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Hodge Theory of MatroidsEditorâs Note: Matt Baker is speaking on this topic in the Current Events Bulletin Lecture at the January 2017 JointMathematics Meetings.
Karim Adiprasito, June Huh, and Eric KatzCommunicated by Benjamin Braun
IntroductionLogarithmic concavity is a property of a sequence ofreal numbers, occurring throughout algebraic geometry,convex geometry, and combinatorics. A sequence ofpositive numbers ð0,⊠,ðð is log-concave if
ð2ð ⥠ððâ1ðð+1 for all ð.
This means that the logarithms, log(ðð), form a concavesequence. The condition implies unimodality of the se-quence (ðð), a property easier to visualize: the sequenceis unimodal if there is an index ð such that
ð0 †⯠†ððâ1 †ðð ⥠ðð+1 ⥠⯠⥠ðð.We will discuss our work on establishing log-concavity
of various combinatorial sequences, such as the coeffi-cients of the chromatic polynomial of graphs and theface numbers of matroid complexes. Our method is moti-vated by complex algebraic geometry, in particular Hodgetheory. From a given combinatorial object ð (a matroid),we construct a graded commutative algebra over the realnumbers
ðŽâ(ð) =ð
âšð=0
ðŽð(ð),
Karim Adiprasito is assistant professor of mathematics at HebrewUniversity of Jerusalem. His e-mail address is [email protected].
June Huh is research fellow at the Clay Mathematics Institute andVeblen Fellow at Princeton University and the Institute for Ad-vanced Study. His e-mail is [email protected].
Eric Katz is assistant professor of mathematics at the Ohio StateUniversity. His e-mail address is [email protected].
For permission to reprint this article, please contact:[email protected]: http://dx.doi.org/10.1090/noti1463
which satisfies analogues of Poincaré duality, the hard Lef-schetz theorem, and the Hodge-Riemann relations for thecohomology of smooth projective varieties. Log-concavitywill be deduced from the Hodge-Riemann relations forð. We believe that behind any log-concave sequencethat appears in nature there is such a âHodge structureâresponsible for the log-concavity.
Coloring GraphsGeneralizing earlier work of George Birkhoff, in 1932Hassler Whitney introduced the chromatic polynomial ofa connected graph ðº as the function on â defined by
ððº(ð) = |{proper colorings of ðº using ð colors}|.In other words, ððº(ð) is the number of ways to colorthe vertices of ðº using ð colors so that the endpoints ofevery edge have different colors. Whitney noticed that thechromatic polynomial is indeed a polynomial. In fact, wecan write
ððº(ð)/ð = ð0(ðº)ðð âð1(ðº)ððâ1 +â¯+ (â1)ððð(ðº)for some positive integers ð0(ðº),⊠,ðð(ðº), where ð isone less than the number of vertices of ðº.
Example 1. The square graph
⢠â¢
⢠â¢has the chromatic polynomial 1ð4 â4ð3 + 6ð2 â 3ð.
The chromatic polynomial was originally devised asa tool for attacking the Four Color Problem, but soonit attracted attention in its own right. Ronald Readconjectured in 1968 that the coefficients of the chromaticpolynomial form a unimodal sequence for any graph.
26 Notices of the AMS Volume 64, Number 1
A few years later, Stuart Hoggar conjectured that thecoefficients in fact form a log-concave sequence:
ðð(ðº)2 ⥠ððâ1(ðº)ðð+1(ðº) for any ð and ðº.The graph theorist William Tutte quipped, âIn compensa-tion for its failure to settle the FourColourConjecture, [thechromatic polynomial] offers us the Unimodal Conjecturefor our further bafflement.â
The chromatic polynomial can be computed using thedeletion-contraction relation: if ðº\ð is the deletion of anedge ð from ðº and ðº/ð is the contraction of the sameedge, then
ððº(ð) = ððº\ð(ð) â ððº/ð(ð).The first term counts the proper colorings of ðº, thesecond term counts the otherwise-proper colorings of ðºwhere the endpoints of ð are permitted to have the samecolor, and the third term counts the otherwise-propercolorings of ðº where the endpoints of ð are mandated tohave the same color. Note that, in general, the sum of twolog-concave sequences is not a log-concave sequence.
