Hodge Theory of Matroids

Editor’s Note: Matt Baker is speaking on this topic in the Current Events Bulletin Lecture at the January 2017 Joint Mathematics Meetings.

Karim Adiprasito, June Huh, and Eric Katz

Communicated by Benjamin Braun

Introduction which satisfies analogues of Poincaré duality, the hard Lef- Logarithmic concavity is a property of a sequence of schetz theorem, and the Hodge-Riemann relations for the real numbers, occurring throughout algebraic geometry, cohomology of smooth projective varieties. Log-concavity convex geometry, and combinatorics. A sequence of will be deduced from the Hodge-Riemann relations for positive numbers 푎0, … , 푎푑 is log-concave if 푀. We believe that behind any log-concave sequence that appears in nature there is such a “Hodge structure” 푎2 ≥ 푎 푎 for all 푖. 푖 푖−1 푖+1 responsible for the log-concavity. This means that the logarithms, log(푎푖), form a concave sequence. The condition implies unimodality of the se- Coloring Graphs quence (푎 ), a property easier to visualize: the sequence 푖 Generalizing earlier work of George Birkhoff, in 1932 is unimodal if there is an index 푖 such that Hassler Whitney introduced the chromatic polynomial of 푎0 ≤ ⋯ ≤ 푎푖−1 ≤ 푎푖 ≥ 푎푖+1 ≥ ⋯ ≥ 푎푑. a connected graph 퐺 as the function on ℕ defined by

We will discuss our work on establishing log-concavity 휒퐺(푞) = |{proper colorings of 퐺 using 푞 colors}|. of various combinatorial sequences, such as the coeffi- cients of the chromatic polynomial of graphs and the In other words, 휒퐺(푞) is the number of ways to color face numbers of matroid complexes. Our method is moti- the vertices of 퐺 using 푞 colors so that the endpoints of vated by complex algebraic geometry, in particular Hodge every edge have different colors. Whitney noticed that the theory. From a given combinatorial object 푀 (a matroid), chromatic polynomial is indeed a polynomial. In fact, we we construct a graded commutative algebra over the real can write 푑 푑−1 푑 numbers 휒퐺(푞)/푞 = 푎0(퐺)푞 − 푎1(퐺)푞 + ⋯ + (−1) 푎푑(퐺) 푑 ∗ 푞 퐴 (푀) = ⨁ 퐴 (푀), for some positive integers 푎0(퐺), … , 푎푑(퐺), where 푑 is 푞=0 one less than the number of vertices of 퐺.

Karim Adiprasito is assistant professor of mathematics at Hebrew Example 1. The square graph University of Jerusalem. His e-mail address is adiprasito@math. • • huji.ac.il. June Huh is research fellow at the Clay Mathematics Institute and • • Veblen Fellow at Princeton University and the Institute for Ad- 4 3 2 vanced Study. His e-mail is [email protected]. has the chromatic polynomial 1푞 − 4푞 + 6푞 − 3푞. Eric Katz is assistant professor of mathematics at the Ohio State The chromatic polynomial was originally devised as University. His e-mail address is [email protected]. a tool for attacking the Four Color Problem, but soon For permission to reprint this article, please contact: it attracted attention in its own right. Ronald Read [email protected]. conjectured in 1968 that the coefficients of the chromatic DOI: http://dx.doi.org/10.1090/noti1463 polynomial form a unimodal sequence for any graph.

