Relationships Among Flag f-Vector Inequalities for Polytopes

Discrete & Computational Geometry, Dec 2003

Catherine Stenson

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Relationships Among Flag f-Vector Inequalities for Polytopes

Discrete Comput Geom Geometry Discrete & Computational Catherine Stenson 0 0 Department of Mathematics, Juniata College , Huntingdon, PA 16652 , USA We examine linear inequalities satisfied by the flag f -vectors of polytopes. One source of these inequalities is the toric g-vector; convolutions of its entries are nonnegative for rational polytopes. We prove a conjecture of Meisinger about a redundancy in these inequalities. Another source of inequalities is the cd-index; among all d-polytopes, each cd-index coefficient is minimized on the d-simplex. We show that not all of the cdindex inequalities are implied by the toric g-vector inequalities, and that not all of the toric g-vector inequalities are implied by the cd-index inequalities. Finally, we show that some inequalities from convolutions of cd-index coefficients are implied by other cd-index inequalities. 1. Introduction ( 1 ) All linear relations satisfied by flag f -vectors of polytopes are consequences of the generalized Dehn–Sommerville equations. The flag f -vector has length 2d , but because of these equations the affine span of the flag f -vectors of d-polytopes has dimension ad − 1, where ad is the dth Fibonacci number (a0 = 1, a1 = 1). Let Sp(d) = { | S S ⊂ {0, 1, . . . , d − 2}, {i, i + 1} ⊂ S for all 0 ≤ i ≤ d − 3}. The ad − 1 flag numbers { fS|S ∈ Sp(d)} are known as the sparse basis; every fT can be expressed in terms of this basis via the generalized Dehn–Sommerville equations (see Appendix A of [15] for dimensions 3–8). Since the linear relations are understood, it is natural to try to understand the linear inequalities satisfied by flag f -vectors of polytopes. In dimension 3 the linear inequalities actually tell the whole story. The flag f -vector is determined by the f -vector, and the f -vectors of 3-polytopes are characterized by the following theorem. Theorem 2 (Steinitz). An integer vector ( f0, f1, f2) is the f -vector of a 3-polytope if and only if it satisfies the Euler relation f2 − f1 + f0 = 2 and (i) f0 ≤ 2 f2 − 4, (ii) f2 ≤ 2 f0 − 4. In higher dimensions the situation is not as well understood. One source of linear inequalities is the toric g-vector introduced by Stanley [18]. This vector and the related toric h-vector depend only on the flag f -vector of the polytope, and can be defined recursively via two polynomials: Definition 1. Let h(Q, x ) = d i=0 hi (Q)x d−i and g(Q, x ) = i =d/02 gi (Q)xi where (i) h(∅, x ) = g(∅, x ) = 1, (ii) gi (Q) = hi (Q) − hi−1(Q) (iii) h(Q, x ) = g(F, x )(x − 1)dim(Q)−dim(F)−1, for 0 ≤ i ≤ d/2 , ∅⊆F⊂Q setting h−1 = 0. The coefficients of these polynomials are the entries of the toric h- and g-vectors. The hi ’s and gi ’s are are all linear combinations of entries of the flag f -vector. Notice that h0 = g0 = 1. As a consequence of the generalized Dehn–Sommerville equations we have Theorem 3 [17]. If Q is a polytope (or, more generally, an Eulerian poset), then hi (Q) = hd−i (Q). If Q is a rational polytope, hi (Q) is the rank of the (2d − 2i )th intersection homology group of the toric variety associated with Q. This interpretation of hi allowed Stanley to prove the following: ( 2 ) ( 3 ) ( 4 ) Theorem 4 [18]. If Q is a rational polytope, then gi (Q) ≥ 0. It is conjectured that gi ≥ 0 for all polytopes, not just rational ones. This has been proven in the cases i = 1 (g1 = f0 − (d + 1) ≥ 0 since it takes d + 1 points to affinely span d-space) and i = 2 (g2 = f02 − 3 f2 + f1 − (d + 1) f0 + d +21 ≥ 0 by a rigidity argument due to Kalai [13]), but remains an open question for i ≥ 3. See Appendix B of [15] for a list of the gid ’s for d ≤ 10. We can use Kalai’s convolution operation to combine the gi ’s to create more linear inequalities [14]. Given fSm and fTn, define their convolution to be ( 5 ) f m n S ∗ fT = fSm (F ) fTn(Q/F ), F ⊂ Q dim(F)=m where Q/F is the quotient of Q by the face F (see [21] for a discussion of quotient polytopes). Note the convolution is just fSm∪+{mn+}∪1(T +m+1), where T +m +1 = {i +m +1|i ∈ T }. Now extend this to linear forms M = S αS fSm and N = T βT fTn (αS, βT ∈ R) by M ∗ N = αSβT fSm (F ) fTn(Q/F ) = αSβT fSm ∗ fT . n ( 6 ) F ⊂ Q S,T S,T dim(F)=m If M ≥ 0 for all m-polytopes and N ≥ 0 for all n-polytopes, then M ∗ N ≥ 0 for all (m + n + 1)-polytopes. Although f∅ is 1 for any polytope, it is useful to think of it as a linear form in its own right. This convention allows us to convolve linear forms involving constants. For in terms of f0 and f2 this becomes∅ t=he fs0e32c−ond3 fo23f ≥Ste0infoitrz’asllin3e-pqoulayltiotipees;s.thReewfirristtteins example, g12 ∗ g00 = ( f02 − 3 f∅2) ∗ f 0 g00 ∗ g12 ≥ 0. Every polytope Q has a dual polytope Q∗; see [21] for a geometric construction. There is a bijection between the i -faces of Q and the (d − i − 1)-faces of Q∗. So fi (Q) = fd−i−1(Q∗) and more generally fi1,i2,...,ik (Q) = fd−ik−1,...,d−i2−1,d−i1−1(Q∗). Thus a linear form M has a dual form M where M (Q) is defined to be M (Q∗). In particular, the gi ’s are also non-negative for all rational polytopes. This paper examines the relationships among the currently known flag f -vector inequalities. In Section 2 we show that some of the toric g inequalities are implied by other toric g inequalities. In Section 3 we discuss the cd-index and show that some inequalities derived from the cd-index are not implied by the toric g inequalities. In Section 4 we show that the toric g inequalities are not all implied by the cd-index inequalities, and in Section 5 we show that the convolution of some cd-index inequalities yields inequalities implied by other cd-index inequalities. 2. Redundancy of Some Inequalities From the Toric g-Vector By convolving the gi ’s and gi ’s we can generate many linear inequalities satisfied by flag f -vectors of rational polytopes. Are any of these inequalities implied by the others? f∅0 M∗efi∅0si=ngefr01[1=5]2gsh01oswinecdetahnatedgg00e∗hga00s atwndo gv0e0r∗ticge01s aarnedregd00u∗ndga01n=t. Wf02e =havfe12 g=00 ∗g01g∗00 g=00 since a polygon has equal numbers of vertices and edges. He also proved that if a linear form is redundant, then every convolution containing that form is redundant. Kalai (unpublished) and Bayer and Klapper [4] proved that if d is even, gdd/2 = gdd/2. We also have g0 = 1 = gd0 for all d. Bayer and Ehrenborg [3] showed that these are the d only redundancies within the set {g0d , g1d , . . . , gdd/2 , g0d , g1d , . . . , gdd/2 }. Meisinger [15] conjectured that there are two additional redundancies involving convolutions. We prove this conjecture in the form of Theorem 5. For i ≥ 0, (i + 2)gi2i+1 = (i + 2)gi2i+1 = i j=0 i j=0 ( j + 1)(gj2 j ∗ gi2−(i−j j)), (i − j + 1)(gj2 j ∗ gi2−(i−j j)). ( 7 ) ( 8 ) ( 9 ) The second equation is the dual of the first, so it is enough to prove ( 7 ). We prove it by showing this equation holds for a basis of (2i + 1)-polytopes, and therefore for all (2i + 1)-polytopes. In [1] Bayer and Billera described a basis created by successive application of two operations: pyramid, denoted P, and bipyramid, denoted B. Let Q be a d-polytope contained in H , a hyperplane in Rd+1. The pyramid over Q is the convex hull of Q and a point not in H . The bipyramid over Q is the convex hull of Q and a line segment that intersects the interior of Q and has one endpoint on each side of H . The choices of point and line segment affect the geometry of the pyramid and bipyramid, but do not change their combinatorial types. A word in B and P represents