Documenta Math. 113 Equivariant Fiber Polytopes To the memory of Rodica Simion. Victor Reiner Received: February 15, 2002 Communicated by GunÄ ter M. Ziegler Abstract. The equivariant generalization of Billera and Sturmfels' ¯ber polytope construction is described. This gives a new relation between the associahedron and cyclohedron, a di®erent natural con- struction for the type B permutohedron, and leads to a family of order-preserving maps between the face lattice of the type B permu- tohedron and that of the cyclohedron 2000 Mathematics Subject Classi¯cation: 52B12, 52B15 Keywords and Phrases: ¯ber polytope, associahedron, cyclohedron, equivariant Contents 1. Introduction 114 2. Equivariant polytope bundles 115 2.1. Group actions on polytope bundles 115 2.2. Equivariant ¯ber polytopes 119 2.3. Equivariant secondary polytopes 120 3. Small, visualizable examples 121 3.1. The standard example. 121 3.2. Triangulations of a regular hexagon. 122 4. Application examples. 124 4.1. Cyclohedra and associahedra. 124 4.2. The type B permutohedron. 125 4.3. Maps from the type B permutohedron to the cyclohedron. 126 5. Remarks/Questions 129 Acknowledgements 131 References 131 Documenta Mathematica 7 (2002) 113{132 114 Victor Reiner (a) (b) Figure 1. The 3-dimensional associahedron, containing the 2-dimensional cyclohedron as a planar slice. 1. Introduction Much of this paper is motivated by a new relationship between two families of convex polytopes which have appeared in diverse places within topology, geometry, and combinatorics [4, 10, 12, 13, 16]: the associahedra (or Stashe® polytopes), and the cyclohedra (or type B associahedra). There is already a well-known relation between these two families: any face in a cyclohedron is combinatorially isomorphic to a Cartesian product of a lower- dimensional cyclohedron along with a collection of lower-dimensional associa- hedra [16, 3.3], [6, Proposition 3.2.1]. Our point of departure is a di®erent relationship,x illustrated in Figure 1. The 3-dimensional associahedron is de- picted in Figure 1(a), with its vertices indexed by triangulations of a (centrally symmetric) hexagon. Figure 1(b) shows inside it a hexagonal slice representing the 2-dimensional cyclohedron, with vertices indexed by the subset of triangu- lations possessing central symmetry { this slice is the invariant subpolytope for a reection symmetry acting on the associahedron, which swaps two vertices if they correspond to triangulations that di®er by 180± rotation. Our main result (Theorem 2.10) asserts a similar relationship generally when one considers subdivisions of a polytope which are invariant under some ¯nite group of symmetries. The theory of ¯ber polytopes introduced by Billera and Sturmfels [1] shows that whenever one has a linear surjection of convex poly- topes P ¼ Q, there is an associated convex polytope §(P ¼ Q), called the ¯ber polytop! e. This ¯ber polytope has dimension dim P dim!Q, and its faces correspond (roughly) to the subdivisions of Q by cells ¡which are projections of families of faces of P . Our result says that when the projection P ¼ Q is G-equivariant for some ¯nite group G acting as symmetries on both P!and Q, then G acts as a group of symmetries on §(P ¼ Q), and the G-invariant ! Documenta Mathematica 7 (2002) 113{132 Equivariant Fiber Polytopes 115 subpolytope §G(P ¼ Q) (the equivariant ¯ber polytope) is a polytope of di- mension dim P G dim! QG whose faces correspond to those subdivisions which are G-invariant. ¡ The paper is structured as follows. Section 2 proves the main results, with Subsection 2.1 containing the technical details needed to generalize ¯ber poly- topes to the equivariant setting. Theorem 2.10 on the existence and dimension of the equivariant ¯ber polytope is deduced in Subsection 2.2. The special case of equivariant secondary polytopes is discussed in Subsection 2.3. Section 3 contains some low-dimensional examples that are easily visualized, while Section 4 gives some general examples as applications. In particular, Example 4.1 explains the above relation between associahedra and cyclohedra, and Example 4.2 explains how the type B permutohedron (that is, the zono- tope generated by the root system of type B) occurs as an equivariant ¯ber polytope. In Section 4.3 we answer a question of R. Simion, by exhibiting a family of natural maps between the face lattices of the Bn-permutohedron and n-dimensional cyclohedron. Section 5 lists some remarks and open questions. 