26 Notices of the AMS Volume 64, Number 1 A few years later, Stuart Hoggar conjectured that the The first term counts the number of independent subsets coefficients in fact form a log-concave sequence: of size 푖, the second term counts the independent subsets 2 of size 푖 not containing 푣, and the third term counts 푎푖(퐺) ≥ 푎푖−1(퐺)푎푖+1(퐺) for any 푖 and 퐺. the independent subsets of size 푖 containing 푣. As in The graph theorist William Tutte quipped, “In compensa- the case of graphs, we notice the apparent interference tion for its failure to settle the Four Colour Conjecture, [the between the log-concavity conjecture and the additive chromatic polynomial] offers us the Unimodal Conjecture nature of 푓 (퐴). The sum of two log-concave sequences is, for our further bafflement.” 푖 in general, not log-concave. The conjecture suggests a new The chromatic polynomial can be computed using the deletion-contraction relation: if 퐺\푒 is the deletion of an description for the numbers 푓푖(퐴) and a corresponding edge 푒 from 퐺 and 퐺/푒 is the contraction of the same structure of 퐴. edge, then Matroids 휒 (푞) = 휒 (푞) − 휒 (푞). 퐺 퐺\푒 퐺/푒 In the 1930s Hassler Whitney observed that several The first term counts the proper colorings of 퐺, the notions in graph theory and linear algebra fit together in second term counts the otherwise-proper colorings of 퐺 a common framework, that of matroids. This observation where the endpoints of 푒 are permitted to have the same started a new subject with applications to a wide range color, and the third term counts the otherwise-proper of topics such as characteristic classes, optimization, and colorings of 퐺 where the endpoints of 푒 are mandated to moduli spaces, to name a few. have the same color. Note that, in general, the sum of two Let 퐸 be a finite set. A matroid 푀 on 퐸 is a collection log-concave sequences is not a log-concave sequence. of subsets of 퐸, called flats of 푀, satisfying the following Example 2. To compute the chromatic polynomial of the axioms: square graph above, we write (1) If 퐹1 and 퐹2 are flats of 푀, then their intersection is • • • • • @ a flat of 푀. @@ = - (2) If 퐹 is a flat of 푀, then any element of 퐸\퐹 is • • • • • • contained in exactly one flat of 푀 covering 퐹. and use (3) The ground set 퐸 is a flat of 푀. 3 휒퐺\푒(푞) = 푞(푞 − 1) , 휒퐺/푒(푞) = 푞(푞 − 1)(푞 − 2). Here, a flat of 푀 is said to cover 퐹 if it is minimal among The Hodge-Riemann relations for the algebra 퐴∗(푀), the flats of 푀 properly containing 퐹. For our purposes, where 푀 is the matroid attached to 퐺 as in the section we may assume that 푀 is loopless: “Matroids” below, imply that the coefficients of the (1) The empty subset of 퐸 is a flat of 푀. chromatic polynomial of 퐺 form a log-concave sequence. Every maximal chain of flats of 푀 has the same length, This is in contrast to a result of Alan Sokal that the set of and this common length roots of the chromatic polynomials of all graphs is dense is called the rank of in the complex plane. 푀. We write 푀\푒 for Matroids encode Counting Independent Subsets the matroid obtained by deleting 푒 from the a combinatorial Linear independence is a fundamental notion in algebra flats of 푀 and 푀/푒 for and geometry: a collection of vectors is linearly indepen- structure dent if no nontrivial linear combination sums to zero. the matroid obtained by How many linearly independent collections of 푖 vectors deleting 푒 from the flats common to are there in a given configuration of vectors? Write 퐴 for of 푀 containing 푒. When 푀 is a matroid on 퐸 , a finite subset of a vector space and 푓푖(퐴) for the number 1 1 graphs and of independent subsets of 퐴 of size 푖. 푀2 is a matroid on 퐸2, and 퐸1 ∩ 퐸2 is empty, vector Example 3. If 퐴 is the set of all nonzero vectors in the the direct sum 푀1 ⊕ 푀2 three-dimensional vector space over the field with two is defined to be the ma- configurations elements, then troid on 퐸1 ∪ 퐸2 whose 푓0 = 1, 푓1 = 7, 푓2 = 21, 푓3 = 28. flats are all sets of the Examples suggest a pattern leading to a conjecture of form 퐹1 ∪ 퐹2, where 퐹1 is a flat of 푀1 and 퐹2 is a flat of 푀2. Dominic Welsh: Matroids encode a combinatorial structure common to graphs and vector configurations. If 퐸 is the set of edges 푓 (퐴)2 ≥ 푓 (퐴)푓 (퐴) for any 푖 and 퐴. 푖 푖−1 푖+1 of a finite graph 퐺, call a subset 퐹 of 퐸 a flat when there is For any small specific case, the conjecture can be verified no edge in 퐸\퐹 whose endpoints are connected by a path by computing the 푓푖(퐴)’s by the deletion-contraction rela- in 퐹. This defines a graphic matroid on 퐸. If 퐸 is a finite tion: if 퐴\푣 is the deletion of a nonzero vector 푣 from 퐴 subset of a vector space, call a subset 퐹 of 퐸 a flat when and 퐴/푣 is the projection of 퐴 in the direction of 푣, then there is no vector in 퐸\퐹 that is contained in the linear 푓푖(퐴) = 푓푖(퐴\푣) + 푓푖−1(퐴/푣). span of 퐹. This defines a linear matroid on 퐸.

