FLAG ENUMERATIONS OF MATROID BASE POLYTOPES SANGWOOK KIM 9 0 0 Abstract. Inthis paper,westudyflagstructuresofmatroidbasepoly- 2 topes. We describe faces of matroid base polytopes in terms of matroid n data,andgiveconditionsforhyperplanesplitsofmatroidbasepolytopes. a Also, we show how the cd-index of a polytope can be expressed when a J polytope is split by a hyperplane, and apply these to the cd-index of a 0 matroid base polytope of a rank 2 matroid. 3 ] O C 1. Introduction . h For a matroid M on [n], a matroid base polytope Q(M) is the polytope in t a Rn whose vertices are the incidence vectors of the bases of M. The polytope m Q(M) is a face of a matroid polytope first studied by Edmonds [6], whose [ vertices are the incidence vectors of all independent sets in M. In this paper, 2 we study flags of faces and the cd-index of matroid base polytopes. v It is known that a face σ of a matroid base polytope is the matroid base 4 3 polytope Q(M ) for some matroid M on [n] (see [8] and Section 2 be- σ σ 5 low). We show that M can be described using equivalence classes of factor- 3 σ . connected flags of subsets of [n]. As a result, one can describe faces of Q(M) 1 in terms of matroid data: 0 9 0 Theorem 1.1 (Theorem 2.9). Let M be a connected matroid on a ground : set [n]. For a face σ of the matroid base polytope Q(M), one can associate v i a poset P defined as follows: X σ (i) the elements of P are the connected components of M , and r σ σ a (ii) for distinct connected components C and C of M , C < C if and 1 2 σ 1 2 only if C ⊆ S ⊆ [n] and σ ⊆ H implies C ⊆ S, 2 S 1 where H is the hyperplane in Rn defined by x = r(S). S e∈S e The cd-index Ψ(Q) of a polytope Q, a polynomiaPl in the noncommutative variables c and d, is a very compact encoding of the flag numbers of a polytope Q [2]. Ehrenborg and Readdy [7] express the cd-indices of a prism, a pyramid, and a bipyramid of a polytope Q in terms of cd-indices of Q Key words and phrases. Matroid base polytopes, cd-index. 1 and its faces. Also, the cd-index of zonotopes, a special class of polytopes, is well-understood [3, 4]. Generalizing the formulas of the cd-indices of a prism and a pyramid of a polytope, we show how the cd-index of a polytope can be expressed when a polytope is split by a hyperplane in Section 3. In Section 4, we find the conditions when a matroid base polytope is split into two matroid base polytopes by a hyperplane: Theorem 1.2 (Theorem 4.1). Let M be a rank r matroid on [n] and H be a hyperplane in Rn given by x = k. Then H decomposes Q(M) into e∈S e two matroid base polytopes if and only if P (i) r(S) > k and r(Sc) > r −k, (ii) if I and I are k-element independent subsets of S such that (M/I )| 1 2 1 Sc and (M/I )| have rank r −k, then (M/I )| = (M/I )| . 2 Sc 1 Sc 2 Sc We apply this theorem to the cd-index of matroid base polytopes for rank 2 matroids in Section 5. Throughout this paper, we assume familiarity with the basic concepts of matroid theory. For further information, we refer the readers to [13]. 