The polytope algebra of generalized permutahedra

The polytope subalgebra of deformations of a zonotope can be endowed with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. We explore this construction and find relations between statistics on (signed) permutations and the module structure in the case of (type B) generalized permutahedra. In type B, the module structure surprisingly reveals that any family of generators (via signed Minkowski sums) for generalized permutahedra of type B will contain at least $2^{d-1}$ full-dimensional polytopes. We find a generating family of simplices attaining this minimum. Finally, we prove that the relations defining the polytope algebra are compatible with the Hopf monoid structure of generalized permutahedra.


Introduction
Generalized permutahedra serve as a geometric model for many classical (type A) combinatorial objects and have been extensively studied in resent years. Notably, Aguiar and Ardila [1] endowed generalized permutahedra with the structure of a Hopf monoid GP in the category of species and, in so doing, gave a unified framework to study similar algebraic structures over many different families of combinatorial objects. At the same time, McMullen's polytope algebra [21] offers a different algebraic perspective to study generalized permutahedra.
One of the goals of the present paper is to investigate the compatibility between both structures. To achieve this goal, we take a more general approach and study deformations of an arbitrary zonotope. We then specialize our results to deformations Coxeter permutahedra of type A and type B, revealing remarkable connections with (type B) Eulerian polynomials and statistics over (signed) permutations. The results in type B allow us to find a family of generalized permutahedra that generates all other deformations via signed Minkowski sums, thus solving a question posed by Ardila, Castillo, Eur, and Postnikov [7].
Let V be a finite-dimensional real vector space. The polytope algebra Π(V ) is generated by the classes [ For a fixed polytope p ⊆ V , Π(p) denotes the subalgebra of Π(V ) generated by classes of deformations of p; see [22]. We are particularly interested in the case where p is a zonotope corresponding to a linear hyperplane arrangement A. In this case, let Σ[A] denote the Tits algebra of A, see Section 3. It is linearly generated by the elements H F as F runs through the faces of the arrangement. The following is a central result of this paper. Theorem 6.11. Every type B generalized permutahedron in R d can be written uniquely as a signed Minkowski sum of the simplices ∆ 0 S for involution-exclusive subsets S. This document is organized as follows. We review McMullen's construction in Section 2. In Section 3, we recall the definition of the Tits algebra of a hyperplane arrangement and some of its representation theoretic properties. Characteristic elements and Eulerian idempotents are also reviewed in this section. In Section 4, we begin the study of the polytope algebra of deformations of a zonotope as a module over the Tits algebra of the corresponding hyperplane arrangement, which is the central construction for this paper. We study the polytope algebra of generalized permutahedra in Section 5. In particular, we provide a conjectural basis of simultaneous eigenvectors for the action of the Adams element and positive dilations on the module Π(π d ). Section 6 contains the analogous results for type B. We also give explicit sets of signed Minkowski generators for type B generalized permutahedra; one shows that the lower bound on the number of full-dimensional polytopes in such a collection is tight, and the other is invariant under the action of B d . In Section 7 we prove that the valuation and translation invariance relations are compatible with the Hopf monoid structure of GP. Section 8 contains some final remarks and questions. We begin with some preliminaries.
1.1. Preliminaries. Let V be a real vector space of dimension d endowed with an inner product · , · , and let 0 ∈ V denote its zero vector. For a polytope p ⊆ V and a vector v ∈ V , let p v denote the face of p maximized in the direction v. That is, The (outer) normal cone of a face f of p is the polyhedral cone There is a natural order-reversing correspondence between faces of p and cones in Σ p . For F ∈ Σ p , we let p F ≤ p denote the face whose normal cone is F . That is p F = p v for any v in the relative interior of F .  Recall that a fan Σ refines Σ ′ if every cone in Σ ′ is a union of cones in Σ. We say that a polytope q is a deformation of p if Σ p refines Σ q . The Minkowski sum of two polytopes p, q ⊆ V is the polytope p + q := {p + q : p ∈ p, q ∈ q}. We say that a polytope q is a Minkowski summand of p if p = q + q ′ for some polytope q ′ . The normal fan of p + q is the common refinement of Σ p and Σ q . Hence, Σ p refines the normal fan of any of its Minkowski summands, and the Minkowski sum of deformations of p is again a deformation of p.
The f -polynomial of a d-dimensional polytope p is where f i (p) is the number of i-dimensional faces of p. The h-polynomial of p is defined by The sequences (f 0 (p), . . . , f d (p)) and (h 0 (p), . . . , h d (p)) are the f -vector and h-vector of p, respectively. These polynomials behave nicely with respect to the Cartesian product of polytopes: where p × q = {(p, q) : p ∈ p, q ∈ q}.