the polytope given by applying these operations from right to left, where the rightmost P stands for the initial point (or pyramid over the empty polytope). For example, PP = BP is a line segment, PPP is a triangle, BPP is a square, and PBPP is a pyramid over a square. Words of length 2i + 2 with no two adjacent B’s and ending in PP are a basis of (2i + 1) polytopes. Before we begin, we need three lemmas. Kalai notes that g( P Q, x ) = g(Q, x ) [14, Remark 3], where Q is any polytope. So we have gid ( P Q) = gid−1(Q). gj2 j ( P Q) = 0. Lemma 1. If d ≥ 1, then Lemma 2. If j ≥ 1, then If dim( P Q) = 2 j , j ≥ 1, then g(Q) has degree at most j − 1, so the coefficient of x j in g( P Q) must be 0. This gives us In the case j = 0, g00 = 1. Because Lemma 2 does not apply, we consider the cases j = 0 and i − j = 0 separately when we look at the right-hand side of ( 7 ). We will frequently encounter polytopes of the form ( P Q)/F and (BQ)/F , where F is a face of Q. The proof of the following lemma is straightforward. Lemma 3. If F is a face of Q, then (PQ)/F = P(Q/F ) and (BQ)/F = B(Q/F ). Because of these equalities, we often omit the parentheses and write PQ/F and BQ/F . Proof of Theorem 5. Our proof is by induction on i . As we saw at the beginning of Section 2, g00 ∗ g00 = 2g01, so the theorem holds for the base case i = 0. Suppose the conjecture is true for all (2k + 1)-polytopes, k < i . Consider the basis of (2i + 1)-polytopes. Each word begins with either P or B P; we consider these two cases separately. Case 1: PQ. Suppose the polytope is of the form PQ, where Q is a (2i )-polytope. On the left-hand side of ( 7 ) we have (i + 2)gi2i+1(PQ) = (i + 2)gi2i (Q) by Lemma 1. Now consider the right-hand side of ( 7 ) in the case j = 0, i , g2 j j ∗ gi2−(i−j j)(PQ) = gj2 j (F )gi2−(i−j j)( P(Q/F )) = 0 g2i i ∗ g00(PQ) = gi2i (Q) + gi2i (F )g00(PQ/PF) = gi2i (Q), F ⊂ Q, dim(F)=2i−1 again by Lemma 2. So the j = i term contributes (i + 1)gi2i (Q) to the right-hand side of ( 7 ). Now consider the case j = 0. The vertices of PQ are the vertex added in taking the pyramid over Q and the vertices of Q itself. Thus g0 ∗ gi2i (PQ) = gi2i (Q) + 0 gi2i (PQ/ V ) = gi2i (Q) again by Lemma 2. So the right-hand side of ( 7 ) is (i + 2)gi2i (Q), as desired, and the theorem holds for polytopes of the form PQ. Case 2: BPQ. Now consider polytopes of the form BPQ, where Q is a (2i − 1)-polytope. To show that the conjecture holds for them, we will need a pair of lemmas. Given that h(B X, x ) = (1 + x )h(X ) [14, Remark 3], their proofs are straightforward. Lemma 4. For k ≥ 1, Lemma 5. For i ≥ 1, gik (B X ) = hik−1(X ) − hik−−21(X ) = gik−1(X ) + gik−−11(X ). gi2i (B X ) = gi2−i−11(X ) and in particular gi2i (BPQ) = gi2−i−11(PQ) = gi2−i−12(Q). Proof of Case 2. Consider the left-hand side of ( 7 ). By Lemmas 4, 2, and 1, (i + 2)gi2i+1(BPQ) = (i + 2)(gi2i ( P Q) + gi2−i1(PQ)) = (i + 2)gi2−i1(PQ) = (i + 2)gi2−i−11(Q) = (i + 2)gi2−(i1−1)+1(Q), = = ( 16 ) ( 17 ) ( 18 ) ( 19 ) ( 20 ) ( 21 ) so we are set up for induction. Consider the right-hand side of ( 7 ) for j = 0, i . Omitting the factor of ( j + 1) we have g2 j j ∗ gi2−(i−j j)(BPQ) = We get ( 18 ) by noting that each 2 j -face of BPQ is either a 2 j -face of PQ or a pyramid over a (2 j −1)-face of PQ, and the second set of faces contributes 0 to the sum. Similarly, expression ( 19 ) comes from noting that each face of PQ is either a 2 j -face of Q or a pyramid over a (2 j − 1)-face of Q, and the second set of faces contributes 0 to the sum. Expression ( 20 ) follows from Lemmas 5 and 1. So each j = 0, i contributes ( j + 1)(gj2 j ∗ gi2−(i−j−j1−1)(Q)) to the right-hand side. When j = i ,   (i + 1)(gi2i ∗ g00)(BPQ) = (i + 1)    F ⊂ BPQ, dim(F)=2i gi2i (F )g00(BPQ/F ) = 0  (22) by Lemma 2 since every facet of BPQ is a pyramid over a facet of PQ. When j = 0, g0 ∗ gi2i (BPQ) = 2gi2i (PQ) + 0 gi2i (BPQ/ V ) = gi2i (B Q) + = gi2−i−11(Q) + V a vert of Q = gi2−i−11(Q) + g0 ∗ gi2−(i1−1)(Q). 