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CHARACTERIZING F-VECTOR Contents 1. Prehistory 1 2. Klee CHARACTERIZING f-VECTOR AHMED UMER ASHRAF Abstract. Given a d-dimensional convex polytope, its kth face number is the number of (k − 1)-dimensional faces it has. The f-vector of a polytope is the sequence of its face numbers. Beside Euler's formula, these numbers satisfy further equalities and inequalities. Characterization of f-vector of d-dimensional convex polytope is already known for d ≤ 3. For d ≥ 3, we do not have a complete answer, but g-theorem gives us a characterization for simplicial (and dually simple) case. Here we review g-theorem and its various proofs. Contents 1. Prehistory 1 2. Klee-Dehn-Sommerville Theorem4 3. Kruskal-Katona-Sch¨utzenberger Theorem5 4. Stanley-Macaulay Theorem6 5. Necessity of g-theorem7 5.1. Stanley's approach7 5.2. McMullen's approach8 5.3. Timorin's presentation9 6. Frontiers 10 References 11 1. Prehistory There has been an interest in convex polytopes from the time of Greeks as far as we know. This is evident from complete characterization of Platonic solids in Euclid's Elements, which are examples of 3-dimensional polytopes. Euler in 1752 observed the remarkable relation between number of vertices v, edges e and faces f of a convex 3-dimensional polytope v − e + f = 2 Though there is evidence that Descartes already knew this around 1640. This formula was generalized by Schl¨afli[Sch50] for arbitrary d-dimensional convex polytopes in 1852. Schl¨afli in his proof assumed shellability of convex d-polytopes which was not proved until 1971 by Bruggesser and Mani [MB71]. Though before that Poincar´e[Poi93] developed the machinery of Algebraic topology and gave the first proof in 1893 of what now is known as Euler-Poincar´e- Schl¨afliformula d X k+1 (−1) fk = 0 k=0 for any convex d-polytope P , where fk is number of k − 1 dimensional faces of P . 1 2 AHMED UMER ASHRAF Definition 1.1. Given an abstract simplicial complex or a convex polytope K, we can define the f-vector of K as f(K) = (f0; f1; f2;::: ) where fk = # σ 2 K : dim(σ) = k − 1 i.e. the number of (k − 1)-dimensional faces. We call fk's the face numbers of K. We also define the f-polynomial of K to be dim(K)+1 X dim(σ)+1 X k fK(t) = t = fkt σ2K k=0 Example 1.2. Following are some face numbers computed via elementary counting arguments • For 3-simplex ∆3, we have f(∆3) = (1; 4; 6; 4; 1) and in general d + 1 d + 1 d + 1 d + 1 f(∆ ) = 1; ; ;:::; ; ; 1 d 1 2 d − 1 d d+1 f∆d (t) = (1 + t) • For 3-cube 3, we have f(3) = (1; 8; 12; 6; 1), and in general d d d f( ) = 1; 2d ; 2d−1 ;:::; 2 ; 1 d 0 1 d − 1 f (t) = 1 + t(2 + t)d d • For octahedron ♦3, we have f(♦3) = (1; 6; 12; 8; 1) and in general d d d f( ) = 1; 2 ; 22 ;:::; 2d ; 1 ♦d 1 2 0 f (t) = (2t + 1)d + td+1 ♦d Notice that the duality between the f(d) and f(♦d) is the consequence of the duality between d and ♦d in general. ♦ d+2 Let's ask ourselves what vector (f0; f1; ··· ; fd; fd+1) 2 Z can arise as f-vectos of a d- dimensional convex polytope. Note that f0; fd+1 are always 1. For the sake of brevity, let's denote v := f1, e := f2 and f := f3, for the immediate discussion • For d = 0, the only convex polytopes are points, which means the only f-vector is (1; 1). • For d = 1, the only convex polytopes are closed intervals, which implies the only f-vector is (1; 2; 1). • For d = 2, the only convex polytopes are convex polygons, whose f-vector can be characterized by vectors of the form (1; v; e; 1) such that v = e v; e ≥ 3 CHARACTERIZING f-VECTOR 3 • For d = 3, we have a characterization of f-vector of 3-polytopes due to Ernst Steinitz [Ste10] from 1922. It says that any vector of the form (1; v; e; f; 1), satisfying v − e + f = 2 (Euler's formula) v; f ≥ 4 2e ≥ 3f; 2e ≥ 3v is an f-vector of a 3-polytope and vice versa. • For d = 4, we don't have a complete answer yet. Notice that the Euler relation is a constrained from topology and the inequalities are the constrained from convexity of polytopes. Definition 1.3. The h-vector of K as h(K) = (h0; h1; h2;::: ) where k X d − i h = (−1)k−if k i k − i