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The Geometry of Matroids∗

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The Geometry of Matroids∗ The geometry of matroids∗ Federico Ardilay 1 Introduction 2 Definitions Matroid theory is a combinatorial theory of independence Matroids were defined independently in the 1930s by which has its origins in linear algebra and graph theory, Nakasawa [19] and Whitney [22]. A matroid M = (E; I) and turns out to have deep connections with many other consists of a finite set E and a collection I of subsets of fields. There are natural notions of independence in lin- E, called the independent sets, such that ear algebra, graph theory, matching theory, the theory (I1) ; 2 I of field extensions, and the theory of routings, among (I2) If J 2 I and I ⊆ J then I 2 I. others. Matroids capture the combinatorial essence that (I3) If I;J 2 I and jIj < jJj then there exists j 2 J −I those notions share. such that I [ j 2 I. Gian-Carlo Rota, who helped lay down the founda- We will assume that every singleton feg is independent. tions of the field and was one of its most energetic am- Thanks to (I2), it is enough to list the collection B of bassadors, rejected the “ineffably cacophonous" name of maximal independent sets; these are called the bases of matroids. He proposed to call them combinatorial geome- M. By (I3), they have the same size r = r(M), which tries instead.1 [10] This alternative name never really we call the rank of M. Our running example will be the caught on, but the geometric roots of the field have since matroid with grown much deeper, bearing many new fruits. The geometric approach to matroid theory has re- E = abcde; B = fabc; abd; abe; acd; aceg; (1) cently led to the solution of long-standing questions, and to the development of fascinating mathematics at the in- omitting brackets for easier readability. tersection of combinatorics, algebra, and geometry. This Let us now discuss the two most important motivating note is a selection of some recent successes. examples of matroids; there are many others. ∗To appear in the Notices of the American Mathematical Society, Sept., 2018. This version includes a full bibliography. yProfessor of Mathematics, San Francisco State University. Simons Research Professor, Mathematical Sciences Research Institute, Berkeley. Profesor Adjunto, Universidad de Los Andes, Colombia. [email protected]. This work was supported in part by NSF Award DMS-1600609, Award DMS-1440140 to MSRI, and the Simons Foundation. 1It was tempting to call this note The geometry of geometries. 1 1. Vector configurations. Let F be a field, let E be a set d of vectors in a vector space over F, and let I be the col- b lection of linearly independent subsets of E. Then (E; I) a is a linear matroid (over F). 2. Graphs. Let E be the set of edges of a graph G and I e be the collection of forests of G; that is, the subsets of E containing no cycle. Then (E; I) is a graphical matroid. c Figure 2: The planar graph of Figure1 and its dual a b graph, whose set of bases is B? = fbd; be; cd; ce; deg. a d e d e 3 Enumerative invariants b c c Two matroids M1 = (E1; I1) and M2 = (E2; I2) are iso- Figure 1: A linear and a graphical representation of the morphic if there is a relabeling bijection φ : E1 ! E2 matroid of (1) with B = fabc; abd; abe; acd; aceg. that maps I1 to I2.A matroid invariant is a function f on matroids such that f(M1) = f(M2) whenever M1 and M are isomorphic. Let us introduce a few important There are several natural operations on matroids. For 2 examples. S ⊆ E, the restriction MjS and the contraction M=S are matroids on the ground sets S and E − S, respectively, The f-vector and the h vector. The independent sets with independent sets of M form a simplicial complex I by (I2); its f-vector counts the number fk(M) of independent sets of M of IjS = fI ⊆ S : I 2 Ig size k + 1 for each k. The h-vector of M, defined by r r I=S = fI ⊆ E − S : I [ IS 2 Ig X r−k X r−k fk−1(q − 1) = hkq ; k=0 k=0 for any maximal independent subset IS of S. When M is a linear matroid in a vector space V , MjS and M=S are stores this information more compactly. For example, the the linear matroids on S and E −S that M determines on matroid of (1) has the vector spaces span(S) and