0 V a vert of P Q (23) (24) (25) (26) (27) (28) (29) i−1 j=0 We get (23) by considering first the two vertices added in bipyramiding over PQ and then the vertices of PQ itself. By Lemma 2 the first term is 0. We get the first term of (24) from the vertex V of PQ added in pyramiding over Q; note PQ/ V = Q in this case. The second term comes from the remaining vertices of PQ, which are just vertices of Q. Expression (25) follows from Lemmas 5 and 1. Putting this all together, we see that the right-hand side of ( 7 ) is gi2−i−11(Q) + ( j + 1)gj2 j ∗ gi2−(i−j−j1−1)(Q) = g2(i−1)+1(Q) + ((i − 1) + 2)(gi2−i−11(Q)) i−1 = (i + 2)(gi2−i−11(Q)) by induction. We also have that the left-hand side of ( 7 ) is (i + 2)gi2−(i1−1)+1(Q). So the theorem holds for polytopes of the form B P Q. The theorem holds for all (2i + 1)-polytopes in the basis, and therefore it is true for all (2i + 1)-polytopes. We note that this result makes sense in light of a theorem of Braden and MacPherson [8]. They introduced the relative g-polynomial g(Q, F ) as a generalization of the toric g-vector. Here we follow their notation and let g(Q) = g(Q, x ). Definition 2. The family of polynomials g(Q,F) associated to a polytope Q and a face F of Q is defined inductively by the following relation: for all Q, F we have F⊆E⊆Q g(E , F )g(Q/E ) = g(Q). Note the induction begins with Q = F , giving g(F, F ) = g(F ). In general the left-hand side gives g(Q, F ) · 1 plus terms involving g(E , F ) with dim(E ) < dim(Q). Theorem 6 [8]. g(Q, F ) = (−1)dimQ−dimF g(F )g(Q/F ). Consider this equality in the case dim(Q) = 2i + 1 and F = ∅. When F = ∅, we have g(Q) on the right-hand side, and when F = Q, we have g(Q), and both of these polynomials have degree at most i . Similarly, if dim(F ) = 2 j , the F term has degree at most i . However, if dim(F ) = 2 j + 1, j = −1, i , the F term has degree at most i − 1. So g(Q, ∅) has degree at most i . Recall that gk2k = gk2k . Then we have that the coefficient of xi in g(Q, ∅) is i j=0 ∅d⊂imFF =⊂2 Qj gi2i+1(Q) + gi2i+1(Q) + (−1)2i+1−2 j gj2 j (F )gi2−i−j2 j (Q/F ) = gi2i+1(Q) + gi2i+1(Q) − i j=0 g2 j j ∗ gi2−i−j2 j (Q). However, we also have Theorem 7 [19]. For any polytope Q = ∅, This gives us g(Q, ∅) = 0 . Thus (30) becomes ∅⊆F ⊆Q (−1)dimF g(F )g(Q/F ) = 0. gi2i+1(Q) + gi2i+1(Q) = g2 j j ∗ gi2−i−j2 j (Q), i j=0 (30) (31) which is just the sum of ( 7 ) and ( 8 ) divided by (i + 2). 3. Some cd-Index Inequalities That Are Not Implied by Toric g-Vector Inequalities We have seen that convolutions of gid ’s and gid ’s are non-negative for rational polytopes. Kalai [14] conjectured that all non-negative forms arise in this way, but Meisinger [15] gave a counterexample. He showed that f26 − 35 ≥ 0, which holds for all 6-polytopes, is not implied by the non-negativity of the g convolutions. He further conjectured d+1 Conjecture 1 [15]. For d ≥ 6 the linear inequalities fj − j+1 ≥ 0 (2 ≤ j ≤ d − 3) are not a non-negative combination of convolutions of the linear forms gik and gik (0 ≤ k ≤ d, 0 ≤ i ≤ k/2 ). We will prove this conjecture and show that the non-negativity of these forms is actually implied by inequalities derived from the cd-index. This means that these cdindex inequalities are not implied by the non-negativity of the g convolutions. The cd-index is another way to encode the information in the flag f -vector. It was introduced by Fine and developed by Bayer and Klapper [4]. To define it, we first need the flag h-vector, given by h S = T ⊆S(−1)|S\T | fT for S ⊆ {0, 1, . . . d − 1}. Now let (32) a and b be non-commuting variables and let u S = u0u1 · · · ud−1, where ui = a if i ∈ S and ui = b if i ∈ S. Then we can define the ab-index of a polytope Q by (Q) = S h Su S. As a