The polytope algebra
We briefly review the construction of McMullen's polytope algebra [21] and its main properties. A hands on introduction to this topic can be found in the survey [14]. The subalgebra relative to a fixed polytope [22] is studied at the end of this section.
2.1. Definition and structure theorem. As an abelian group, the polytope algebra Π(V ) is generated by elements [p], one for each polytope p ⊆ V . These generators satisfy the relations For each scalar λ ∈ R, the dilation morphism δ λ : Π(V ) → Π(V ) is defined on generators by δ λ [p] = [λp]. Recall that for any subset S ⊆ V and λ ∈ R, λS := {λv : v ∈ S}. One can easily verify that δ λ preserves the valuation (2.1) and translation invariance (2.2) relations, and that it defines a morphism of rings. The main structural result for the polytope algebra is the following. i. as an abelian group, Π(V ) admits a direct sum decomposition iii. Ξ 0 (V ) ∼ = Z, and for r = 1, . . . , d, Ξ r (V ) is a real vector space; iv. the product of elements in r≥1 Ξ r (V ) is bilinear. v. the dilations δ λ are algebra endomorphisms, and for r = 0, 1, . . . , d, if x ∈ Ξ r (V ) and λ ≥ 0, then δ λ x = λ r x.
We discuss the definition of the graded components Ξ r (V ) below. The component Ξ 0 (V ) of degree 0 is simply the subring of Π(V ) generated by 1, and thus Ξ 0 (V ) ∼ = Z. Let Z 1 be the subgroup of Π(V ) generated by all elements of the form [p] − 1. Observe that δ 0 [p] = 1 for every polytope p, so Z 1 = ker δ 0 is an ideal. As an abelian group, Π(V ) has a direct sum decomposition Π(V ) = Ξ 0 (V ) ⊕ Z 1 . Moreover, Z 1 is a nil ideal, since for any k-dimensional polytope p, This is [21,Lemma 13]. McMullen also shows that Z 1 has the structure of a vector space (first over Q and then over R). Therefore, we can define inverse maps log exp by means of their usual power series. In particular, we can define the log-class of a k-dimensional polytope p by Using the exponential map, we recover [p] from log[p]: The log and exp maps satisfy the standard properties of logarithms and exponentials. In particular, Consider the polytope z = l 1 + l 1 + · · · + l k , a Minkowski sum of segments. Using the logarithm property (2.5), we get (log The last equality follows since k! log[l i ] is the only square-free term in the expansion of ( log[l i ]) k , and (log[l]) 2 = ([l] − 1) 2 = 0 for any line segment l. Finally, (log[z]) k = 0 if and only if k ≤ dim(z), and z being the sum of k line segments has dimension at most k, with equality precisely when the vectors v 1 , . . . , v k are linearly independent.
For r ≥ 1 let Ξ r (V ) be the subgroup (or subspace) of Π(V ) generated by elements of the form (log[p]) r . The following result implies that the sum Π(V ) = Ξ 0 (V ) + Ξ 1 (V ) + · · · + Ξ d (V ) is direct, and characterizes each graded component as the space of eigenvectors for the positive dilations δ λ . Lemma 20]). Let x ∈ Π(V ) and λ > 0, with λ = 1. Then, It is clear by the definition of the graded components Ξ r (V ) that the class of a half-open parallelogram in Figure 2.1 is in Ξ 2 (V ). We can also verify this using (2.6) with λ = 2: To simplify notation, we drop the subscript R and sometimes wirte Ξ r and Π instead of Ξ r (V ) and Π(V ) R .
For an arbitrary vector v ∈ V , we can define a maximization operator p → p v on the space of all polytopes p ⊆ V . The next result shows that it induces a well-defined map on Π(V ). This endomorphism commutes with nonnegative dilations.
In particular, the morphism x → x v restricts to each graded component Ξ r .
Lastly, the Euler map x → x * is the linear operator defined on generators by The sum runs over all nonempty faces q of p. Up to a sign, the element [p] * corresponds to the class of the interior of p.

2.2.
Subalgebra relative to a fixed polytope. Fix a polytope p ⊆ V . The subalgebra relative to p, denoted Π(p), is the subalgebra of Π(V ) generated by classes [q] of deformations q of p.
It is worth pointing out that if q, q ′ are deformations of p such that q ∪ q ′ is a polytope, then both q ∪ q ′ and q ∩ q ′ are deformations of p. This follows since, in this case, q ∪ q ′ + q ∩ q ′ = q + q ′ and Minkowski summands of a deformation of p are again deformations of p. Thus, the valuation property (2.1) is not introducing classes of new polytopes to Π(p).  (2.4) show that Π(p) is generated by homogeneous elements. Thus, the grading of Π(V ) induces a grading of Π(p). We let Ξ r (p) = Π(p) ∩ Ξ r (V ) denote the component of Π(p) in degree r. Unlike the full algebra Π(V ), the subalgebra Π(p) has finite dimension. McMullen described the dimension of the graded components of Π(p) when p is a simple polytope.
Let f be a face of p and v ∈ relint N (f, p) . The maximization operator x → x v defines a morphism of graded algebras that only depends on the face f and not on the particular choice of v ∈ relint N (f, p) . First observe that this map is well defined; that is, Minkowski summand of f. Moreover, since the normal fan of p refines that of q, then q w = q v for any other w ∈ relint N (f, p) . Therefore the morphism (2.8) only depends on f. , the arrangement under X is the following hyperplane arrangement in ambient space X: The characteristic polynomial of A is is surjective and order preserving. We view L[A] as a commutative monoid with the join operation ∨ for the product. This makes the support map (3.1) a morphism of monoids. We let H X denote the basis element of RL[A] associated to the flat X of A, so that H X · H Y = H X∨Y .
A result of Solomon [28,Theorem 1] shows that the monoid algebra RL[A] is split-semisimple. This rests on the fact that the unique complete system of orthogonal idempotents for RL[A] consists of elements Q X uniquely determined by In particular, RL[A] is the maximal split-semisimple quotient of Σ[A] via the support map and the simple modules of Σ[A] are indexed by flats. The character χ X of the simple module associated with the flat X evaluated on an element Let t be a fixed scalar. An element w of the Tits algebra is characteristic of parameter t if for each flat X with χ X (w) as in (3.3). Characteristic elements determine the characteristic polynomial of the arrangement and also determine the characteristic polynomial of the arrangements under each flat. See [5,Section 12.4] and [2] for more information.
3.3. Eulerian idempotents and diagonalization. An Eulerian family of A is a collection of idempotent and mutually orthogonal elements with a F = 0 for at least one F with s(F ) = X. It follows that {E X } X is a complete system of primitive orthogonal idempotents and that s(E X ) = Q X [5,Theorem 11.20]. That is, and E X cannot be written as the sum of two non-trivial idempotents.
Example 3.1. Let C 2 be the coordinate arrangement in R 2 . The following is an Eulerian family of C 2 . Observe that in this example only faces in the first quadrant have non-zero coefficients.
A characteristic element w of non-critical 1 parameter t uniquely determines an Eulerian fam- This is a consequence of [5,Propositions 11.9,12.59]. It follows that the action of such a characteristic elements w on any be a characteristic element of non-critical parameter t and {E X } X be the corresponding Eulerian family. Then, we have a decomposition of vector spaces. Expression (3.4) shows that w acts on M · E X by multiplication by t dim(X) . We define is independent of the characteristic element w. Furthermore, using relations (3.2) and the linearity of χ M we deduce Moreover, the number of composition factors M i+1 /M i isomorphic to the simple module indexed by X in a composition series 0 ⊂ M 1 ⊂ M 2 ⊂ · · · ⊂ M k = M of M is precisely η X (M ). See [5, Section 9.5 and Theorem D.37] for details.