V=span(S), respectively. f(M) = (1; 5; 9; 5); h(M) = (1; 2; 2; 0): The direct sum M1 ⊕ M2 of two matroids M1 = The characteristic polynomial. We define the rank func- (E1; I1) and M2 = (E2; I2) on disjoint ground sets is tion r : 2E ! of a matroid M by the matroid on E1 [ E2 with independent sets Z r(A) = largest size of an independent subset of A; I ⊕ I = fI [ I : I 2 I ;I 2 I g: 1 2 1 2 1 1 2 2 for A ⊆ E. Let r = r(M) = r(E) be the rank of M. When M is a linear matroid, r(A) = dim span(A). The Every matroid decomposes uniquely as a direct sum of its characteristic polynomial of M is connected components. X jAj r(M)−r(A) Finally, the orthogonal matroid of M, denoted M ?, is χM (q) = (−1) q : the matroid on E with bases A⊆E The sequence w(M) of Whitney numbers of the first kind ? r r−1 r 0 B = fE − B : B 2 Bg: is defined by χM (q) = w0q − w1q + ··· + (−1) wrq . For example, the matroid of (1) has Remarkably, this simple notion simultaneously general- w(M) = (1; 4; 5; 2): izes orthogonal complements and dual graphs. If M is the matroid for the columns of a matrix whose rowspan The characteristic polynomial of a matroid is one of is U ⊆ V , then M ? is the matroid for the columns of its most fundamental invariants. For graphical and linear any matrix whose rowspan is U ?. If M is the matroid matroids, it has the following interpretations. [10, 20, 23] for a planar graph G, drawn on the plane without edge 1. Graphs. If M is the matroid of a connected graph G, ? intersections, then M is the matroid for the dual graph then q χM (q) is the chromatic polynomial of G; it counts G?, whose vertices and edges correspond to the faces and the colorings of the vertices of G with q given colors such edges of G, respectively, as shown in Figure2. that no two neighbors have the same color. 2 2. Hyperplane arrangements. Suppose M is the matroid • c(B) = f(vB) for each B 2 B. d of non-zero vectors v1; : : : ; vn 2 F , and consider the ar- If one can do this, then the optimal object(s) B corre- rangement A of hyperplanes spond to the vertices of the face of the polytope PB where the linear function f is minimized. This simple, beautiful Hi : vi · x = 0; 1 ≤ i ≤ n idea is the foundation of linear programming. There are many techniques to optimize f, whose efficiency depends d and its complement V (A) = − (H1 [···[ Hn). De- F on the complexity of the polytope PB. pending on the underlying field, χM (q) stores different Edmonds observed that, given a matroid M and a information about V (A): cost function c : E ! R on its ground set, the bases (a) ( = q) V (A) consists of χM (q) points. F F B = fb1; : : : ; brg of M of minimum cost c(B) := c(b1) + (b) ( = ) V (A) consists of jχM (−1)j regions. F R ··· + c(br) can be found via linear programming on the (c) ( = ) The Poincar´epolynomial of V (A) F C matroid polytope PM . As a sample application, Edmonds [11] used these X k k d rank H (V (A); Z)q = (−1) χM (−1=q): ideas to solve the matroid intersection problem for ma- k≥0 troids M and N on the same ground set. This problem asks to find the size of the largest set which is independent 4 Geometric Model 1. Matroid polytopes in both M and N. Algebraic Geometry. Instead of studying the r- A crucial insight on the geometry of matroids came from n two seemingly unrelated places: combinatorial optimiza- dimensional subspaces of C one at a time, it is often use- ful to study them all at once. They can be conveniently tion and algebraic geometry. From both points of view, n it is natural to model a matroid in terms of the following organized into the space of r-subspaces of C called the polytope. Grassmannian Gr(r; n); each point of Gr(r; n) represents an r-subspace of Cn. Definition 1. (Edmonds, 1970, [11]) Let M be a matroid A choice of a coordinate system on Cn gives rise to on the ground set E. The matroid polytope the Pl¨uckerembedding of p n (r)−1 PM = convfeB : B is a basis of Mg; Gr(r; n) ,−! PC n E as follows. For an r-subspace V ⊂ , choose an r × n where fei : i 2 Eg is the standard basis of R and we C matrix A with V = rowspan(A). Then for each of the n write eB = eb + ··· + eb for B = fb1; : : : ; brg. r 1 r r-subsets B of [n] let Figure3 shows the matroid polytope for example (1).
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