2. Matroid base polytopes This section contains the description of faces of matroid base polytopes. In particular, we associate a poset to each face of a matroid base polytope. We start with a precise characterization of matroid base polytopes. Let B be a collection of r-element subsets of [n]. For each subset B = {b ,...,b }, 1 r let e = e +···+e ∈ Rn, B b1 br wheree istheithstandardbasisvectorofRn. ThecollectionBisrepresented i by the convex hull of these points Q(B) = conv{e : B ∈ B}. B This is a convex polytope of dimension ≤ n−1 and is a subset of the (n−1)- simplex ∆ = {(x ,...,x ) ∈ Rn : x ≥ 0,...,x ≥ 0,x +···+x = r}. n 1 n 1 n 1 n Gelfand, Goresky, MacPherson, and Serganova [9, Thm. 4.1] show the following characterization of matroid base polytopes. Theorem 2.1. B is the collection of bases of a matroid if and only if every edge of the polytope Q(B) is parallel to a difference e − e of two distinct α β standard basis vectors. 2 For a rank r matroid M on a ground set [n] with a set of bases B(M), the polytope Q(M) := Q(B(M)) is called the matroid base polytope of M. By the definition, the vertices of Q(M) represent the bases of M. For two bases B and B′ in B(M), eB and eB′ are connected by an edge if and only if eB−eB′ = eα−eβ for some α,β ∈ [n]. Since the latter condition is equivalent to B \B′ = {α} and B′ \B = {β}, the edges of Q(M) represent the basis exchange axiom. The basis exchange axiom gives the following equivalence relation on the ground set [n] of the matroid M: α and β are equivalent if thereexist bases B andB′ inB(M) withB\B′ = {α}andB′\B = {β}. The equivalence classes are called the connected components of M. The matroid M is called connected if it has only one connected component. Feichtner and Sturmfels [8, Prop. 2.4] express the dimension of the matroid base polytope Q(M) in terms of the number of connected components of M. Proposition 2.2. Let M be a matroid on [n]. The dimension of the matroid base polytope Q(M) equals n−c(M), where c(M) is the number of connected components of M. Theorem 2.1 implies that every face of a matroid base polytope is also a matroid base polytope. For a face σ of Q(M), let M denote the matroid on σ [n]whose matroidbasepolytopeisσ. Forω ∈ Rn, let M denotethematroid ω whosebasesB(M )isthecollectionofbasesofM having minimum ω-weight. ω n Then Q(M ) is the face of Q(M) at which the linear form ω x attains ω i=1 i i its minimum. Let F(ω) denote the unique flag of subsets P {∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]} 0 1 k k+1 for which ω is constant on each set S \S and ω| < ω| . Ardila i i−1 Si\Si−1 Si+1\Si and Klivans [1] show that M depends only on F(ω), and hence one can call ω it M . They also give the following description of M . F F Proposition 2.3. [1, Prop. 2] Let M be a matroid on [n] and F be a flag of subsets {∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]}, 0 1 k k+1 then k+1 M = (M| )/S , F Si i−1 i=1 M where M| is the restriction of M on S and M/S is the contraction of M S on S. A flag F = {∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]} is called factor- 0 1 k k+1 connected (with respect to M) if the matroids (M| )/S are connected for Si i−1 all i = 1,...,k+1. If F is factor-connected, then the connected components 3 of M are S \S ,S \S ,...,S \S . Proposition 2.2 and Proposition 2.3 F 1 0 2 1 k+1 k show that the dimension of Q(M ) is n−k−1 when F is factor-connected. F Since Q(M ⊕M ) = Q(M )×Q(M ), it is enough to restrict to attention 1 2 1 2 to connected matroid M. For a connected matroid M on [n], facets of Q(M) correspond to factor-connected flags of the form {∅ ⊆ S ⊆ [n]}. Feichtner and Sturmfels [8] show that there are two types of facets of Q(M): (i) a facet corresponding to a factor-connected flag {∅ ⊆ F ⊆ [n]} for some flat F of M (in this case, the facet is called a flacet), (ii) a facet corresponding to a factor-connected flag {∅ ⊆ S ⊆ [n]} for an (n−1)-subset S of [n]. Lemma 2.4. Let M be a connected matroid on [n] and F = {∅ =: S ⊆ S ⊆ ··· ⊂ S ⊆ S := [n]} 0 1 k k+1 a factor-connected flag with respect to M. Then the matroid (M| )/S Sj+1 j−1 has at most two connected components for 1 ≤ j ≤ k. (i) If it has one connected component, the flag G = {∅ =: S ⊆ ··· ⊆ S ⊆ S ⊆ ··· ⊆ S := [n]} 0 j−1 j+1 k+1 is factor-connected and Q(M ) covers Q(M ) in the face lattice of G F Q(M). (ii) If it has two connected components, then they are S \S and S \S . j j−1 j+1 j Moreover, the flag F′ = {∅ =: S ⊆ ··· ⊆ S ⊆ S′ ⊆ S ⊆ ··· ⊆ S := [n]}, 0 j−1 j j+1 k+1 whereSj′ = Sj−1∪(Sj+1\Sj), is factor-connectedandQ(MF′) = Q(MF). In this case, F and F′ are said to be adjacent. Proof. For j = 1,...,k, we have (M| )/S = [(M| )/S ]/(S \S ) Sj+1 j Sj+1 j−1 j j−1 and (M| )/S = [(M| )/S ]| . The first assertion is obtained from Sj j−1 Sj+1 j−1 Sj [13, Proposition 4.2.10], and the other assertions follow from [13, Proposition (cid:3) 4.2.13] and Proposition 2.3. Two factor-connected flags F and F′ of the same length are said to be equivalent if there is a sequence of factor-connected flags F = F ,F ,...,F = F′ 0 1 r such that F is adjacent to F for i = 1,...,r. We write F ∼ F′ when i i−1 factor-connected flags F and F′ are equivalent. Lemma 2.5. Let M be a connected matroid on [n] and X,Y disjoint subsets of [n] with X ∪Y = [n]. Then B((M| )⊕(M/X)) and B((M| )⊕(M/Y)) X Y are disjoint. 4 Proof. Since X ∩ Y = ∅ and X ∪ Y = [n], one has r(X) + r(Y) ≥ r(M). If r(X) + r(Y) = r(M), then |B ∩ X| = r(X) and |B ∩ Y| = r(Y) for all B ∈ B(M), andhence M = M| ⊕M| . But then M isnot connected. Thus X Y r(X)+r(Y) > r(M). Therefore there is no base which has r(X) elements in X and r(Y) elements in Y, i.e., B((M| )⊕(M/X)) and B((M| )⊕(M/Y)) X Y (cid:3) are disjoint. The following proposition shows that the equivalence classes of factor- connected flags characterize faces of a matroid base polytope. Proposition 2.6. Let M be a connected matroid on [n]. If F and F′ are two factor-connected flags of subsets of [n] given by F ={∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]}, 0 1 k k+1 F′ ={∅ =: T ⊆ T ⊆ ··· ⊆ T ⊆ T := [n]}, 0 1 l l+1 then MF = MF′ if and only if F and F′ are equivalent. Proof. If F ∼ F′, then MF = MF′ from Lemma 2.4. For the other direction, suppose that MF = MF′. Then F and F′ have the same length since dimQ(MF) = n−k−1 and dimQ(MF′) = n−l−1. Without loss of generality, we may assume that S 6= T and S 6= T . 1 1 k k We will use induction on k. We claim that (2.1) T = S \S for some m > 1. 1 m m−1 This claim is going to be used both in the base case and the inductive step. Let m be the smallest index such that T ∩ S 6= ∅ and let α ∈ T ∩ S . 1 m 1 m Suppose that T is not contained in S , i.e., there is an element β ∈ T \S . 1 m 1 m Since M| is connected, there are B′,B′ ∈ B(M| ) such that α ∈ B′ and T1 1 1 T1 1 B′ = (B′ \{α}) ∪{β}. Choose B′ ∈ B((M| )/T ) for j = 2,...,k + 1. 