consequence of the generalized Dehn–Sommerville equations, can always be written in terms of c = a + b and d = ab + ba. When it is written this way, it is known as the cd-index. For example, when d = 1, (Q) = c; when d = 2, (Q) = c2 + ( f0(Q) − 2)d; and when d = 3, (Q) = c3 + ( f0(Q) − 2)dc + ( f1(Q) − f0(Q))cd. If we assign a weight of 1 to each c and 2 to each d, each word in the cd-index of a d-polytope has weight d. Denote the coefficient of the cd word w by [w]. Meisinger [15, Appendix D] generated a list of all [w] for 3 ≤ d ≤ 8. The cd-index behaves nicely under duality; [w](Q∗) = [w](Q), where w is w written backwards [4]. We can see this in dimension 3: [dc] = f0−2 and [cd] = f1 − f0 = f2 − 2, and f0(Q∗) = f2(Q). How are [wc] and [wd] related to [w]? To answer this question, we need to introduce yet another vector, the flag k-vector [6]. For S sparse, let kS = T ⊆S(−2)|S|−|T | fT . Billera, Ehrenborg, and Readdy [5] showed Theorem 8 [6]. Let w = cn1 dcn2 dcn3 · · · cnp dcnp+1 and define m0, . . . , m p by m0 = 0 and mi = mi−1 + ni + 2. Then [w] = (−1)(m1−i1)+(m2−i2)+···+(mp−ip)ki1,i2,...,ip , (33) i1,...,ip where the sum is over all p-tuples (i1, i2, . . . , ip) such that m j−1 ≤ ij ≤ m j − 2. For further discussion of the flag k-vector, see [9]. Lemma 6. If [w] = S∈Sp(d) αS fSd , then [wc] = S∈Sp(d) αS fSd+1. This is clear because n p+1 is not involved in the expression for [w] in terms of the flag k-vector. So, for example, we have [cdck ] = f1k+3 − f0k+3 for all k ≥ 0. Lemma 7. If [w] = S∈Sp(d) αS fSd , then [dw] = S∈Sp(d) αS( f{d0+}∪2(S+2) − 2 f(dS++22)). Proof. In the flag k-vector expression for [dw], n1 = 0, so i1 = 0. Then each (−1)(m1−i1)+(m2−i2)+···+(mp−ip)ki1,i2,...,ip in [w] corresponds to [w] contributes αS( f{d0+}∪2(S+2) − 2 f(dS++22)) to [dw]. in [dw]. Further, since every subset of {0, i1 + 2, i2 + 2, . . . , ip + 2} is either a subset of (S + 2) = {i1 + 2, i2 + 2, . . . , ip + 2} or {0} union such a subset, each fT in ki1,i2,...,ip corresponds to f{0}∪(T +2) − 2 f(T +2) in k0,i1+2,i2+2,...,ip+2. Therefore, αS fSd in For example, [cd] = f13 − f03 and [dcd] = f053 − 2 f35 − f052 + 2 f25. By duality, we have Corollary 1. Given [w] written in the sparse basis, we can find [wd]. Each αT fTd in [w] contributes αT ( fTd∪+{2d+1} − 2 fTd+2) to [wd]. Notice that fTd∪+{2d+1} is not part of the sparse basis, so to write [wd] in terms of that basis we need to apply the generalized Dehn–Sommerville equations. Stanley [20] showed that if w has weight d, then [w] is non-negative for all dpolytopes. Furthermore, Theorem 9 [5]. For Q a d-polytope, where d is the d-simplex. [w](Q) − [w]( d ) ≥ 0, So the cd-index gives us more linear inequalities for flag f -vectors. From Lemma 6, Corollary 1, and the Euler–Poincare´ relation, we can see that so Therefore we have [c j dcd− j−2] = fj − fj−1 + · · · + (−1) j f0 − 1 + (−1) j+1 fj = [c j dcd− j−2] + [c j−1dcd− j−1] + 2[cd ]. Theorem 10. For Q a d-polytope, Billera and Ehrenborg’s cd-index inequalities imply the following previously known (see Section 10.2 of [12]) inequalities: fj (Q) − fj ( d ) = fj (Q) − d + 1 j + 1 ≥ 0. If j ∈ {0, 1, d −2, d −1}, then this inequality can be expressed as a non-negative linear cfodd−m2b−inatddio+−n11 of=g-12co(gn1dv−o1lu∗tiogn00s:+f0dd g−d1 )(,da+nd1)fd=d−1g−1d, (fd1d +− 1d)+21= =gd112 [(1g500]∗. Ag1dd−d1it+iondagl1dly),, f25 − 15 = 23 (g15 + g15) + 21 (g00 ∗ g14 + g14 ∗ g00). We prove the following theorem and in the process prove Meisinger’s conjecture. Theorem 11. For Q a d-polytope, d ≥ 6, and 2 ≤ j ≤ d − 4, the inequality [c j dcd− j−2](Q) − [c j dcd− j−2]( d ) ≥ 0 is not implied by the non-negativity of the g convolutions. In his proof of the case d = 6, j = 2 of his conjecture, Meisinger introduced the point X6 = ( f0, f1, f2, f3, f4, f02, f03, f04, f13, f14, f24, f024) = ( 7, 21, 0, 0, 21, 