Example 2. To compute the chromatic polynomial of thesquare graph above, we write
⢠â¢
⢠â¢=
⢠â¢
⢠â¢-
â¢@@
@
⢠â¢and use
ððº\ð(ð) = ð(ð â 1)3, ððº/ð(ð) = ð(ð â 1)(ð â 2).The Hodge-Riemann relations for the algebra ðŽâ(ð),
where ð is the matroid attached to ðº as in the sectionâMatroidsâ below, imply that the coefficients of thechromatic polynomial of ðº form a log-concave sequence.This is in contrast to a result of Alan Sokal that the set ofroots of the chromatic polynomials of all graphs is densein the complex plane.
Counting Independent SubsetsLinear independence is a fundamental notion in algebraand geometry: a collection of vectors is linearly indepen-dent if no nontrivial linear combination sums to zero.How many linearly independent collections of ð vectorsare there in a given configuration of vectors? Write ðŽ fora finite subset of a vector space and ðð(ðŽ) for the numberof independent subsets of ðŽ of size ð.Example 3. If ðŽ is the set of all nonzero vectors in thethree-dimensional vector space over the field with twoelements, then
ð0 = 1, ð1 = 7, ð2 = 21, ð3 = 28.Examples suggest a pattern leading to a conjecture of
Dominic Welsh:ðð(ðŽ)2 ⥠ððâ1(ðŽ)ðð+1(ðŽ) for any ð and ðŽ.
For any small specific case, the conjecture can be verifiedby computing the ðð(ðŽ)âs by the deletion-contraction rela-tion: if ðŽ\ð£ is the deletion of a nonzero vector ð£ from ðŽand ðŽ/ð£ is the projection of ðŽ in the direction of ð£, then
ðð(ðŽ) = ðð(ðŽ\ð£) + ððâ1(ðŽ/ð£).
The first term counts the number of independent subsetsof size ð, the second term counts the independent subsetsof size ð not containing ð£, and the third term countsthe independent subsets of size ð containing ð£. As inthe case of graphs, we notice the apparent interferencebetween the log-concavity conjecture and the additivenature of ðð(ðŽ). The sum of two log-concave sequences is,in general, not log-concave. The conjecture suggests a newdescription for the numbers ðð(ðŽ) and a correspondingstructure of ðŽ.
MatroidsIn the 1930s Hassler Whitney observed that severalnotions in graph theory and linear algebra fit together ina common framework, that of matroids. This observationstarted a new subject with applications to a wide rangeof topics such as characteristic classes, optimization, andmoduli spaces, to name a few.
Let ðž be a finite set. A matroid ð on ðž is a collectionof subsets of ðž, called flats of ð, satisfying the followingaxioms:(1) If ð¹1 and ð¹2 are flats of ð, then their intersection is
a flat of ð.(2) If ð¹ is a flat of ð, then any element of ðž\ð¹ is
contained in exactly one flat of ð covering ð¹.(3) The ground set ðž is a flat of ð.
Here, a flat of ð is said to cover ð¹ if it is minimal amongthe flats of ð properly containing ð¹. For our purposes,we may assume that ð is loopless:
(1) The empty subset of ðž is a flat of ð.Every maximal chain of flats of ð has the same length,
Matroids encodea combinatorial
structurecommon tographs and
vectorconfigurations
and this common lengthis called the rank ofð. We write ð\ð forthe matroid obtainedby deleting ð from theflats of ð and ð/ð forthe matroid obtained bydeleting ð from the flatsof ð containing ð. Whenð1 is a matroid on ðž1,ð2 is a matroid on ðž2,and ðž1 â© ðž2 is empty,the direct sum ð1 â ð2is defined to be the ma-troid on ðž1 ⪠ðž2 whoseflats are all sets of theform ð¹1 âªð¹2, where ð¹1 is a flat of ð1 and ð¹2 is a flat of ð2.