2. Equivariant polytope bundles 2.1. Group actions on polytope bundles. For convenience, we work with the same notation as in [1, 1] in working out the equivariant versions of the same results. x Let Q be a polytope bundle, that is, Q is a convex polytope in Rd, and B ! n for each x in Q, the set x is a convex polytope in R , such that the graph B n+d x x : x Q is a bounded Borel subset of R . We further assume that fB £ 2 g we have a ¯nite group G acting linearly on both Rd and Rn. S Definition 2.1. Say that Q is a G-equivariant polytope bundle if B ! G acts as symmetries of Q, i.e. g(Q) = Q for all g in G. In particular, ² without loss of generality, the centroid of Q is the origin 0. for every x in Q and g in G one has = g( ). ² Bg(x) Bx Alternatively, equivariance of a polytope bundle is equivalent to G-invariance of Q, along with G-invariance of with respect to a natural G-action on polytope bundles: given Q and g inBG, let g( Q) be the polytope bundle de¯ned B ! B ! by g( Q) := g( 1 ). B ! x Bg¡ x The linear action of G on Rn induces a contragredient action on the dual space n 1 n n (R )¤ of functionals: g(Ã)(x) := Ã(g¡ (x)) for g G; y R ; à (R )¤. 2 2 2 A section of Q is a choice of (x) x for each x. If Q is B ! 2 B B !1 equivariant, there is a G-action on sections de¯ned by g()(x) := g(g¡ x). For any of these G-actions, de¯ne the averaging (or Reynolds) operator 1 ¼ = g G G g G j j X2 which is an idempotent projector onto the subset (or subspace) of G-invariants, n n G n n G e.g. ¼ maps R (R ) and maps (R )¤ ((R )¤) . ! ! Documenta Mathematica 7 (2002) 113{132 116 Victor Reiner It turns out that much of the reason that ¯ber polytopes interact well with ¯nite group actions boils down to ¼G being a linear operator which is a convex combination of the group operations g in G. Recall that for a polytope bundle Q, the Minkowski integral is the B ! Q B subset of n consisting of all integrals of measurable sections , and that R Q R this is a non-empty compact, convex subset of n. R R Proposition 2.2. Integration commutes with the G-action on sections of a G-equivariant polytope bundle Q: B ! g = g ZQ ZQ ¶ for all g in G. Proof. 1 g = g((g¡ x))dx ZQ ZQ 1 = g (g¡ x)dx ZQ ¶ = g (u)du ZQ ¶ Here the second equality uses linearity of g and linearity of integration. The third equality comes from the change of variable x = g(u), using the fact that the Jacobian determinant for this change of variable is det(g), which must be 1, since g is an element of ¯nite order in GL(Rn). § ¤ Corollary 2.3. For any G-equvariant polytope bundle Q, the group G acts on the convex set . B ! Q B Furthermore, one has R G n G := (R ) B B \ ZQ ¶ ZQ ¶ (2.1) = ¼ G B ZQ ¶ = : G-equivariant, measurable sections ; ½ZQ Proof. Proposition 2.2 implies the ¯rst assertion. For the second equality in (2.1), we claim more generally that n G C (R ) = ¼G(C) \ for any convex G-invariant subset C Rn. To see this, note that the left-hand ½ side is contained in the right because of idempotence of ¼G. The right-hand side is obviously contained in (Rn)G. It also lies in C since for any x in C, the 1 convex combination ¼G(x) = G g G g(x) will also lie in C. j j 2 P Documenta Mathematica 7 (2002) 113{132 Equivariant Fiber Polytopes 117 For the third equality in (2.1), note that the right-hand side is contained in the left since the integral Q of any G-equivariant section will be a G-invariant point of n by Proposition 2.2. Conversely, a typical point on the left-hand R R side is where is a section such that g( ) = for all g G, and Q Q Q 2 one can check using Proposition 2.2 that the G-equivariant section ¼ has the R R R G same integral: 1 ¼ = g G G Q g G Q Z j j X2 Z 1 = g G g G Q ¶ j j X2 Z = ZQ ¤ G We wish to interpret faces of in terms of face bundles. Recall that a Q B face bundle Q of Q ³isRa p´olytope bundle in which x is a face of x for every x inF Q!. A coheB r!ent face bundle of Q is one of theF form à BQ à à n B ! ÃB ! having x := ( x) , where à (R )¤ is any linear functional and P denotes the faceBof P onB which the functio2 nal à is maximized. When Q is G-equivariant, the G-action on bundles restricts to a G-action on faceBbundles,! and as before, a face bundle is G-equivariant if and only if it is invariant under the G-action.