105, 350, 315, 620, 840, 630, 2520 ), (34) (35) (36) which satifies all the inequalities generated by the g convolutions but obviously violates the inequality f2 − 35 ≥ 0. It also violates [ccdcc](Q) − [ccdcc]( d ) = f2 − f1 + f0 − 21 ≥ 0. Therefore, neither f2 − 35 nor f2 − f1 + f0 − 21 can be a non-negative combination of g convolutions. If we know the flag f -vector of a polytope Q, we can find the flag f -vector of P Q, the pyramid over Q. Using the conventions f−1 = 1 = fd and fS∪{i}∪{i} = fS∪{i} we have Lemma 8. If PQ is the pyramid over the polytope Q, then fi1,i2,...,ik (PQ) = fi1,i2,...,ik (Q) + fi1,...,ij ,ij+1−1,...,ik−1(Q) k j=1 and in particular fj ( P Q) = fj (Q) + fj−1(Q). Proof. Every face of P Q is either a face of Q or the pyramid over a face of Q. If a face Fl ⊂ P Q is the pyramid over Fl ⊂ Q and Fl ⊂ Fm ⊂ P Q, then Fm = P Fm for some Fm ⊂ Q. So every chain Fi1 ⊂ Fi2 ⊂ · · · ⊂ Fik in P Q corresponds to a chain Fi1 ⊂ · · · ⊂ Fij ⊂ Fij+1 ⊂ · · · ⊂ Fik in Q for some j , 0 ≤ j ≤ k − 1. We use Lemma 8 to define P yr (X ), the flag f -vector of the “pyramid” over any vector X of length ad − 1. Definition 3. Let X be a vector in the affine span of the flag f -vectors of d-polytopes. Define P yr (X ) by fi1,i2,...,ik ( P yr (X )) = fi1,i2,...,ik (X ) + fi1,...,ij ,ij+1−1,...,ik−1(X ). (39) k j=1 Lemma 9. Let X be a vector in the span of the flag f -vectors of (d − 1)-polytopes. Then gid11 ∗· · ·∗gidjj ∗gidjj++11−1∗gidjj++22 ∗· · ·∗gidkk−−11 ∗gidkk (X ), (40) gid11 ∗ i2 ∗· · ·∗gidkk ( P yr (X )) = gd2 where k − 1 + k j=1 dj = d. k−1 j=0 Proof. Let Q be a (d − 1)-polytope. Consider i1 ∗ gid22 ∗ · · · ∗ gidkk−−11 ∗ gidkk (PQ) gd1 where dim(Fl ) = l − 1 + (37) (38) (41) From the proof of Lemma 8, we know that for each chain F1 ⊂ F2 ⊂ · · · ⊂ Fk−1 ⊂ P Q there is a 0 ≤ j ≤ k − 1 such that (F1 ⊂ · · · ⊂ Fj ⊂ Fj+1 · · · ⊂ Fk−1 ⊂ PQ) = (F1 ⊂ · · · ⊂ Fj ⊂ PFj+1 · · · ⊂ PFk−1 ⊂ PQ), (42) where F1 ⊂ · · · ⊂ Fj ⊆ Fj+1 ⊂ · · · ⊂ Fk−1 ⊂ Q. It is easy to see that (PFl )/(PFl−1) = Fl /Fl−1. Therefore by Lemmas 3 and 1 we have gid1 (F1) · · · gijj (Fj /Fj−1)gidjj++11 (Fj+1/Fj ) · · · gidkk−−11 (Fk−1/Fk−2)gidkk (PQ/Fk−1) d 1 we get gid11 ∗ · · · ∗ gidjj ∗ gidjj++11−1 ∗ · · · ∗ gidkk−−11 ∗ gidkk (Q). Finally, if we sum over all j , we get i1 ∗ gid22 ∗ · · · ∗ gidkk−−11 ∗ gidk ( P Q) = gd1 k gid11 ∗ · · · ∗ gidjj ∗ gidjj++11−1 ∗ · · · ∗ gidkk−−11 ∗ gidkk (Q). (44) k−1 j=0 We have shown this equality for Q a (d − 1)-polytope. However, since it is true for all (d − 1)-polytopes, it is true for all points in the span of their flag f -vectors. Proof of Theorem 11. Starting with X6, recursively define Xd = P yr (Xd−1). We will show that Xd satisfies all the inequalities generated by the g convolutions but violates [c j dcd− j−2](Q)−[c j dcd− j−2]( d ) ≥ 0 for 2 ≤ j ≤ d −4, and therefore these cd-index inequalities are not implied by the g inequalities. Our proof is by induction on d. We have already shown that the statement holds for dimension 6, our base case. Suppose it is true for dimension d − 1. We know Lemma 9 holds for Xd−1. Since gid11 ∗ · · · ∗ gidjj ∗ gidjj++11−1 ∗ · · · ∗ gidkk−−11 ∗ gidkk (Xd−1) ≥ 0 forN0o≤te jth≤at kL−em1m,wae9 haalsvoe hgoid11ld∗s gifid22s∗o m··e· o∗fgtidhkk−e−11g∗ gidkk (Xd ) ≥ 0. i are replaced by gi , since we know gin( P Q) = gin(( P Q)∗) = gin( P(Q∗)) = gin−1(Q∗) = gin−1(Q). So for each dimension d ≥ 6, we have constructed a point Xd that satisfies all the d+1 g inequalities. Now we will show it violates fj − j+1 ≥ 0 for 2 ≤ j ≤ d − 3 and [c j dcd− j−2](Q) − [c j dcd− j−2]( d ) ≥ 0 for 2 ≤ j ≤ d − 4. In the case d = 6 we have fj (Xd ) = fj ( d ) for j ∈ {−1d, 0s,a1ti,sfdy−(328,),din−du1c,tdio}nanodn fj (Xd ) < fj ( d ) for 2 ≤ j ≤ d − 3 . Since both Xd and d gives us these statements for all d ≥ 6, so we have proved Meisinger’s conjecture. By (34) and (38), we have [c j dcd− j−2](Xd ) = fj (Xd ) − fj−1(Xd ) + · · · + (−1) j f0(Xd ) − 1 + (−1) j+1 Similarly, = ( fj (Xd−1) + fj−1(Xd−1)) − ( fj−1(Xd−1) + fj−2(Xd−1)) + · · · + (−1) j ( f0(Xd−1) + 1) − 1 + (−1) j+1 = fj (Xd−1) − 1. [c j dcd− j−2]( d ) = fj ( d−1) − 1. However, we have just shown that fj (Xd−1) < fj ( d−1) for 2 ≤ j ≤ (d − 1) − 3. So Xd satisfies all the g inequalities but violates [c j dcd− j−2](Q) − [c j dcd− j−2]( d ) ≥ 0 for 2 ≤ j ≤ d − 4, and therefore these inequalities are not implied by the non-negativity of the g convolutions. Meisinger [15, Theorem 4.6] asserts that for 3 ≤ d ≤ 10 every coefficient in the cdindex is a non-negative combination of g convolutions. Our Theorem 11 does not directly contradict this, since it involves [w](Q) − [w]( d ) ≥ 0 rather than [w](Q) ≥ 0. However, computations with Maple and PORTA indicate that [ccdcc], [ccdccc], [ccdccd], [ccdcccc], [cccdccc], and their duals are not non-negative combinations of g convolutions. 4. Some Toric g-Vector Inequalities That Are Not Implied by cd-Index Inequalities We now know that for d ≥ 6 not all of the cd-index inequalities are implied by the g inequalities. What about implications in the other direction? For large enough d, are all of the g inequalities implied by the cd inequalities? The short answer is “no.” The longer answer is Theorem 12. For d ≥ 3, the inequality g12 ∗ g0d−3 = f0d2 − 3 f2d ≥ 0 is not implied by the cd-index inequalities. (45) (46) Proof. Our proof follows the same outline as the proof of Theorem 11. We work by induction on d. Our base case is dimension d = 3. The cd inequalities are [ccc](Q) − [ccc]( d ) = 1 − 1 ≥ 0, [cd](Q) − [cd]( d ) = f1(Q) − f0(Q) − 2 ≥ 0, and [dc](Q) − [dc]( d ) = f0(Q) − 4 ≥ 0. Let Y3 = ( f03, f13) = ( 5, 10 ). Notice that Y3 satisfies the cd-index inequalities. Using the generalized Dehn–Sommerville equations, f032 − 3 f23 = 2 f13 − 0 + 2), so g12 ∗ g0d−3(Y3) ≥ 0 and this g inequality is not implied by the cd-index 3( f13 − f 3 inequalities. Now recursively define Yd = P yr (Yd−1). Suppose t1h∗atgY(dd−−11)−s3atisfies the (d − 1)dimensional cd-index inequalities but does not satisfy g2 0 ≥ 0. We will sh2ow that Yd satisfies all the d-dimensional cd-index inequalities but does not satisfy g1 ∗ g0d−3 ≥ 0. Ehrenborg and Readdy [11] studied the effect of pyramiding on the cd-index of a polytope and found Theorem 13 [11]. Let Q be a (d − 1)-polytope. Then 1 (PQ) = 2 ( (Q)c + c (Q) + D( (Q))), (47) where D is a derivation given by D(c) = 2d, D(d) = cd + dc, and D(wx) = D(w)x + wD(x) for w and x any words in c and d. As in our proof of Lemma 9, since this is true for all (d − 1)-polytopes, it is true for all points in the span of their flag f -vectors. Hence it is true for P yr (Yd−1) = Yd . By the induction hypothesis, we know that (Yd −d−11s)ahtiassfiensonal-lneogfatthiveecdco-ienfdfiecxieinntes.qTuahleitoireesm,an1d3 hence the polynomial (Yd−1) − snheogwatsivtehalitntehaerccooemffibciineanttisonosf of(tYhde)c−oeffi(ciedn)ts=of (P(Yydr−(Y1)d−−1)) −( d−(1P), adn−d1)heanrecenoanrealso non-negative. So Yd satisfies all of the cd-index inequalities. Now consider the g inequality. Using Definition 3 and the generalized Dehn–Sommerville equation f0d1−1 = 2 f1d−1, we have g12 ∗ g0d−3(Yd ) = f0d2(P yr (Yd−1)) − 3 f2d (P yr (Yd−1)) = f0d2−1(Yd−1) + f0d1−1(Yd−1) + f1d−1(Yd−1) − 3 f2d−1(Yd−1) − 3 f1d−1(Yd−1) = f02(Yd−1) − 3 f2(Yd−1), which is negative by the induction hypothesis. 