Matroids encode a combinatorial structure common tographs and vector configurations. If ðž is the set of edgesof a finite graph ðº, call a subset ð¹ of ðž a flat when there isno edge in ðž\ð¹ whose endpoints are connected by a pathin ð¹. This defines a graphic matroid on ðž. If ðž is a finitesubset of a vector space, call a subset ð¹ of ðž a flat whenthere is no vector in ðž\ð¹ that is contained in the linearspan of ð¹. This defines a linear matroid on ðž.
January 2017 Notices of the AMS 27
Example 4. Write ðž = {0, 1, 2, 3} for the set of edges ofthe square graph ðº in Example 1. The graphic matroid ðon ðž attached to ðº has flats
â ,{0}, {1}, {2}, {3}, {0, 1}, {0, 2}, {0, 3},{1, 2}, {1, 3}, {2, 3}, {0, 1, 2, 3}.
Example 5. Write ðž = {0, 1, 2, 3, 4, 5, 6} for the configura-tion of vectors ðŽ in Example 3. The linear matroid ð onðž attached to ðŽ has flats
â ,{0}, {1}, {2}, {3}, {4}, {5}, {6},{1, 2, 3}, {3, 4, 5}, {1, 5, 6}, {0, 1, 4}, {0, 2, 5}, {0, 3, 6},
{2, 4, 6}, {0, 1, 2, 3, 4, 5, 6}.A linear matroid that arises from a subset of a vector
space over a field ð is said to be realizable over ð.Not surprisingly, this notion is sensitive to the field ð.A matroid may arise from a vector configuration overone field, while no such vector configuration exists overanother field. Many matroids are not realizable over anyfield.
Among the rank 3 loopless matroids pictured above,where rank 1 flats are represented by points and rank2 flats containing more than two points are representedby lines, the first is realizable over ð if and only if thecharacteristic of ð is 2, the second is realizable over ð ifand only if the characteristic of ð is not 2, and the thirdis not realizable over any field.
The characteristic polynomial ðð(ð) of a matroid ð isa generalization of the chromatic polynomial ððº(ð) of agraph ðº. It can be recursively defined using the followingrules:(1) If ð is the direct sum ð1 âð2, then
ðð(ð) = ðð1(ð) ðð2(ð).(2) If ð is not a direct sum, then, for any ð,
ðð(ð) = ðð\ð(ð) â ðð/ð(ð).(3) If ð is the rank 1 matroid on {ð}, then
ðð(ð) = ðâ 1.(4) If ð is the rank 0 matroid on {ð}, then
ðð(ð) = 0.It is a consequence of the Möbius inversion for partiallyordered sets that the characteristic polynomial of ð iswell defined.
We may now state our result in [AHK], which confirmsa conjecture of Gian-Carlo Rota and Dominic Welsh.Theorem 6 ([AHK, Theorem 9.9]). The coefficients of thecharacteristic polynomial form a log-concave sequencefor any matroid ð.
This implies the log-concavity of the sequence ðð(ðº)[Huh12] and the log-concavity of the sequence ðð(ðŽ)[Len12].
