomials Definition The volume polynomial VP M ( t ) ∈ R [ t F : F ∈ L M ] of M � � � d VP M ( t ) = deg M x F t F F ∈ L M (where deg M : A d ( M ) → R is extended to A d [ t F ’s] → R [ t F ’s]).

7. Formula for VP M ◮ M be a matroid of rank r = d + 1 on a ground set E , ◮ ∅ = F 0 � F 1 � · · · � F k � F k +1 = E a chain of flats of M with ranks r i := rk F i , ◮ d 1 , . . . , d k ∈ Z > 0 such that � d i := � i i d i = d , and � j =1 d j Theorem [E ’18] The coefficient of t d 1 F 1 · · · t d k F k in VP M ( t ) is � � � d i − 1 � k � d � ( − 1) d − k d i − r i ( M | F i +1 / F i ) , µ � d 1 , . . . , d k d i − r i i =1 { µ i ( M ′ ) } = unsigned coefficients of the reduced characteristic polynomial χ M ′ ( t ) = µ 0 ( M ′ ) t rk M ′ − 1 − µ 1 ( M ′ ) t rk M ′ − 2 + · · · +( − 1) rk M ′ − 1 µ rk M ′ − 1 ( M ′ ) of a matroid M ′ .

8. First applications 1. M = U n , n : VP M → volumes of generalized permutohedra ◮ Relation to [Postnikov ’09]? 2. M = M ( K n − 1 ): VP M → embedding degrees of M 0 , n ◮ not a Mori dream space [Castravet-Tevelev ’15] 3. The operation M �→ VP M ∈ R [ t S | S ⊆ E ] is valuative. ◮ “The construction of the Chow ring of a matroid respects its type A structure.” ◮ Hodge theory of matroids of arbitrary Lie type?

9. Shifted rank volume I Nef divisors “=” submodular functions Definition The shifted rank divisor of M : � D M := (rk M F ) x F F ∈ L M The shifted rank volume of M : shRVol( M ) := deg M ( D d M ) = VP M ( t F = rk M ( F )) . Remark the Tutte polynomial Unrelated to: volume of the matroid polytope

10. Shifted rank volume II ◮ uniform matroid U r , n : n general vectors in r -space. Theorem [E. ’18] For M a realizable matroid of rank r = d + 1 on n elements, shRVol( M ) ≤ shRVol( U r , n ) = n d , with equality iff M = U r , n . π : X M → P d the wonderful compactification Proof: D M = n � H − E , where � H = π ∗ O P d (1) the pullback of the hyperplane class, and E an effective divisor such that E = 0 iff M = U r , n H 0 ( m ( n � H − E )) ⊂ H 0 ( m ( n � H )) for any m ∈ Z ≥ 0 → counting sections of divisors in tropical setting?

Thanks Thanks: Federico Ardila, Justin Chen, David Eisenbud, Alex Fink, June Huh, Vic Reiner, Bernd Sturmfels, Mengyuan Zhang. Thank you for listening!