i=0 where fi's are face numbers of K. More compactly, we can define i+j d − j Md = (−1) i − j 0≤i;j≤d then the definition read as h(K) = Mdf(K) We also define the h-polynomial of K to be dim(K)+1 X k hK(t) = hkt k=0 Example 1.4. Going back to our running example of simplex, cube and octahedron, we have • For 3-simplex ∆3, we have h(∆3) = (1; 0; 0; 0; 0) and in general h(∆d) = (1; 0;:::; 0) h∆d (t) = 1 Notice that h(@∆d) = (1; 1;:::; 1). • For 3-cube 3, we have h(3) = (1; 4; −6; 2; 0), and h(@3) = (1; 5; −1; 1) • For octahedron ♦3, we have f(♦3) = (1; 2; 0; −2; 0) and h(@♦3) = (1; 3; 3; 1) ♦ Remark 1.5. Keeping in view the definition of f-vector and h-vector of a simplicial complex, we should notice the following facts: 4 AHMED UMER ASHRAF • f0 = h0 = 1 for all simplicial complexes. Further, one may notice f∆(−1) = −χe(∆) d hd = (−1) χe(∆) • The transformation matrix Md is invertible with inverse given by −1 d − j Md = i − j 0≤i;j≤d • Another way to state the relationship between f-polynomial and h-polynomial is d X t f(t) = h ti(1 + t)d−i = (1 + t)d · h i 1 + t i=0 d X t h(t) = f ti(1 − t)d−i = (1 − t)d · f i 1 − t i=0 2. Klee-Dehn-Sommerville Theorem Noticing the palindromicity of h(K), whenever K happen to be the boundary of a simplical polytope, Max Dehn [Deh04] (in 1904) and Duncan Sommerville [Som27] (in 1927), showed that hk(K) = hd−k(K) for all k = 0; : : : ; d, whenever K is a simplicial sphere of dimension (d − 1). Victor Klee [Kle64a] generalized this result in 1964, to any Eulerian complex of dimension (d − 1). Here, we state the formulation of Klee. Theorem 2.1. (Klee-Dehn-Sommerville Equations, KDS equations) [Kle64a, Deh04, Som27, Sta12] Suppose ∆ is a pure Eulerian complex of dimension d − 1 and h(∆) = (h0; h1; : : : ; hd), where h0 = 1, then for each k ≥ 0 d h − h = (−1)k 1 − (−1)dχ(∆) k d−k k e d−1 In paricular, when χe(∆) = (−1) , then hk = hd−k We also mention the generalization of KDS for Eulerian posets. Since face lattice of convex polytopes is also Eulerian, this can be viewed as KDS for convex polytopes. For its formulation, we need to define the graded version of f-vector Definition 2.2. Let P be a finite graded poset of rank n, with rank function rk : P −! [n]. • The flag f-vector (fS : S ⊆ [n]) is defined as fS = number of chains C such that rk(C) is given by S • The flag h-vector (hS : S ⊆ [n]) is defined as X jS|−|T j hS = (−1) fT T ⊆S • The ab-index of P is defined as X ΦP (a; b) = hSuS S⊆[n] CHARACTERIZING f-VECTOR 5 where uS = u1u2 ··· un and ( a if i2 = S ui = b if i 2 S Theorem 2.3. [BB85, BK91, Sta94] The ab-index of an Eulerian poset P , can be expressed as a unique polynomial Ψ(c; d) called the cd-index of P , where c = a + b d = ab + ba Furthermore, if P is the face poset of a polytope, then the coefficients of Ψ(c; d) are positive. So now we can state the analogue of KDS equations in case of general convex polytope. Corollary 2.4. For an Eulerian poset, we have hS = hSc where Sc denote the complement of S in [n]. 3. Kruskal-Katona-Schutzenberger¨ Theorem There is a complete characterization of f-vector of abstract simplicial complexes given by Kruskal-Katona-Sch¨utzenberger Theorem. These are worked out by Joseph Kruskal in 1963 [Kru63] and Gyula Katona in 1966 [Kat68]. Though Marcel-Pail Sch¨utzenberger already described them in 1959 in a technical report for MIT's Research Laboratory of Electronics [Sch59]. To state these, we need a couple of definitions: Theorem 3.1. Given n; k 2 Z+, there exists a unique representation of n in the form a a a n = k + k−1 + ··· + r k k − 1 r where ak > ak−1 > : : : > ar ≥ r ≥ 1. This representation of n is call the k-cascade representation of n. Given such a representation of n, define • kth Kruskal-Katona function @k @k : N −! N 0 7! 0 a a a a a a n = k + k−1 + ··· + r 7! k + k−1 + ··· + r k k − 1 r k − 1 k − 2 r − 1 • kth Macaulay function @k k @ : N −! N 0 7! 0 a a a a − 1 a − 1 a − 1 n = k + k−1 + ··· + r 7! k + k−1 + ··· + r k k − 1 r k − 1 k − 2 r − 1 • We call n 2 N(1) a K-sequence if @k(nk) ≤ nk−1 for all k ≥ 1.
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