Hodge-Riemann Relations for MatroidsLet ð be a mathematical object of âdimensionâ ð. Often itis possible to construct from ð in a natural way a gradedvector space over the real numbers
ðŽâ(ð) =ð
âšð=0
ðŽð(ð),
equipped with a graded bilinear pairing
ð ⶠðŽâ(ð) ÃðŽðââ(ð) ⶠâ,and a graded linear map
ð¿ ⶠðŽâ(ð) ⶠðŽâ+1(ð), ð¥ ⌠Lð¥(âðâ is for Poincaré, and âð¿â is for Lefschetz). For example,ðŽâ(ð) may be the cohomology of real (ð, ð)-forms on acompact KÀhler manifold ð or the ring of algebraic cyclesmodulo homological equivalence on a smooth projectivevariety ð or the combinatorial intersection cohomologyof a convex polytope ð [Kar04] or the Soergel bimoduleof an element of a Coxeter group ð [EW14] or the Chowring of a matroid ð defined below. We expect that, forevery nonnegative integer ð†ð
2 :(1) the bilinear pairing
ð ⶠðŽð(ð) ÃðŽðâð(ð) ⶠâis nondegenerate (Poincaré duality for ð),
(2) the composition of linear maps
ð¿ðâ2ð ⶠðŽð(ð) ⶠðŽðâð(ð)is bijective (the hard Lefschetz theorem for ð), and
(3) the bilinear form on ðŽð(ð) defined by
(ð¥1, ð¥2) ⌠(â1)ð ð(ð¥1, ð¿ðâ2ðð¥2)is symmetric and is positive definite on the kernelof
ð¿ðâ2ð+1 ⶠðŽð(ð) ⶠðŽðâð+1(ð)(the Hodge-Riemann relations for ð).
All threeproperties areknowntohold for theobjects listedabove except one, which is the subject of Grothendieckâsstandard conjectures on algebraic cycles.
For a loopless matroid ð on ðž, the vector space ðŽâ(ð)has the structure of a graded algebra that can be describedexplicitly.
Definition 7. We introduce variables ð¥ð¹, one for eachnonempty proper flat ð¹ of ð, and set
ðâ(ð) = â[ð¥ð¹]ð¹â â ,ð¹â ðž.The Chow ring ðŽâ(ð) of ð is the quotient of ðâ(ð) bythe ideal generated by the linear forms
âð1âð¹
ð¥ð¹ â âð2âð¹
ð¥ð¹,
one for each pair of distinct elements ð1 and ð2 of ðž, andthe quadratic monomials
ð¥ð¹1ð¥ð¹2 ,one for each pair of incomparable nonempty proper flatsð¹1 and ð¹2 of ð.
28 Notices of the AMS Volume 64, Number 1
The Chow ring of ð was introduced by Eva MariaFeichtner and Sergey Yuzvinsky. When ð is realizableover a field ð, it is the Chow ring of the âwonderfulâcompactification of the complement of a hyperplanearrangement defined over ð as described by Corrado deConcini and Claudio Procesi.
Let ð be the integer one less than the rank of ð.
Theorem 8 ([AHK, Proposition 5.10]). There is a linearbijection
degⶠðŽð(ð) ⶠâuniquely determined by the property that
deg(ð¥ð¹1ð¥ð¹2 â¯ð¥ð¹ð) = 1for every maximal chain of nonempty proper flats
ð¹1 â ð¹2 â ⯠â ð¹ð.In addition, the bilinear pairing
ð ⶠðŽð(ð) ÃðŽðâð(ð) â â, (ð¥, ðŠ) ⊠deg(ð¥ðŠ)is nondegenerate for every nonnegative ð †ð.
What should be the linear operator ð¿ for ð? We collectall valid choices of ð¿ in a nonempty open convex cone.The cone is an analogue of the KÀhler cone in complexgeometry.
Definition 9. A real-valued function ð on 2ðž is said to bestrictly submodular if
ðâ = 0, ððž = 0,and, for any two incomparable subsets ðŒ1, ðŒ2 â ðž,
ððŒ1 + ððŒ2 > ððŒ1â©ðŒ2 + ððŒ1âªðŒ2 .A strictly submodular function ð defines an element
ð¿(ð) = âð¹ðð¹ð¥ð¹ â ðŽ1(ð)
that acts as a linear operator by multiplication
ðŽâ(ð) ⶠðŽâ+1(ð), ð¥ ⌠ð¿(ð)ð¥.The set of all such elements is a convex cone in ðŽ1(ð).
The main result of [AHK] states that the triple(ðŽâ(ð),ð, ð¿(ð)) satisfies the hard Lefschetz theoremand the Hodge-Riemann relations for every strictly sub-modular function ð:
Theorem 10. Let ð be a nonnegative integer less than ð2 .