The polytope algebra as a module
Fix a hyperplane arrangement A in V . Take a normal vector v H for each hyperplane H ∈ A, and consider the zonotope (Minkowski sum of segments): (4.1) Its normal fan Σ z coincides with the collection of faces Σ[A] of the arrangement A. We say that a polytope q is a generalized zonotope of A it is a deformation of z. We now consider the algebra Π(z) introduced in Section 2.2. It is generated by the classes of generalized zonotopes of A. It only depends on the arrangement A and not on the particular choice of normal vectors v H . We start with a simple yet interesting result. Recall the morphism Proof. Let f be a face of z, F ∈ Σ[A] be its normal cone, and X = s(F ) be the flat orthogonal to f. It follows from (4.1) that f = z F is a translate of In particular, f is a Minkowski summand of z. Being a Minkowski summand is a transitive relation. Hence, any generator [q] of Π(f) is also in Π(z). That is, Π(f) is a subalgebra of Π(z). Moreover, if v ∈ relint(F ) and q is a Minkowski summand of f, then q v = q. Therefore, the composition Compare with Theorem 2.9 and note that we do not assume the zonotope z to be simple. For an arbitrary polytope p and a face f of p, there is no natural morphism Π(f) → Π(p), unlike in the previous case. This is a particular property of zonotopes. Indeed, a polytope p is a zonotope if and only if every face f ≤ p is a Minkowski summand of p, see [11, Proposition 2.2.14] for a proof.
Let F be a face of A and f = z F the corresponding face of z. We define right multiplication by the basis element where v ∈ relint(F ).

Moreover, each graded component Ξ r (z) is a Σ[A]-submodule and the action of basis elements {H
Proof. The zero vector belongs to the central face O, so the action is clearly unital. Associativity follows from the following fact about polytopes [18, Section 3.
It follows that this product gives Π(z) the structure of a right Σ[A]-module.
The second statement follows directly from Theorem 2.5 and the characterization of the graded components Ξ r in (2.6). Indeed, for any x ∈ Ξ r (z) and λ > 0, 4.1. Simultaneous diagonalization. Let λ > 0 and let w ∈ Σ[A] be a characteristic element of non-critical parameter t. We know that the dilation morphism δ λ and the action of w are diagonalizable. Moreover, since δ λ and the action of w commute, they are simultaneously diagonalizable. A natural question is to determine the eigenvalues of δ λ and of the action of w in their simultaneous eigenspaces. We completely answer this question in the case of the Coxeter arrangements of type A and B in the next two sections. The following result holds in the general case.
Proposition 4.4. Let x ∈ Π(z) be a (nonzero) simultaneous eigenvector for δ λ and w with eigenvalues λ r and t k , respectively. Then, r + k ≤ d.
Proof. Let {E X } X be the Eulerian family associated to w. Using the characterization of the graded components Ξ r as the eigenspaces of δ λ , and the decomposition of Σ[A]-modules in (3.5), we deduce that the common eigenspace for δ λ and w with the given eigenvalues is X Ξ r (z) · E X , where the sum is over all k-dimensional flats of A. Without loss of generality, we assume that x ∈ Ξ r (z) · E X for a single k-dimensional flat X.
is any face of support Y. Hence, formula (3.6) yields If in addition A is a simplicial arrangement, like in the case of reflection arrangements, then z and each of its faces are simple polytopes. In that case, Theorem 2.8 allows us to replace dim R (Ξ r (z Y )) by h r (z Y ) in expression (4.2). Multiplying by z r and taking the sum over all values of r, we obtain Observe that X S ≤ X T if and only if T ⊆ S, and in this case Hence, In particular, a series decomposition of Ξ r (c d ) contains exactly one copy of the simple module indexed by X S for every S ∈ [d] r . Let us now consider the characteristic element γ t ∈ Σ[C d ] introduced in [2, Section 5.3] for t = 1. It is defined by A simple computation shows that the Eulerian family corresponding to the characteristic element γ t is determined by In dimension 2, this is the Eulerian family in Example 3.1. For each S ⊆ [d], define Example 2.2 shows that y S is a nonzero element of Π(c d ).
We claim that {y S } S⊆[d] is a basis of simultaneous eigenvectors of Π(c d ). Explicitly, y S is an eigenvector for the action of γ t of eigenvalue t d−|S| , and for the action of δ λ of eigenvalue λ |S| (λ > 0). The second statement is clear, since log[l i ] ∈ Ξ 1 (c d ). Moreover, using that log On the other hand, observe that Therefore, The claim follows since dim(X S ) = d − |S|.

4.3.
The zonotope module of a product of arrangements. The Cartesian product of two arrangements A in V and A ′ in W is the following collection of hyperplanes in V ⊕ W : In fact, it is also true that where z and z ′ are zonotopes of A and A ′ , respectively, and therefore z × z ′ is a zonotope of A × A ′ . Indeed, every generalized zonotope of A × A ′ is the Cartesian product of generalized zonotopes of A and A ′ . The corresponding isomorphism is induced by The fact that this map is well-defined and a morphism of Σ[A × A ′ ]-modules follows from the ideas in Section 7.2.