1 1 j Tj j−1 Then B′ = B′ ∪B′ ∪···∪B′ and B′e= B′ ∪B′ ∪···∪B′ are bases of 1 2 k+1 1 2 k+1 e MF′. But either B′ or B′ is not a base of MF since e e |B′ ∩(S \S )| = |B′ ∩(S \S )|−1. m m−1 m m−1 e This contradicts the assumption MF = MF′. Therefore T1 ⊆ Sm. e Now suppose (S \ S ) \ T 6= ∅, i.e., there is γ ∈ (S \ S ) \ T . m m−1 1 m m−1 1 Since (M| )/S is connected, there are bases B ,B of (M| )/S Sm m−1 m m Sm m−1 satisfying α ∈ B and B = (B \{α})∪{γ}. If B ∈ B((M| )/S ) for m m m j Sj j−1 all j 6= m, then e e B = B ∪···∪B and 1 k+1 B = B ∪···∪B ∪B ∪B ∪···∪B 1 m−1 m m+1 k+1 5 e e are contained in B(MF). But at least one of them is not in B(MF′) because |B ∩T1| = |B ∩T1|−1. This is impossible because MF = MF′. Therefore T = S \S . 1 m m−1 eBase case: k = 1. Equation (2.1) shows that T = S \S . Since T ∪S = 1 2 1 1 1 [n], M is not connected by Lemma 2.5. Thus, F and F′ are equivalent by Lemma 2.4(ii). Inductive step. Now suppose k > 1. By Equation (2.1), T = S \S 1 m m−1 for some m > 1. Let F be a flag ∅ =: S ⊆ S ⊆ S ∪T ⊆ S ∪T ⊆ ··· ⊆ S ∪T ⊆ S ··· ⊆ S := [n]. 0 1 1 1 2 1 m−1 1 m+1 k+1 e =Sm WeclaimthatM| = M| ⊕M| forj = 1,...,m−1. LetB ∈ B(M| ) Sj∪T1 Sj T1 | {z } j Sj and B′ ∈ B(M| ). Since 1 T1 MF = ⊕ki=+11(M|Si)/Si−1 = ⊕ki=+11(M|Ti)/Ti−1 = MF′ and M| = (M| )/S , there is a base B ∈ B(M ) such that B∩S = B T1 Sm m−1 F j j and B ∩T = B′. Thus B ∪B′ ∈ B(M| ), and hence 1 1 j 1 Sj∪T1 r(M| ) = r(M| )+r(M| ). Sj∪T1 Sj T1 By [12, Lemma 3], we have M| = M| ⊕M| . Therefore Sj∪T1 Sj T1 (M| )/S = [M| ⊕M| ]/S S1∪T1 1 T1 S1 1 = (M| )/S Sm m−1 and (M| )/(S ∪T ) = [M| ⊕M| ]/(S ∪T )) Si∪T1 i−1 1 T1 Si i−1 1 = (M| )/S Si i−1 for i = 2,...,m − 1. Thus F is factor-connected and MFe = MF. By the induction assumption, F ∼ F. Similarly, one can show S = T \ T for 1 l l−1 some l > 1 and the chain F′ geiven by e ∅ =: T ⊆ T ⊆ S ∪T ⊆ S ∪T ⊆ ··· ⊆ S ∪T ⊆ T ··· ⊆ T := [n] 0 1 1 1 1 2 1 l−1 l+1 k+1 e =Tl is factor-connected and MFe′ = MF′. Thu|s th{ezind}uction assumption gives F′ ∼ F′. By the induction assumption again, we have F ∼ F′ and hence F and F′ (cid:3) aere equivalent. e e If M is a matroid on [n] and S is a subset of [n], then the hyperplane H defined by x = r(S) is a supporting hyperplane of Q(M) and S e∈S e Q(M) ∩ H is the matroid base polytope for (M| ) ⊕ (M/S). The next S S P lemma tells us when a face of Q(M) is contained in H . S 6 Lemma 2.7. Let M be a matroid on [n] and S be a subset of [n]. A face σ of Q(M) is contained in H if and only if there is a factor-connected flag S F = {∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]} 0 1 k k+1 containing S and σ = Q(M ). F Proof. If there is a flag F containing S and σ = Q(M ), then every base of F M is contained in B((M| )⊕(M/S)) and hence σ is contained in H . F S S For the converse, suppose σ is contained in H . Since σ is a face of Q(M), S there is a factor-connected flag F′ = {∅ =: T ⊆ T ⊆ ··· ⊆ T ⊆ T := [n]}, 0 1 k k+1 such that σ = Q(MF′) and k = n − dimσ − 1. We claim that Ti \ Ti−1 is contained in either S or [n] \ S for i = 1,...,k + 1. Suppose there are α ∈ (T \T )∩S andβ ∈ (T \T )\S. Since(M| )/T isconnected, there i i−1 i i−1 Ti i−1 exist bases B1 and B2 of MF′ with α ∈ B1 and B2 = (B1\{α})∪{β}. Then |B ∩S| > |B ∩S| and so σ is not contained in H , which is a contradiction. 