(1) The multiplication by ð¿(ð) defines an isomorphism
ðŽð(ð) ⶠðŽðâð(ð), ð¥ ⌠ð¿(ð)ðâ2ð ð¥.(2) The symmetric bilinear form on ðŽð(ð) defined by
(ð¥1, ð¥2) ⌠(â1)ð ð(ð¥1, ð¿(ð)ðâ2ðð¥2)is positive definite on the kernel of ð¿(ð)ðâ2ð+1.
The known proofs of the hard Lefschetz theorem andthe Hodge-Riemann relations for the different types ofobjects listed above have certain structural similarities,but there is no known way of deducing one from theothers.
Sketch of Proof of Log-ConcavityWe now explain why the Hodge-Riemann relations forð imply log-concavity for ðð(ð). The Hodge-Riemannrelations for ð, in fact, imply that the sequence (ðð) inthe expression
ðð(ð)/(ð â 1) = ð0ðð âð1ððâ1 +â¯+ (â1)ððð
is log-concave, which is stronger.We define two elements of ðŽ1(ð): for any ð â ðž, set
ðŒð = âðâð¹
ð¥ð¹, ðœð = âðâð¹
ð¥ð¹.
The two elements do not depend on the choice of ð, andthey are limits of elements of the form ð¿(ð) for strictlysubmodular ð. A short combinatorial argument showsthat ðð is a mixed degree of ðŒð and ðœð:
ðð = deg(ðŒðð ðœðâð
ð ).Thus, it is enough to prove for every ð that
deg(ðŒðâð+1ð ðœðâ1
ð )deg(ðŒðâðâ1ð ðœð+1
ð ) †deg(ðŒðâðð ðœð
ð)2.This is an analogue of the Teisser-Khovanskii inequal-ity for intersection numbers in algebraic geometry andthe Alexandrov-Fenchel inequality for mixed volumes inconvex geometry. The main case is when ð = ðâ 1.
By a continuity argument, we may replace ðœð byð¿ = ð¿(ð) sufficiently close to ðœð. The desired inequalityin the main case then becomes
deg(ðŒ2ðð¿ðâ2)deg(ð¿ð) †deg(ðŒðð¿ðâ1)2.
This follows from the fact that the signature of the bilinearform
ðŽ1(ð) ÃðŽ1(ð) â â, (ð¥1, ð¥2) ⊠deg(ð¥1ð¿ðâ2ð¥2)restricted to the span of ðŒð and ð¿ is semi-indefinite,which, in turn, is a consequence of the Hodge-Riemannrelations for ð in the cases ð = 0, 1.
This application uses only a small piece of the Hodge-Riemann relations for ð. The general Hodge-Riemannrelations for ð may be used to extract other interestingcombinatorial information about ð.
References[AHK] Karim Adiprasito, June Huh, and Eric Katz, Hodge
theory for combinatorial geometries, arXiv:1511.02888.[EW14] Elias-Williamson, Ben Elias, and Geordie
Williamson, The Hodge theory of Soergel bi-modules, Ann. Math. (2) 180 (2014), 1089â1136.MR3245013
[Huh12] June Huh, Milnor numbers of projective hypersurfacesand the chromatic polynomial of graphs, J. Amer. Math.Soc. 25 (2012), 907â927. MR2904577
[Kar04] Kalle Karu, Hard Lefschetz theorem for nonra-tional polytopes, Invent. Math. 157 (2004), 419â447.MR2076929
[Len12] Matthias Lenz, The f-vector of a representable-matroid complex is log-concave, Adv. in Appl. Math. 51(2013), no. 5, 543â545. MR3118543
January 2017 Notices of the AMS 29
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Photo CreditsPhoto of June Huh is courtesy of Nayoung Kim.Photo of Eric Katz is courtesy of Joseph Rabinoff.
THE AUTHORS
Karim Adiprasito
June Huh
Eric Katz
30 Notices of the AMS Volume 64, Number 1