The module of generalized permutahedra
Generalized permutahedra are the deformations of the standard permutahedron π d ⊆ R d . Edmonds first introduced them under a different name in [15], where he studied their relation to submodular functions and optimization. For a thorough study of the combinatorics of these polytopes, see [1,25,26].
In this section, we study the algebra Π(π d ) of generalized permutahedra and its structure as a module over the Tits algebra of the braid arrangement A d . We begin with a brief review of the braid arrangement, its relation with the symmetric group, and some statistics on permutations.
5.1. The braid arrangement. The braid arrangement A d in R d consists of the diagonal hyperplanes x i = x j for 1 ≤ i < j ≤ d. Its central face is the line perpendicular to the hyperplane x 1 + · · · + x d = 0. Intersecting A d with this hyperplane and a sphere around the origin we obtain the Coxeter complex of type A d−1 . The pictures below show the cases d = 3 and 4.
Flats and faces of A d are in one-to-one correspondence with set partitions and set compositions of [d] := {1, 2, . . . , d}, respectively. We proceed to review this correspondence.
A weak set partition of a finite set I is a collection X = {S 1 , . . . , S k } of pairwise disjoint subsets S i ⊆ I such that I = S 1 ∪ · · · ∪ S k . The subsets S i are the blocks of X. A set partition is a weak set partition with no empty blocks. We write X ⊢ I to denote that X is a set partition of I. Given a partition X ⊢ [d], the corresponding flat of A d is the intersection of the hyperplanes x a = x b for all a, b that belong to the same block of X, as illustrated in the following example for d = 8: If S ⊆ I is a union of blocks of a partition X ⊢ I, we let X| S ⊢ S denote the partition of S formed by the blocks of X whose union is S. Let X = {S 1 , . . . , S k } ⊢ [d] be a partition. Then, the choice of Y ≥ X is equivalent to the choice of partitions Y| S i ⊢ S i for each block of X. With X and Y as above, the Möbius function of L[A d ] is determined by where in each factor, ⊥ denotes the minimum partition of S i . A set composition of I is an ordered set partition F = (S 1 , . . . , S k ). We write F I to denote that F is a composition of I, and let s(F ) ⊢ I be the underlying (unordered) set partition. Given a set composition F [d], the corresponding face of A d is obtained by intersecting the hyperplanes x a = x b whenever a, b are in the same block of F , and the halfspaces x a ≥ x b whenever the block containing a precedes the block containing b. For example, For a permutation σ ∈ S d , we let s(σ) denote the subspace of points fixed by the action of σ; it is a flat of A d . In view of the identification between flats of A d and partitions of [d], s(σ) can equivalently be defined as the partition of [d] into the disjoint cycles of σ. For example, if in cycle notation σ = (13)(2658)(4) (7), then s(σ) = {13, 2568, 4, 7}.
We can similarly define descents and excedances for permutations of any set S with a total order ≺, we denote the corresponding statistics by des ≺ and exc ≺ .
It is a classical result that descents and excedances are equidistributed in S d . That is, for all possible values of k. Foata's fundamental transformation provides a simple proof of this result. The numbers A d,k are the classical Eulerian numbers (OEIS: A008292). The Eulerian polynomial A d (z) is: The exponential generating function for these polynomials was originally given by Euler himself: See [16, Section 3] for a derivation of this formula. Let C(S) the collection of cyclic permutations on a finite set S, and . Given a permutation σ ∈ S d and a block S ∈ s(σ), the restriction σ| S of σ to S is a cyclic permutation. For example, with σ as before and S = {2, 5, 6} ∈ s(σ), we have σ| S = (265) ∈ C({2, 5, 6}). A very simple but important observation is that the number of excedances of σ can be computed by adding up the excedances in each cycle in its cycle decomposition. That is, exc(σ) = S∈s(σ) exc(σ| S ). The number of excedances in each cycle σ| S is computed with respect to the natural order in S ⊆ [d].
It is a zonotope of the braid arrangement A d and has dimension d − 1. Deformations of π d are called generalized permutahedra. We consider the module Π(π d ) as in Section 4. The main goal of this section will be to prove the following result.   The relation between Π(π d ) and statistics on S d is via the h-polynomial of π d . Brenti [12,Theorem 2.3] showed that h(π d , z) = A d (z). Moreover, for a flat/partition X = {S 1 , . . . , S k } of A d , the face (π d ) X is a translate of π |S 1 | × · · · × π |S k | , a product of lower-dimensional permutahedra. Thus, 3) Lemma 5.3 below is an essential ingredient in the proof of Theorem 5.1. Its proof uses the Compositional Formula, for which a type B analog is proved in Proposition 6.3.
Proof. We will show that the exponential generating function of both sides of (5.4) are equal to log(A(z, x)), where A(z, x) is the generating function for the Eulerian polynomials in (5.2). First, recall that if X = {S 1 , . . . , S k }, then µ(⊥, X) = (−1) k−1 (k − 1)!. Thus, a direct application of the Compositional Formula shows that the exponential generating function of the LHS of (5.4) is the composition of which is precisely log (A(z, x)).
On the other hand, grouping permutations with the same underlying partition s(σ), we obtain Since a permutation σ with s(σ) = X is the product of cyclic permutations σ S ∈ C(S) for each block S ∈ X, and in this case exc(σ) = S∈X exc(σ S ), Taking logarithms on both sides yields the result.
A small modification in the proof of the previous Lemma immediately gives the following result, which was first discovered by Brenti.
An analogous formula for the type B Coxeter group is described in Proposition 6.7. We are now ready to prove the main result of this section.
Proof of Theorem 5.1. We will compute the values η X (Ξ r (π d )) using formula (4.3), which in this case reads Using (5.1) and (5.3), we can rewrite the expression above as Now, an application of Lemma 5.3 and relation (5.5) gives Finally, taking the coefficient of z r on both sides of the last equality yields the result.
Adding over all flats with the same dimension in Theorem 5.1, we conclude the following.

5.4.
Simultaneous-eigenbasis for the Adams element. Perhaps the most natural characteristic elements for the braid arrangement are the Adams elements, defined for any parameter t by: where dim(G/F ) = dim(G) − dim(F ) and, deg(G/F ) = S∈F G| S . Theorem 5.1 suggest the existence of a natural basis for Ξ r (π d ) · E X indexed by permutations σ with r excedances and s(σ) = X. In this section we will construct a candidate for such basis.
The   We will use a bijection between increasing rooted forests on [d] and permutations in S d . An increasing rooted forest is a disjoint union of planar rooted trees where each child is larger than its parent and the children are in increasing order from left to right. Given a rooted forest t, the corresponding permutation σ(t) is read as follows. Each connected component of t corresponds to a cycle of σ(t). To form a cycle, traverse the corresponding tree counterclockwise and record a node the second time you pass by it 2 . The inverse can be described inductively by writing each cycle with its minimum element in the last position, and using right to left minima. We omit the details, but provide an example σ → t(σ) to illustrate the idea. This bijection is such that the connected components of the forest t(σ) are the blocks of s(σ). Moreover, the number of leaves of t(σ) in S ∈ s(σ) is exc(σ| S ) (a tree consisting only of its root has zero leaves). Consequently, the total number of leaves of t(σ) is exc(σ). Let σ ∈ S d be a permutation with r excedances and let X = s(σ). For 1 ≤ i ≤ r, let J i be the elements on the path from the i th leaf of t(σ) to the root of the corresponding tree. Define the element For instance, if σ is the permutation in (5.6), then It follows from the definition (5.7) that x σ ∈ Ξ r (π d ) · E X . The content of the conjecture is that these elements are linearly independent. Explicit computations show that this is the case for d = 2, 3, 4. Propositions 5.8 and 5.9 below prove the extremal cases r = 1 and r = d − |X| of this conjecture, respectively.

Proposition 5.8. For a subset J ⊆ [d] of cardinality at least 2, let X J ⊢ [d] be the partition whose only non-singleton block is
Proof. First, observe that any cyclic permutation on a set with more than one element has at least one excedance, and only one cyclic permutation attains this minimum. Namely, the only cyclic permutation in S d having one excedance is (d d − 1 . . . 2 1). Hence, a permutation σ ∈ S d has at least as many excedances as non-singleton blocks in s(σ). It then follows from Theorem 5.1 that has exactly one non-singleton block, 0 otherwise.
Thus, the second statement follows from the first. Since {log[∆ J ] : J ⊆ [d], J ≥ 2} is a linear basis for Ξ 1 (π d ), it is enough to write log[∆ J ] · E X J as a non-trivial linear combination of these basis elements. Observe that if F = (S 1 , S 2 , . . . , S k ), then [∆ J ] · H F = [∆ J∩S i ] where i is the first index for which the intersection J ∩ S i is nonempty. Thus, Using that the action of H F is an algebra morphism, we have log[ The equality follows since for any flat X of the braid arrangement, A X d has dim(X)! chambers. Note that the element log[∆ J ] · E X J in the proposition is precisely the element x σ for the unique permutation σ with s(σ) = X J and exc(σ) = 1. Indeed, t(σ) consists of a increasing path whose nodes are the elements in J and isolated roots indexed by the elements in [d] \ J.