1 2 S Construct aflag F asfollows: S = S ∪(T \T ), where l isthesmallest j j−1 l l−1 index such that (i) T \T ⊆ S \S if S \S 6= ∅, l l−1 j−1 j−1 (ii) T \T ⊆ [n]\S if S \S = ∅. l l−1 j−1 j−1 By the construction of F, S is contained in F. Since σ is contained in H , S MF = ⊕ki=+11(M|Si)/Si−1 = ⊕ki=+11(M|Ti)/Ti−1 = MF′ and hence σ = Q(M ). (cid:3) F For a face σ of Q(M), let L be the poset of all subsets of [n] which σ are contained in some factor-connected flag F with σ = Q(M ) ordered by F inclusion. The following lemma shows that L is a lattice. σ Lemma 2.8. LetM be a matroidon [n] andS,T ⊆ [n]. IfQ(M) is contained in H and H , then it is also contained in H . S T S∩T Proof. Since Q(M) is contained in H and H , we have S T r(S ∪T)+r(S ∩T) = r(S)+r(T) (see [13, Lemma 1.3.1]). Suppose that Q(M) is not contained in H , i.e., S∩T there is a base B of M such that |B ∩(S ∩T)| < r(S ∩T). Then |B ∩(S ∪T)| = |B ∩S|+|B ∩T|−|B ∩(S ∩T)| > r(S)+r(T)−r(S ∩T) = r(S ∪T) which is impossible. Therefore Q(M) is contained in H . (cid:3) S∩T 7 Since L is a sublattice of the Boolean lattice B , it is distributive. The σ n fundamental theorem for finite distributive lattices [17] shows that there is a finite poset P for which L is the lattice of order ideals of P . The following σ σ σ theorem describes the poset P in terms of matroid data. σ Theorem 2.9. Let M be a matroid on [n] and σ be a face of Q(M). Then L is the lattice of order ideals of P , where P is the poset defined as follows: σ σ σ (i) The elements of P are the connected components of M , and σ σ (ii) for distinct connected components C and C of M , C < C if and 1 2 σ 1 2 only if C ⊆ S ⊆ [n] and σ ⊆ H implies C ⊆ S. 2 S 1 Note that P is a well-defined poset. Reflexivity and transitivity are clear. σ Suppose C and C are distinct connected components of M with C < C . 1 2 σ 1 2 Consider a minimal subset S such that C ⊆ S and σ ⊆ H . C < C implies 2 S 1 2 C is contained in S. Since C is a connected component of M , Lemma 2.7 1 2 σ implies that there is a flag containing S and S \C such that Q(M ) = σ. 2 F Since C ⊆ S \C , σ ⊆ H , and C * S \C , we have C ≮ C . 1 2 S\C2 2 2 2 1 Example 2.10. Let M be the rank 2 matroid on [4] = {1,2,3,4} whose 2,1,1 unique non-base is 12 and let σ be an edge of Q(M ) connecting e and 2,1,1 14 e . Then the connected components of M are 12, 3 and 4. Since {1,2,3,4} 24 σ is the only subset S containing {3} such that σ ⊆ H , 12 < 3 and 4 < 3 S in P . One can see that there are no other relations in P . Figure 1 is the σ σ proper part of the face poset of Q(M ) whose faces (shown on the right- 2,1,1 hand side of each box) are labeled by corresponding posets (shown on the left-hand side of each box) and P is shown in the shaded box. σ Proof of Theorem 2.9. Let S be a set in L . By Lemma 2.7, there is a factor- σ connected flag F = {∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]} 0 1 k k+1 such that M = M and S = S for some i. Then S = ∪i (S \ S ). σ F i j=1 j j−1 SupposeS \S < S \S inP forsomej ≤ i. Sinceσ ⊆ H , S \S ⊆ S l l−1 j j−1 σ S l l−1 and hence S is an order ideal of P . σ Conversely, suppose T is an order ideal of P . Let U be the intersection σ of all subsets T such that σ ⊆ HTe and T ⊆ T. By Lemma 2.8, U lies in Lσ. Suppose T 6= U. By Lemma 2.7, there is a factor-connected flag e e F = {∅ =: U ⊆ U ⊆ ··· ⊆ U ⊆ U := [n]} 0 1 k k+1 such that σ = Q(M ) and U = U for some j. Since T 6= U and T is an F j order