Pure and Applied Mathematics Quarterly Volume 2, Number 3 (Special Issue: In honor of Robert MacPherson, Part 1 of 3 ) 655|671, 2006 Nested Complexes and their Polyhedral Realizations Andrei Zelevinsky To Bob MacPherson on the occasion of his 60th birthday 1. Introduction This note which can be viewed as a complement to [9], presents a self-contained overview of basic properties of nested complexes and their two dual polyhedral realizations: as complete simplicial fans, and as simple polytopes. Most of the results are not new; our aim is to bring into focus a striking similarity between nested complexes and associated fans and polytopes on one side, and cluster complexes and generalized associahedra introduced and studied in [7, 2] on the other side. First a very brief history (more details and references can be found in [10, 9]). Nested complexes appeared in the work by De Concini - Procesi [3] in the context of subspace arrangements. More general complexes associated with arbitrary ¯nite graphs were introduced and studied in [1] and more recently in [10]; these papers also present a construction of associated simple convex polytopes (dubbed graph associahedra in [1], and De Concini - Procesi associahedra in [10]). An even more general setup developed by Feichtner - Yuzvinsky in [6] associates a nested complex and a simplicial fan to an arbitrary ¯nite atomic lattice. In the present note we follow [5] and [9] in adopting an intermediate level of generality, and studying the nested complexes associated to building sets (see De¯nition 2.1 below). Feichtner and Sturmfels in [5] show that the simplicial fan associated to a building set is the normal fan of a simple convex polytope, while Postnikov in [9] gives an elegant construction of this polytope as a Minkowski sum of simplices. Received July 29, 2005. 2000 Mathematics Subject Classi¯cation.Primary 52B. Research supported by NSF grant DMS-0500534. 656 Andrei Zelevinsky In this note we develop an alternative approach, constructing the polytope di- rectly from the fan; this construction is completely analogous to that in [2]. The main role in this approach is played by the link decomposition property (Propo- sition 3.2 below) asserting that the link of any element in a nested complex can be naturally identi¯ed with a nested complex of smaller rank. This result pro- vides a uni¯ed method for establishing properties of nested complexes and their polyhedral realizations by induction on the rank of a building set. The note is structured as follows. Section 2 introduces (basically following [9]) building sets and associated nested complexes. Section 3 presents the link decomposition property (Proposition 3.2). In this generality it seems to be new; for the graph associahedra it appeared in [10]. Section 4 contains some struc- tural properties of nested complexes, all of them derived in a uni¯ed way from Proposition 3.2. In Section 5 we show that any nested complex can be realized as a smooth complete simplicial fan (Theorem 5.1 and Corollary 5.2); we refer to this fan as the nested fan. This construction has appeared in [6, 4, 5]. However, it is not so easy to extract from these papers a proof that the nested fan is well-de¯ned; this is why we prefer to give a simple self-contained proof. In Section 6, we prove (Theorem 6.1) that the nested fan is the normal fan of a simple polytope ¦, which we call the nested polytope. The argument is completely parallel to that in [2]; in particular, it leads to a speci¯c realization of ¦ which can be shown to be the same as the realization given in [9]. The concluding Section 7 discusses the special class of graphical building sets, those giving rise to graph associahedra [1, 10]. We give a simple characterization of graphical building sets (Proposition 7.3). For them an analogy with cluster complexes becomes sharper: in particular, the corresponding nested complex is a clique complex (Corollary 7.4). 