Proposition 5.9. For any
Proof. Observe that any cyclic permutation on a set with s elements has at most s − 1 excedances, and only one cyclic permutation attains this maximum. Namely, the only cyclic permutation in S d having d − 1 excedances is (1 2 . . . d − 1 d). Hence, for any X = {S 1 , . . . , S k } ⊢ [d] there is exactly one permutation with s(σ) = X and d − k excedances. Theorem 5.1 then implies that dim R (Ξ d−k (π d ) · E X ) = 1. It follows from Example 2.2 that the element x X is nonzero, and counting the number of factors in (5.8) shows that x X ∈ Ξ d−k (π d ). Thus, we are only left to prove that Then, for some block S i ∈ X and some a ∈ S i , a and min(S i ) are not in the same block of s(G). Hence [∆ {min(S i ),a} ] · H G is the class of a point, and log[∆ {min(S i ),a} ] · H G = log[{0}] = 0. Since the action of H G is an algebra morphism, we get that x X · H G = 0. Therefore, as we wanted to show.
The element x X in the previous result is x σ for the only permutation σ with s(σ) = X and exc(σ) = d − |X|. In this case, the forest t(σ) consists of a k trees with node sets S 1 , . . . , S k , respectively. Each tree has min(S i ) as a root and every other element in S i as a child of the root.

The module of type B generalized permutahedra
In this section, we study the algebra Π(π B d ) of type B generalized permutahedra and its structure as a module over the Tits algebra of the Coxeter arrangement of type B.
6.1. The type B Coxeter arrangement. The type B Coxeter arrangement A ± d in R d consists of the hyperplanes x i = x j , x i = −x j for 1 ≤ i < j ≤ d and x k = 0 for 1 ≤ k ≤ d. Its central face is the trivial cone {0}. The Coxeter complex of type B d , obtained by intersecting the arrangement with a sphere around the origin in R d , is shown below for d = 2 and 3. In contrast, S ⊆ I is said to be involution-inclusive if S = S. Given an arbitrary subset S ⊆ I, we let ±S be the involution-inclusive set S ∪ S.
A signed set partition of I is a weak set partition of the form X = {S 0 , S 1 , S 1 , . . . , S k , S k }, where S 0 is involution-inclusive and allowed to be empty, and each S i for i = 0 is nonempty and involution-exclusive. We call S 0 the zero block of X. We write X ⊢ B I to denote that X is a signed partition of I. Given a signed partition X ⊢ B [±d], the corresponding flat of A ± d is the intersection of the hyperplanes x i = x j for each i, j in the same block of X, where for k ∈ [d], we let x k denote −x k . In particular, if k ∈ [d] is in the zero block of X, the corresponding flat lies in the hyperplane x k = 0. For instance, consider the following two examples for d = 7: ←→ {∅, 13,13, 245,245, 67,67}, The zero block in the first partition is empty since the corresponding flat is not contained in any coordinate hyperplane. We use X to denote both a flat of A ± d and the corresponding signed partition of [±d]. Observe that the number of nonzero blocks of X is 2 dim(X If S is an involution-inclusive union of blocks of X ⊢ B I, we let X| S ⊢ B S denote the corresponding signed partition. If, on the other hand, S is an involution-exclusive union of blocks of X ⊢ B I, we let X| S ⊢ S denote the corresponding (type A) partition.
Let X = {S 0 , S 1 , S 1 , . . . , S k , S k }. A choice of Y ≥ X is equivalent to the choice of a signed partition Y| S 0 ⊢ B S 0 of the zero block and of (type A) partitions Y| S i ⊢ S i for i = 1, . . . , k. Note that in this case, Y| S i is automatically determined by Y| S i .
With X and Y as above, the Möbius function of . . . 1, and in each factor ⊥ denotes the minimum (signed) partition of (S 0 ) S i . A signed composition is an ordered signed set partition with the property that S i precedes S j if and only if S j precedes S i . The following examples illustrate the identification between signed compositions of ±[d] and faces of A ± d : ←→ (67,245, 13, ∅,13, 245,67) Recall that, for instance, x 1 = −x 1 . For a signed permutation σ ∈ B d , we let s(σ) denote the subspace of points fixed by the action of σ; it is a flat of A ± d . Under the identification above, s(σ) is the signed partition of [±d] obtained from the underlying the cycle decomposition of σ by merging all the blocks that contain an element i and its negative i. For example, if in cycle notation σ = (1)(1)(22)(3434)(56)(56), then s(σ) = {223344, 1,1, 56,56}.
Let σ ∈ B n . The restriction σ| S 0 to the zero block S 0 ∈ s(σ) is a signed permutation of S 0 . Its action on R |S 0 |/2 does not fix any nonzero vector, so s(σ| S 0 ) =⊥. For a nonzero block S ∈ s(σ), σ| S ∈ C(S) is a cyclic permutation of the elements in S. The restriction σ| ±S is again a signed permutation, and it is completely determined by either σ| S or σ| S . We present some statistics on signed permutations. For σ ∈ B d , let where we set σ(0) = 0. Elements in the sets above are descents, excedances and negations of σ, respectively. The last statistic is called the flag-excedance of a signed permutation. We define one last statistic, the B-excedance of σ: Foata and Han [17,Section 9] show that descents and B-excedances are equidistributed. That is, The exponential generating function of these polynomials is first due to Brenti [12,Theorem 3.4]. We will be interested in the type B exponential generating function of these polynomials:  As in type A, the relation between Π(π B d ) and statistics on B d is due to Brenti's result showing that h(π B d , z) = B d (z). For a flat X = {S 0 , S 1 , S 1 , . . . , S k , S k } of A ± d , the face (π B d ) X is a translate of π B |S 0 |/2 × π |S 1 | × · · · × π |S k | , a product of lower-dimensional permutahedra of type A and B, where exactly one factor is of type B. Thus, The following result is the analogous in type B of Lemma 5.3.
In the same spirit as the proof of Lemma 5.3, we will establish (6.5) by comparing the type B exponential generating function of both sides of the equality. An important tool in this proof is the following analog of the Compositional Formula (Theorem 5.2) for type B generating functions.
Proof. Using the usual Compositional Formula, the coefficient of To verify the last equality, note that choosing a type B partition {S 0 , S 1 , S 1 , . . . , S k , S k } ⊢ B [±d] with |S 0 | = 2r is equivalent to: (1) choosing a subset K 0 ∈ [d] r and setting S 0 = ±K 0 , (2) choosing a partition {K 1 , . . . , K k } of [d] \ K 0 , and (3) constructing blocks {S i , S i } from K i as follows: for each j ∈ K i \{max K i }, choose whether j and max K i will be in the same or in opposite blocks. There are precisely 2 d−r−k possible choices in the last step.
Taking g d = 1 in the Type B Compositional Formula we deduce the following.