ideal, we may choose F so that U \ U * T. Then U ∈ L and j j−1 j−1 σ T ⊆ U which is a contradiction. (cid:3) j−1 8 2 34 123 34 124 34 34 23 24 1 34 134 13 14 4 14 24 3 13 23 12 13 14 234 23 24 2 34 2 34 14 24 1 34 23 1 34 24 2 34 3 4 1 34 13 14 23 24 1 12 4 12 3 2 4 14 3 13 13 14 4 24 13 3 23 23 2 3 2 4 1 2 1 3 1 4 14 13 34 24 23 1 4 1 3 3 4 2 4 2 3 Figure 1. The proper part of the face poset of Q(M ) 2,1,1 The posets P coincide with the posets obtained from preposets corre- σ sponding to normal cones of the matroid base polytope Q(M) studied by Postnikov, Reiner, and Williams [14]. Billera, Jia, and Reiner [5] define posets related to bases of matroids: For each e ∈ B ∈ B(M) the basic bond for e in B is the set of e′ ∈ [n]\ B for which (B \ {e}) ∪ {e′} is another base of M. Dually, for each e′ ∈ [n] \ B the basic circuit for e in B is the set of e ∈ B for which (B \{e})∪{e′} is another base of M. Since e′ lies in the basic bond for e if and only if e lies in the basic circuit of e′, one can define the poset P to be the poset defined B on [n] where e < e′ if and only if e′ is in the basic bond for e. The next PB proposition shows that our poset P is the same as the poset P of Billera, σ B Jia, and Reiner if σ is a vertex e . B Proposition 2.11. If B is a base of a matroid M on [n] and σ is a vertex e of Q(M), then P = P . B σ B Proof. Since all connected components of M are singletons, P and P have σ σ B the same elements. If e ,e ∈ B, then they are not comparable in P since 1 2 σ e ⊆ H and e ⊆ H . Also, e ,e ∈/ B are not comparable in P because B e1 B e2 1 2 σ 9 e ⊆ H and e ⊆ H . Now assume e ∈ B and e ∈/ B. We claim B B∪{e1} B B∪{e2} 1 2 that e < e if and only if B′ := (B\{e })∪{e } ∈ B(M), i.e., e < e . 1 Pσ 2 1 2 1 PB 2 Assume e < e , i.e., e ∈ S and σ ⊆ H implies e ∈ S. Suppose B′ is not 1 Pσ 2 2 S 1 a base of M. Then r(B′) = r(M)−1. Thus we have e2 ∈ B′,σ ⊆ HB′, but e ∈/ B′, which is a contradiction. Therefore B′ is a base of M. Conversely, 1 assume B′ ∈ B(M). Let S be a subset of [n] such that e ∈ S and σ ⊆ H , 2 S i.e., |B∩S| = r(S). If e ∈/ S, then |B′∩S| = |B∩S|+1 = r(S)+1, which 1 is impossible. Thus e < e . (cid:3) 1 Pσ 2 3. The cd-index In this section, we define the cd-index for Eulerian posets and give the relationship among cd-indices of polytopes when a polytope is split by a hyperplane. Let P be a graded poset of rank n + 1 with the rank function ρ. For a subset S of [n], define f (S) to be the number of chains of P whose ranks P are exactly given by the set S. The function f : 2[n] → N is called the flag P f-vector of P. The flag h-vector of P is defined by the identity h (S) = (−1)|S\T| ·f (T). P P T⊆S X Since this identity is equivalent to the relation f (S) = h (T), P P T⊆S X the flag f-vector and the flag h-vector contain the same information. For S ⊆ [n], define the noncommutative ab-monomial u = u ...u , S 1 n where a if i ∈/ S, u = i b if i ∈ S. (cid:26) The ab-index of the poset P is defined to be the sum Ψ(P) = h (S)·u . P S S⊆[n] X An alternative way of defining the ab-index is as follows. For a chain c := {ˆ0 < x < ··· < x < ˆ1}, 1 k we give a weight w (c) = w(c) = z ···z , where P 1 n b if i ∈ {ρ(x ),...,ρ(x )}, z = 1 k i a−b otherwise. (cid:26) 10

Flag enumerations of matroid base polytopes

Similar Flag enumerations of matroid base polytopes