January 2017 Notices of the AMS 27 Example 4. Write 퐸 = {0, 1, 2, 3} for the set of edges of Hodge-Riemann Relations for Matroids the square graph 퐺 in Example 1. The graphic matroid 푀 Let 푋 be a mathematical object of “dimension” 푑. Often it on 퐸 attached to 퐺 has flats is possible to construct from 푋 in a natural way a graded vector space over the real numbers ∅, {0}, {1}, {2}, {3}, {0, 1}, {0, 2}, {0, 3}, 푑 {1, 2}, {1, 3}, {2, 3}, {0, 1, 2, 3}. 퐴∗(푋) = ⨁ 퐴푞(푋), Example 5. Write 퐸 = {0, 1, 2, 3, 4, 5, 6} for the configura- 푞=0 tion of vectors 퐴 in Example 3. The linear matroid 푀 on equipped with a graded bilinear pairing 퐸 attached to 퐴 has flats 푃 ∶ 퐴∗(푋) × 퐴푑−∗(푋) ⟶ ℝ, ∅, {0}, {1}, {2}, {3}, {4}, {5}, {6}, and a graded linear map {1, 2, 3}, {3, 4, 5}, {1, 5, 6}, {0, 1, 4}, {0, 2, 5}, {0, 3, 6}, 퐿 ∶ 퐴∗(푋) ⟶ 퐴∗+1(푋), 푥 ⟼ L푥 {2, 4, 6}, {0, 1, 2, 3, 4, 5, 6}. (“푃” is for Poincaré, and “퐿” is for Lefschetz). For example, A linear matroid that arises from a subset of a vector 퐴∗(푋) may be the cohomology of real (푝, 푝)-forms on a space over a field 푘 is said to be realizable over 푘. compact Kähler manifold 푋 or the ring of algebraic cycles Not surprisingly, this notion is sensitive to the field 푘. modulo homological equivalence on a smooth projective A matroid may arise from a vector configuration over variety 푋 or the combinatorial intersection cohomology one field, while no such vector configuration exists over of a convex polytope 푋 [Kar04] or the Soergel bimodule another field. Many matroids are not realizable over any field. of an element of a Coxeter group 푋 [EW14] or the Chow ring of a matroid 푋 defined below. We expect that, for 푑 every nonnegative integer 푞 ≤ 2 : (1) the bilinear pairing 푃 ∶ 퐴푞(푋) × 퐴푑−푞(푋) ⟶ ℝ is nondegenerate (Poincaré duality for 푋), Among the rank 3 loopless matroids pictured above, (2) the composition of linear maps where rank 1 flats are represented by points and rank 2 flats containing more than two points are represented 퐿푑−2푞 ∶ 퐴푞(푋) ⟶ 퐴푑−푞(푋) by lines, the first is realizable over 푘 if and only if the is bijective (the hard Lefschetz theorem for 푋), and characteristic of 푘 is 2, the second is realizable over 푘 if (3) the bilinear form on 퐴푞(푋) defined by and only if the characteristic of 푘 is not 2, and the third 푞 푑−2푞 is not realizable over any field. (푥1, 푥2) ⟼ (−1) 푃(푥1, 퐿 푥2) The characteristic polynomial 휒 (푞) of a matroid 푀 is 푀 is symmetric and is positive definite on the kernel a generalization of the chromatic polynomial 휒 (푞) of a 퐺 of graph 퐺. It can be recursively defined using the following 푑−2푞+1 푞 푑−푞+1 rules: 퐿 ∶ 퐴 (푋) ⟶ 퐴 (푋)