Corollary 6.4 (Type B Exponential Formula). Let f (x) and a(x) be as before. If
In the proof of Theorem 5.1, we used that for (type A) permutations σ ∈ S d , exc(σ) equals the sum of exc(σ| S ) as S runs through the blocks of s(σ). For signed permutations, one easily checks this also holds for the statistics exc and neg. However, it is not obvious at all that the same is true for exc B , since its definition uses the floor function.
Consider the order ≺ of the elements of any involution-exclusive subset S ⊆ [±d] defined by: where exc ≺ (σ| S i ) is the number of usual (type A) excedances of σ| S i with respect to the order ≺.
Observe that according to the three cases of definition (6.6), a ≺-excedance of σ| S i corresponds to either      an excedance of σ| ±S i occurring in S i , or a negation of σ| ±S i occurring in S i , or an excedance of σ| ±S i occurring in S i , respectively. Since exactly half of the negations of σ| ±S i occur in S i , we deduce that Thus, in view of (6.2) and using that is always an integer, = exc B (σ| S 0 ) + exc ≺ (σ| S 1 ) + · · · + exc ≺ (σ| S k ), as we wanted to show. Corollary 6.6. Let w ∈ Σ[A ± d ] be a characteristic element of non-critical parameter t and λ > 0. The dimension of the simultaneous eigenspace for w and δ λ with eigenvalues t k and λ r is {σ ∈ B d : dim(s(σ)) = k, exc B (σ) = r} .
As in the type A case, we can modify the proof of the previous Lemma to obtain the generating function for the bivariate polynomials σ∈B d t dim(s(σ)) z exc B (σ) . To the best of our knowledge, this is a new result.
Proposition 6.7. The following identity holds Proof. We can slightly modify (6.7) to obtain Using the (type B) generating functions of the factors deduced in the proofs of Lemmas 5.3 and 6.2, and the type B compositional formula, we deduce 2 . Substituting the expressions for A(z, x) and B(z, x) in (5.2) and (6.3) yields the result.
Specializing t := 0 and x := 2x gives an alternative expression for the exponential generating function of the OEIS sequence A156919. In our context, these coefficients count the number of signed permutations whose action on R d has no nonzero fixed point weighted by the statistic exc B . 6.4. Two bases basis for type B generalized permutahedra. Faces of the standard simplex ∆ [d] correspond to linearly independent rays of the cone of generalized permutahedra in R d . In the language of the present work, this means that {log[∆ S ] : S ⊆ [d], |S| ≥ 2} forms a linear basis of Ξ 1 (π d ) and that log[∆ S ] is not the sum of the log-classes of other generalized permutahedra (other than trivial dilations of itself). The goal of this section is to establish an analogous result for the type B case.
The standard simplex coincides with the weight polytope P S d (λ 1 ) in the sense of Ardila, Castillo, Eur, and Postnikov [7], where λ 1 is the fundamental weight (1, 0, . . . , 0) of S d . In this manner, the cross-polytope P B d (λ 1 ) = Conv{±e i : i ∈ [d]} is the type B analog of the standard simplex. However, in the same paper the authors point out that the faces of the cross-polytope span a space of roughly half the desired dimension. This is intuitively clear once we notice that the collection of faces of P B d (λ 1 ) entirely contained in one orthant span the same space in Ξ 1 (π B d ) as the faces contained in the opposite orthant. The following result shows that it is not possible to find a single type B generalized permutahedron whose faces generate Ξ 1 (π B d ). Proposition 6.8. Let P = {p α } α be a collection of type B generalized permutahedra such that {log[p α ]} α spans Ξ 1 (π B d ). Then, P contains at least 2 d−1 full dimensional polytopes.
be an Eulerian family of A ± d . The result follows from the following two facts, which we justify below.
(1) The projection of log[p] to the subspace Ξ 1 (π B d ) · E ⊥ is zero unless p is full-dimensional.
(1). Let p be a type B generalized permutahedron that is not full-dimensional. Then we can choose a face F = O of A ± d such that p F = p, for instance any maximal face in N (p, p). It follows from [1, Lemma 11.12] that H F · E ⊥ = 0. Therefore, ). Using Theorem 6.1, this is the number of signed permutations σ ∈ B d with s(σ) =⊥ and exc B (σ) = 1. A signed permutation σ ∈ B d with s(σ) =⊥ is the product of cycles on involution-inclusive subsets S ⊆ [±d]. Moreover, each such cycle adds at least 1 to the number of negations of σ. Thus, a signed permutation with s(σ) =⊥ and exc B (σ) = 1 must be the product of either 1 or 2 cycles and have no excedances. Such cycles are necessarily of the form (d d − 1 . . . 1d d − 1 . . . 1). Thus, the permutations counted by η ⊥ (Ξ 1 (π B d )) are in correspondence with unordered pairs {S, [±d] \ S} of involution-inclusive subsets of [±d], and there are precisely 2 d−1 many of them.
Alternatively, one can manipulate the generating function in Proposition 6.7 ( ∂ ∂z t,z=0 ) to deduce that the coefficient of t 0 z 1 x d is 2 d−1 .
In [24], Padrol, Pilaud, and Ritter construct a family of type B generalized permutahedra called shard polytopes. They show that any type B generalized permutahedron can be written uniquely as a signed Minkowski sum of these polytopes (up to translation). Already in R 3 , there are 14 full-dimensional shard polytopes. We proceed to construct a family of generators that achieves the minimum imposed by the previous proposition.
For a nonempty involution-exclusive subset S ⊆ [±d], define the simplices Observe that the only full-dimensional simplices in this collection are ∆ 0 S with |S| = d. We say that an involution-exclusive subset S ⊆ [±d] is special if in addition min{|i| : i ∈ S} ∈ S. From now on, we will only consider simplices ∆ S and ∆ 0 S for special sets S. Theorem 6.9. Every type B generalized permutahedron can be written uniquely as a signed Minkowski sum of the simplices ∆ S and ∆ 0 S with S as above. is linearly independent in Ξ 1 (π B d ). To prove this, we will use the following linear map, whose existence is guaranteed by [21,Theorem 5]: , we will conclude that all the coefficients α S and β S are necessarily 0, and therefore the collection (6.8) is linearly independent.
We will start with the coefficients α. Fix k ∈ [d− 1], we prove by induction on |S| that α S = 0 for all S such that k = min{|i| : i ∈ S}. The base case is |S| = 2. Let S = {k, j} with j ∈ ±[k + 1, d] and consider the following face of A ± d x 1 > x 2 > · · · > x k−1 > x k = x j > x k+1 > · · · > x |j| > · · · > x d > 0, or equivalently, The hat over x |j| denotes that the corresponding inequality is missing (since we already imposed x |j| = ±x k ). By the description of the normal cones in (6.9), we observe that this face only appears in a normal cone of either: Since we are only considering special sets S ′ (that is, such that min{|i| : i ∈ S ′ } ∈ S ′ ) and k > 0, the only possibility is S ′ = {k, j} = S. Thus, the coefficient of this face in Ψ 1 (ϕ) is α {k,j} = 0. Now suppose S = {k, j 1 , . . . , j r } with j 1 , . . . , j r ∈ ±[k + 1, d]. If we proceed with the previous analysis for the face we conclude that it appears in a normal cone of ∆ S ′ if and only if {k, j 1 } ⊆ S ′ ⊆ S. Thus, the coefficient of this face in Ψ 1 (ϕ) is {k,j 1 }⊆S ′ ⊆S α S ′ = 0. By induction we have α S ′ = 0 for all {k, j 1 } ⊆ S ′ S, and so we conclude α S = 0.
We proceed now with the coefficients β. Fix k ∈ [d − 1] and j ∈ ±[k + 1, d], and let S = S(k, j) be the collection of special subsets S such that k, j ∈ S and k = min{|i| : i ∈ S}. We first prove that β S = 0 for any S ∈ S. Let S = {k, j, s 1 , . . . , s r } ∈ S and let {t 1 , . . . , t r ′ } = [k + 1, d] \ {|j|, |s 1 |, . . . , |s r |}. Consider the following (d − 1)-dimensional face of A ± d : We claim that these relations, one for each S ∈ S, imply that β S = 0 for all S ∈ S. This follows since for any S ∈ S, The last equality follows from the next observations: The coefficient of this face in Ψ 1 (ϕ) is {k}⊆S⊆[d] β S = 0. Since we have shown that β S = 0 for any S with |S| ≥ 2, we conclude that β {k} = 0. Remark 6.10. Observe that the generating collection {∆ S : S ⊆ [d]} for generalized permutahedra is invariant under the action of S d . In contrast, the collection of generators for type B generalized permutahedra presented in the previous theorem fails to be invariant under the action of B d . This is not by accident. Already in R 2 , we see that any collection of type B generalized permutahedra that contains a triangle p (full-dimensional simplex) and that is invariant under the action of B d , will contain the rotations of p by 90 • , 180 • , and 270 • , all of which are necessarily different. Thus, such a collection will not attain the minimum number of full-dimensional polytopes required by Proposition 6.8. In the following theorem, we present a different collection of generators for type B generalized permutahedra that is invariant under the action of B d . is a basis of Ξ 1 (π B d ). By comparing the number of elements, we see that it is enough to write the elements in the basis (6.8) as a linear combination of the elements in the proposed basis above. Since the proposed basis already contains all the elements of the form log[∆ 0 S ], we are only left to write log[∆ S ] for |S| ≥ 2 as a linear combination of the proposed basis. We do this by induction on |S|.
We use λ = −1 and r = 1 in Theorem 2.6 to conclude that for any polytope p: Applying this identity to p = ∆ 0 Observe that all of the terms in the right are in the proposed basis, so we have completed the base case of the induction. If |S| > 2, we use (6.11) and observe that, by induction hypothesis, all the terms in the right can be written as a linear combination of the proposed basis.