5. Convolutions of cd-Index Inequalities It is natural to ask if convolutions of cd coefficients might generate more inequalities whose non-negativity is not implied by previously known inequalities. We will show that convolutions of cd coefficients can be written as non-negative linear combinations of other cd coefficients. This means that convolutions of inequalities of the form [w](Q) ≥ 0 yield nothing new. Convolutions of the [w](Q) ≥ [w]( d ) inequalities are still open to investigation. Theorem 14. For w and x words in c and d, we have [wd] ∗ [dx] = 2[wdcdx], [wc] ∗ [dx] = [wddx] + 2[wccdx], [wd] ∗ [cx] = [wddx] + 2[wdccx], [wc] ∗ [cx] = [wcdx] + [wdcx] + 2[wcccx]. Proof. We begin with the base cases: [d] ∗ [d] = ( f02 − 2 f∅2) ∗ ( f02 − 2 f 2) = −2 f052 + 2 f053 − 4 f35 + 4 f25 = 2[dcd], ∅ [c] ∗ [d] = ( f∅1) ∗ ( f02 − 2 f 2) = f042 − 2 f14 = [dd] + 2[ccd], ∅ [d] ∗ [c] = ( f02 − 2 f∅2) ∗ ( f 1) = f042 − 2 f24 = [dd] + 2[dcc], ∅ [c] ∗ [c] = ( f∅1) ∗ ( f∅1) = f13 = [cd] + [dc] + 2[ccc]. Now we prove that [u] ∗ [vc] = [([u] ∗ [v])c] and [du] ∗ [v] = [d([u] ∗ [v])]. Say [u] = S∈Sp(m) αS fSm and [v] = T ∈Sp(n) βT fTn. By Lemma 6, we know that [vc] = T ∈Sp(n) βT fTn+1. So [u] ∗ [v] = αSβT fSm ∗ fT = n S∈Sp(m) T ∈Sp(n) S∈Sp(m) T ∈Sp(n) αSβT fSm∪+{mn+}∪1(T +m+1) (52) and S∈Sp(m) T ∈Sp(n) [u] ∗ [vc] = αSβT fSm∪+{mn+}∪2(T +m+1). (53) Since S and T are sparse, if 0 ∈ T , then S ∪ {m} ∪ (T + m + 1) is sparse. If T = {0} ∪ T , the generalized Dehn–Sommerville equations that express f m+n+1 and f m+n+2 S∪{m,m+1}∪(T +m+1) S∪{m,m+1}∪(T +m+1) in terms of the sparse basis are the same. So if [u] ∗ [v] = U∈Sp(m+n+1) γU fUm+n+1, then [u] ∗ [vc] = Lemma 6 we also have [([u] ∗ [v])c] = [([u] ∗ [v])c]. Now consider [du] ∗ [v]. By Lemma 7, if [u] = U∈Sp(m+n+1) γU fUm+n+2. However, by U∈Sp(m+n+1) γU fUm+n+2, so [u] ∗ [vc] = S∈Sp(m) αS fSm , then [du] = S∈Sp(m) αS( f{m0}+∪2(S+2) −2 f(mS++22)). So we have [u]∗[v] = S∈Sp(m) T ∈Sp(n) αSβT fSm ∗ fTn and [du] ∗ [v] = S∈Sp(m) T ∈Sp(n) αSβT ( f{m0}+∪2(S+2) − 2 f(mS++22)) ∗ fTn. As in the [u] ∗ [vc] case, we apply the generalized Dehn–Sommerville equations if 0 ∈ T . So if [u] ∗ [v] = U∈Sp(m+n+1) γU fUm+n+1, then we have [du] ∗ [v] = U∈Sp(m+n+1) γU ( f{m0}+∪n(U++32) − 2 f(mU++n2+)3). However, by Lemma 7 we also have [d([u] ∗ [v])] = U∈Sp(m+n+1) γU ( f{m0}+∪n(U++32) − 2 f(mU++n2+)3), so [du] ∗ [v] = [d([u] ∗ [v])]. By duality, we also have [cu] ∗ [v] = [c([u] ∗ [v])] and [u] ∗ [vd] = [([u] ∗ [v])d]. To show that [wd] ∗ [dx] = 2[wdcdx], we start with our base case [d] ∗ [d] = 2[dcd] and multiply by appropriate c’s and d’s on the right and left until we have w and x. The same holds for [wc] ∗ [dx], [wd] ∗ [cx], and [wc] ∗ [cx], and thus we have proved the theorem. This means that the coefficients of the cd-index form an algebra; in fact, they generate the algebra defined by Billera and Liu in [7]. Reading [16, Proposition 21] also proved (48) using a change of basis argument. Ehrenborg [10] suggested that all of Theorem 14 could be proved using coproducts. Let δw(Q) = [w](Q) if wei ght (w) = di m(Q) and 0 otherwise. Then we can write [wd] ∗ [dx](Q) = ∅⊂F⊂Q δwd(F )δdx(Q/F ) and similar statements for the remaining three cases. Ehrenborg and Readdy [11] showed that the cd-index is a Newtonian coalgebra map, and this plus some straightforward calculations will also prove Theorem 14. Acknowledgments References Many thanks to Lou Billera for his generous feedback on this work. Thanks also to Richard Ehrenborg and Margaret Bayer for their comments on earlier versions of this paper, and thanks to the anonymous referee for posing the question addressed in Section 4 and considerably shortening the proof of Theorem 5. 1. M. Bayer and L. J. Billera , Counting faces and chains in polytopes and posets . Contemp. Math. 34 ( 1984 ), 207 - 252 . 2. M. Bayer and L. J. Billera , Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets . Invent. Math . 79 ( 1985 ), 143 - 157 . 3. M. Bayer and R. Ehrenborg , On the toric h-vectors of partially ordered sets . Trans. Amer. 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Catherine Stenson. Relationships Among Flag f-Vector Inequalities for Polytopes, Discrete & Computational Geometry, 2003, 257-273, DOI: 10.1007/s00454-003-0756-0