(1) If 푀 is the direct sum 푀1 ⊕ 푀2, then (the Hodge-Riemann relations for 푋). All three properties are known to hold for the objects listed 휒푀(푞) = 휒푀1 (푞) 휒푀2 (푞). above except one, which is the subject of Grothendieck’s (2) If 푀 is not a direct sum, then, for any 푒, standard conjectures on algebraic cycles. ∗ 휒푀(푞) = 휒푀\푒(푞) − 휒푀/푒(푞). For a loopless matroid 푀 on 퐸, the vector space 퐴 (푀) (3) If 푀 is the rank 1 matroid on {푒}, then has the structure of a graded algebra that can be described explicitly. 휒푀(푞) = 푞 − 1. (4) If 푀 is the rank 0 matroid on {푒}, then Definition 7. We introduce variables 푥퐹, one for each nonempty proper flat 퐹 of 푀, and set 휒푀(푞) = 0. 푆∗(푀) = ℝ[푥 ] . It is a consequence of the Möbius inversion for partially 퐹 퐹≠∅,퐹≠퐸 ordered sets that the characteristic polynomial of 푀 is The Chow ring 퐴∗(푀) of 푀 is the quotient of 푆∗(푀) by well defined. the ideal generated by the linear forms We may now state our result in [AHK], which confirms a conjecture of Gian-Carlo Rota and Dominic Welsh. ∑ 푥퐹 − ∑ 푥퐹, 푖1∈퐹 푖2∈퐹 Theorem 6 ([AHK, Theorem 9.9]). The coefficients of the one for each pair of distinct elements 푖 and 푖 of 퐸, and characteristic polynomial form a log-concave sequence 1 2 the quadratic monomials for any matroid 푀.

푥퐹1 푥퐹2 , This implies the log-concavity of the sequence 푎푖(퐺) [Huh12] and the log-concavity of the sequence 푓푖(퐴) one for each pair of incomparable nonempty proper flats [Len12]. 퐹1 and 퐹2 of 푀.

28 Notices of the AMS Volume 64, Number 1 The Chow ring of 푀 was introduced by Eva Maria Sketch of Proof of Log-Concavity Feichtner and Sergey Yuzvinsky. When 푀 is realizable We now explain why the Hodge-Riemann relations for over a field 푘, it is the Chow ring of the “wonderful” 푀 imply log-concavity for 휒푀(푞). The Hodge-Riemann compactification of the complement of a hyperplane relations for 푀, in fact, imply that the sequence (푚푖) in arrangement defined over 푘 as described by Corrado de the expression Concini and Claudio Procesi. 푑 푑−1 푑 휒푀(푞)/(푞 − 1) = 푚0푞 − 푚1푞 + ⋯ + (−1) 푚푑 Let 푑 be the integer one less than the rank of 푀. is log-concave, which is stronger. Theorem 8 ([AHK, Proposition 5.10]). There is a linear We define two elements of 퐴1(푀): for any 푗 ∈ 퐸, set bijection 푑 훼푀 = ∑ 푥퐹, 훽푀 = ∑ 푥퐹. deg∶ 퐴 (푀) ⟶ ℝ 푗∈퐹 푗∉퐹 uniquely determined by the property that The two elements do not depend on the choice of 푗, and they are limits of elements of the form 퐿(푐) for strictly deg(푥퐹 푥퐹 ⋯ 푥퐹 ) = 1 1 2 푑 submodular 푐. A short combinatorial argument shows for every maximal chain of nonempty proper flats that 푚푖 is a mixed degree of 훼푀 and 훽푀: 퐹 ⊊ 퐹 ⊊ ⋯ ⊊ 퐹 . 푖 푑−푖 1 2 푑 푚푖 = deg(훼푀 훽푀 ). In addition, the bilinear pairing Thus, it is enough to prove for every 푖 that 푞 푑−푞 푑−푖+1 푖−1 푑−푖−1 푖+1 푑−푖 푖 2 푃 ∶ 퐴 (푀) × 퐴 (푀) → ℝ, (푥, 푦) ↦ deg(푥푦) deg(훼푀 훽푀 )deg(훼푀 훽푀 ) ≤ deg(훼푀 훽푀) . is nondegenerate for every nonnegative 푞 ≤ 푑. This is an analogue of the Teisser-Khovanskii inequal- ity for intersection numbers in algebraic geometry and What should be the linear operator 퐿 for 푀? We collect the Alexandrov-Fenchel inequality for mixed volumes in all valid choices of 퐿 in a nonempty open convex cone. convex geometry. The main case is when 푖 = 푑 − 1. The cone is an analogue of the Kähler cone in complex By a continuity argument, we may replace 훽푀 by geometry. 퐿 = 퐿(푐) sufficiently close to 훽푀. The desired inequality in the main case then becomes Definition 9. A real-valued function 푐 on 2퐸 is said to be 2 푑−2 푑 푑−1 2 strictly submodular if deg(훼푀퐿 )deg(퐿 ) ≤ deg(훼푀퐿 ) . This follows from the fact that the signature of the bilinear 푐∅ = 0, 푐퐸 = 0, form and, for any two incomparable subsets 퐼1, 퐼2 ⊆ 퐸, 1 1 푑−2 퐴 (푀) × 퐴 (푀) → ℝ, (푥1, 푥2) ↦ deg(푥1퐿 푥2)