Hopf monoid structure
Combinatorial species were originally introduced by Joyal [20] as a tool for studying generating power series from a combinatorial perspective. A comprehensive introduction to the theory of species can be found in the work by Bergeron, Labelle, and Leroux [9]. The category of species possesses more than one monoidal structure. Of central interest for the present work are the Cauchy and Hadamard product. Aguiar and Mahajan [3,4] have explored these structures extensively, and have exploited this rich algebraic structure to obtain outstanding combinatorial results. The first of these structures leads to the definition of Hopf monoids in species, a very active topic of research in recent years.
Aguiar and Ardila introduced the Hopf monoid of generalized permutahedra GP in [1]. It contains many other interesting combinatorial Hopf monoids as submonoids. In this section we show that the valuation (2.1) and translation invariance (2.2) properties define a Hopf monoid quotient of GP. We say a species h is a Hopf monoid if it is a bimonoid with an antipode in this monoidal category.
Let us make these definitions explicit. A species p consists of the following data: i.  Proof. For generators of Mc of the form (7.2), the result follows from the following two observations. If p ∈ GP[S], r ∈ GP[T ] and t ∈ R S , then If p ∈ GP[I] and t ∈ R T , then where t S and t T denote the projections of t to R S and R T , respectively.
We will now focus on generators of Mc of the form (7.1). Fix an arbitrary finite set I and a nontrivial decomposition I = S ⊔ T . Let v ∈ R I be any vector in the interior of the corresponding face of the braid arrangement.
Suppose p, q, p ∪ q ∈ GP[S] and r ∈ GP[T ]. Then, and (p ∪ q) × r = (p × r) ∪ (q × r) is a polytope if and only if p ∪ q is. It follows that Since GP is commutative, this proves that Mc is an ideal. Now, let p, q, p ∪ q ∈ GP[I]. There are two possibilities: i. The face (p ∪ q) v of p ∪ q is completely contained in p or in q. Without loss of generality, suppose the former. Then (p ∪ q) v = p v and, necessarily, (p ∩ q) v = q v . Hence, Expanding the first equality we have The union of two Cartesian products A × B and C × D is again a Cartesian product if and only if one contains the other or either A = C or B = D. By assumption, the is no containment between p v and q v . We can therefore assume without loss of generality that Projecting to R S and R T , we further see that In particular, p/ S ∪q/ S is a generalized permutahedron. On the other hand, expanding (p∩q Comparing factors, we deduce (p ∩ q)| S = p| S and (p ∩ q)/ S = p/ S ∩ q/ S . (7.5) Putting together (7.3), (7.4) and (7.5), we conclude Thus, in either case we get ∆ S, Comparing the generators of the (co)ideal Mc with the relations defining McMullen's polytope algebra, it is natural to ask if Π[I] agrees with Π(π I ), where π I ⊆ R I is the standard permutahedron. The answer is no. For instance, in the polytope algebra, the structure of R-vector space is defined so that α where the numbers over the segments denote their length. However, if α is an irrational number, then Nevertheless, the proof of the previous theorem works verbatim to show that the Hopf monoid operations are well-defined in Π(π I ). for all generalized permutahedra p ⊆ R d . Thus, in the case of generalized permutahedra, the module structure of Section 5 is precisely the one induced from the Hopf monoid structure above.
The antipode formula of GP descends to the quotient Π, but it is no longer grouping-free in general. The Euler map (2.7) allows us to write the antipode formula of Π in a very compact form: 7.3. Higher monoidal structures. We have just proved that Π is a Hopf monoid in the symmetric monoidal category (Sp, ·). The algebra structure of each space Π[I] defined by McMullen can also be defined for GP. In both cases, this endows the species with the structure of a monoid in the symmetric monoidal category (Sp, ×) of species with the Hadamard product. The Hadamard product of two species p and q is defined by Hence, a monoid in (Sp, ×) consists of a species p with an algebra structure on each space p[I]. For generalized permutahedra, these structures are compatible in a very special way.
See [3,Chapter 7] for the definition of higher monoidal categories and of monoids in such categories. The notation (2,1) indicates that GP is a monoid with respect to the first two monoidal structures (Cartesian product and Minkowski sum, respectively) and a comonoid with respect to the last (coproduct maps ∆ S,T ).
Proof. We only discuss the remaining compatibility axioms: the compatibility between Cartesian product and Minkowski sum, and the compatibility between Minkowski sum and the coproduct.
The compatibility between Cartesian product and Minkowski sum boils down to the identity (p 1 + p 2 ) × (q 1 + q 2 ) = (p 1 × q 1 ) + (p 2 × q 2 ) for p 1 , p 2 ∈ GP[S] and q 1 , q 2 ∈ GP[T ], which one easily verifies for arbitrary sets p 1 , p 2 ⊆ R S and q 1 , q 2 ⊆ R T . On the other hand, the compatibility between Minkowski sum and the coproduct is equivalent to the following identity for generalized permutahedra p, q ∈ GP[I]: (p + q)| S ⊗ (p + q)/ S = (p| S + q| S ) ⊗ (p/ S + q/ S ).
This follows by projecting the identity (p + q) v = p v + q v to R S and R T , respectively, where v is any vector in the interior of the face of the braid arrangement corresponding to the composition (S, T ).
The compatibility between Minkowski sum and the Hopf monoid operations refines the last statement in Theorem 4.3; which, in the language of this section, states that the maps µ S,T • ∆ S,T are compatible with the Minkowski sum operation.