2. Nested complexes We start by recalling basic de¯nitions from [9, Section 7]. De¯nition 2.1 ([9], De¯nition 7.1). Let S be a nonempty ¯nite set. A building set (or simply a building) on S is a collection B of nonempty subsets of S satisfying the conditions: (B1) If I;J 2 B and I \ J 6= ;, then I [ J 2 B. (B2) B contains all singletons fig, for i 2 S. The nested complex associated to a building B is de¯ned as follows. Let Bmax ½ B be the set of maximal (by inclusion) elements of B. We refer to the elements of Bmax as the B-components. In view of (B1) and (B2), the B-components Nested Complexes 657 are pairwise disjoint, and their union is S. We de¯ne the rank of B by setting rk(B) = jSj ¡ jBmaxj. De¯nition 2.2 (cf. [9], De¯nition 7.3). A subset N ½ B ¡ Bmax is called nested if it satis¯es the following condition: for any k ¸ 2 and any I1;:::;Ik 2 N such that none of the Ii contains another one, the union I1 [ ¢ ¢ ¢ [ Ik is not in B. The simplicial complex on B¡Bmax formed by the nested sets is called the nested complex and denoted N (B). In particular, the 1-simplices of N (B) are pairs fI;Jg ½ B such that: (2.1) either I ½ J, or J ½ I, or I [ J2 = B (in view of (B2), in the last case I and J are disjoint). Note that our de¯nition of nested sets is di®erent from the one in [9, De¯nition 7.3]: Postnikov's nested sets are obtained from ours by adjoining the set Bmax. Of course this does not a®ect the poset structure of the nested complex; an advantage of the present version is that our nested sets form a simplicial complex. 3. Link decomposition Let B be a building on S, and N (B) the associated nested complex. For every C 2 B ¡ Bmax, let 0 0 (3.1) N (B)C = fN ½ B ¡ Bmax ¡ fCg : N [ fCg 2 N (B)g denote the link of C in N (B); this is a simplicial complex on the set fI 2 B¡Bmax ¡ fCg : fC; Ig 2 N (B)g. Our main tool in studying the structure of N (B) is the recursive description of N (B)C given in Proposition 3.2 below. To state it, we need some preparation. De¯nition 3.1. Let C be a nonempty subset of S. ² The restriction of B to C is the building on C de¯ned by (3.2) BjC = fI ½ C : I 2 Bg: ² The contraction of C from B is the building on S ¡ C de¯ned by (3.3) CnB = fI ½ S ¡ C : I 2 B or C [ I 2 Bg: The fact that both BjC and CnB are indeed buildings, is immediate from De¯nition 2.1. Let S1;:::;Sk be disjoint ¯nite sets, and let Bi be a building on Si for i = 1; : : : ; k. The product B1 £ ¢ ¢ ¢ £ Bk is de¯ned as the building on S1 t ¢ ¢ ¢ t Sk formed by the elements of all the Bi. Clearly, the nested complex N (B1 £¢ ¢ ¢£Bk) is naturally isomorphic to the product N (B1) £ ¢ ¢ ¢ £ N (Bk). In particular, every building B on S decomposes as a product BjS1 £¢ ¢ ¢£BjSk , where S1;:::;Sk are the B-components; therefore, N (B) can be identi¯ed with N (BjS1 ) £ ¢ ¢ ¢ £ N (BjSk ). 658 Andrei Zelevinsky Proposition 3.2 (Link Decomposition). For every C 2 B¡Bmax, the link N (B)C is isomorphic to N (BjC £ CnB). 0 Proof. Fix C 2 B ¡ Bmax and abbreviate B = BjC £ CnB. Let B0 = fI 2 B ¡ Bmax ¡ fCg : fC; Ig 2 N (B)g be the ground set of N (B)C . In view of (2.1), B0 is the disjoint union of the following three sets: B1 = fI 2 B ¡ fCg : I ½ Cg;(3.4) B2 = fI 2 B ¡ Bmax ¡ fCg : C ½ Ig;(3.5) (3.6) B3 = fI 2 B ¡ Bmax : C \ I = ;;C [ I2 = Bg: 0 Now to every I 2 B0 = B1 [B2 [B3 we associate a subset I ½ S by setting ( I if I 2 B1 [B3; (3.7) I0 = I ¡ C if I 2 B2. Remembering the de¯nitions, we see that B1 is the ground set of the nested complex N (BjC ), and B3 is a subset of the ground set of N (CnB) formed by those I0 ½ S ¡C for which C [I0 2= B. Furthermore, the correspondence I 7! I0 = I ¡C identi¯es B2 with the remaining part of the ground set of N (CnB), formed by those I0 ½ S ¡ C for which C [ I0 2 B. Thus, the correspondence I 7! I0 is a 0 bijection between the ground sets of N (B)C and N (B ). To prove Proposition 3.2, it remains to show the following: The correspondence I 7! I0 induces an(3.8) 0 isomorphism of complexes N (B)C and N (B ). Thus, we need to show the following: for any distinct I1;:::;Ik 2 B0, the 0 0 0 set fC; I1;:::;Ikg is nested for B if and only if fI1;:::;Ikg is nested for B . 0 0 0 First suppose fI1;:::;Ikg is not nested for B . Without loss of generality, we 0 0 can assume that k ¸ 2, none of the Ii contains another one, and I1 [ ¢ ¢ ¢ [ 0 0 Ik 2 B . Choosing the smallest possible k with this property and using property 0 (B1) in De¯nition 2.1, we can further assume that the Ii are pairwise disjoint.