푐퐼1 + 푐퐼2 > 푐퐼1 ∩퐼2 + 푐퐼1 ∪퐼2 . restricted to the span of 훼푀 and 퐿 is semi-indefinite, A strictly submodular function 푐 defines an element which, in turn, is a consequence of the Hodge-Riemann relations for 푀 in the cases 푞 = 0, 1. 퐿(푐) = 푐 푥 ∈ 퐴1(푀) ∑ 퐹 퐹 This application uses only a small piece of the Hodge- 퐹 Riemann relations for 푀. The general Hodge-Riemann that acts as a linear operator by multiplication relations for 푀 may be used to extract other interesting 퐴∗(푀) ⟶ 퐴∗+1(푀), 푥 ⟼ 퐿(푐)푥. combinatorial information about 푀. 1 The set of all such elements is a convex cone in 퐴 (푀). References The main result of [AHK] states that the triple [AHK] Karim Adiprasito, June Huh, and Eric Katz, Hodge (퐴∗(푀), 푃, 퐿(푐)) satisfies the hard Lefschetz theorem theory for combinatorial geometries, arXiv:1511.02888. and the Hodge-Riemann relations for every strictly sub- [EW14] Elias-Williamson, Ben Elias, and Geordie Williamson, The Hodge theory of Soergel bi- modular function 푐: modules, Ann. Math. (2) 180 (2014), 1089–1136. Theorem 10. Let 푞 be a nonnegative integer less than 푑 . MR3245013 2 [Huh12] June Huh, Milnor numbers of projective hypersurfaces (1) The multiplication by 퐿(푐) defines an isomorphism and the chromatic polynomial of graphs, J. Amer. Math. 퐴푞(푀) ⟶ 퐴푑−푞(푀), 푥 ⟼ 퐿(푐)푑−2푞 푥. Soc. 25 (2012), 907–927. MR2904577 [Kar04] Kalle Karu, Hard Lefschetz theorem for nonra- (2) The symmetric bilinear form on 퐴푞(푀) defined by tional polytopes, Invent. Math. 157 (2004), 419–447. 푞 푑−2푞 MR2076929 (푥1, 푥2) ⟼ (−1) 푃(푥1, 퐿(푐) 푥2) [Len12] Matthias Lenz, The f-vector of a representable- is positive definite on the kernel of 퐿(푐)푑−2푞+1. matroid complex is log-concave, Adv. in Appl. Math. 51 (2013), no. 5, 543–545. MR3118543 The known proofs of the hard Lefschetz theorem and the Hodge-Riemann relations for the different types of objects listed above have certain structural similarities, but there is no known way of deducing one from the others.

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30 Notices of the AMS Volume 64, Number 1