Final remarks and questions
1. Eulerian numbers are defined for any Coxeter group W in terms of W -descents. For the Coxeter groups of type A and B, descents and (B-)excedances are equally distributed, so we can interpret the W -Eulerian polynomials as the generating functions for (B-)excedances. However, the joint distributions of | s(·)|, des(·) and | s(·)|, exc(·) do not agree. Therefore, Theorems 5.1 and 6.1 cannot immediately be expressed in terms of descents.
Extending the results of this section to other Coxeter groups W requires to find the correct notion of W -excedance for other types. Is there a nice analog of the result of Ardila, Benedetti, and Doker, and of Theorem 6.9 in type D?

2.
McMullen [21] also studied valuation relations for the collection of polyhedral cones in V . The full cone group of V is generated by the classes [C], one for each polyhedral cone C ⊆ V . They satisfy the following relation: [C 1 ∪ C 2 ] = [C 1 ] + [C 2 ] (8.1) whenever C 1 ∪ C 2 is a cone and C 1 ∩ C 2 is a proper face of C 1 and of C 2 . Note that this is not to say that the class [C 1 ∩ C 2 ] is zero.
A cone of an arrangement A is any convex cone obtained as the union of faces of A. The space of formal linear combinations of cones Ω[A] is a right Σ[A] module under the following operation. If C is a cone of A and F ∈ Σ[A], then whereF ≤ C is the minimum face of C containing F and TF C denotes the tangent cone of C atF . The relation (8.1) is compatible with this action, and defines a quotient module Ω[A].
Restricting to the case of all the braid arrangements, Ω defines a Hopf monoid in species. The product is defined by means of the Cartesian product. Let C be a cone of the braid arrangement in R I and F the face corresponding to the composition (S, T ) I. If F ⊆ C, the tangent cone TF C decomposes as a product C| S × C/ S of cones in R S and R T . The coproduct of Ω is defined as follows: With these operations, Ω is isomorphic to the Hopf monoid of preposets Q considered in [1]. Relation (8.1) defines a Hopf monoid quotient Ω. Under a suitable change of basis, Ω is isomorphic to the dual Hopf monoid of faces Σ * defined in [3, Chapter 12].
3. There is a Hopf monoid morphism GP → Ω, whose components GP[I] → Ω[I] are defined as follows: Moreover, this is a morphism of (2, 1)-monoids in the 3-monoidal category (Sp, ·, ×, ·), where the monoidal structure of Ω under the Hadamard product is given by That this map defines a morphism of monoids under the Hadamard product is equivalent to the following fact: the normal fan of p + q is the common refinement of Σ p and Σ q . The map (8.2) does not induce a well defined morphism Π → Ω. In [21,Theorem 5], McMullen shows that p −→ q≤p vol(q)N (q, p) induces an injective map Π[I] → Ω[I], where vol(q) is the normalized volume of q in the affine space spanned by p. Moreover, one can verify that the induced morphism Π → Ω is a morphism of Hopf monoids. Is it possible to endow Ω with the structure of a (2, 1)-monoid so that the morphism above is a morphism of (2, 1)-monoids? Such a structure onΩ[I] would contain a subalgebra isomorphic to the Möbius algebra B * (M ) introduced by Huh and Wang in [19,Definition 5], where M is the matroid associated with the braid arrangement A I in R I .