The Hopf monoid of generalized permutahedra. SIAM Discrete Mathematics Meeting Austin, TX, June 2010

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1 The Hopf monoid of generalized permutahedra Marcelo Aguiar Texas A+M University Federico Ardila San Francisco State University SIAM Discrete Mathematics Meeting Austin, TX, June 2010

2 The plan. 1. Species. 2. Hopf monoids. Product, coproduct. Three important examples: posets, graphs, matroids. Antipode, characters, and invariants. 3. Generalized permutahedra. The Hopf monoid: product, coproduct. Restriction to posets, graphs, and matroids. The antipode and reciprocity theorems.

3 1. Species. (Joyal) A species P consists of: For each finite set I, a vector space P[I ]. For each bijection σ : I J, a linear map P[σ] : P[I ] P[J] such that P[σ τ] = P[σ] P[τ] and P[id] = id. Think: ways of putting a certain combinatorial structure on a set I. P(I ) = span{combinatorial structures on I } Examples. Q[I ] := span{posets on I }. G[I ] := span{graphs on vertex set I }. M[I ] := span{matroids on ground set I }.

4 2. Hopf monoids. (Aguiar-Mahajan, Monoidal functors, species, and Hopf algebras.) A (connected) Hopf monoid (P, µ, ) consists of: A species P. For each I = S T, product and coproduct maps P[S] P[T ] µ S,T P[I ] and P[I ] S,T P[S] P[T ]. Two inverse maps k P[ ] and P[ ] k. Think: rules for merging and breaking our structures. These maps should satisfy various axioms, including the following:

5 2. Hopf monoids. Compatibility axiom. Fix decompositions S T = I = S T, and let A, B, C, D be: S A B S T T C D Then this diagram must commute: P[A] P[B] P[C] P[D] id switch id P[A] P[C] P[B] P[D] A,B C,D P[S] P[T ] µ S,T P[I ] S,T µ A,C µ B,D P[S ] P[T ]

6 2. Hopf monoids. The Hopf monoid of posets. Q[I ] := vector space with basis the set of posets on I. The species of posets Q is a Hopf monoid, under: Product: direct sum. Coproduct: splitting. µ S,T : Q[S] Q[T ] Q[I ] q S q T q S q T S,T : Q[I ] Q[S] Q[T ] { q S q T if S is an order ideal of q q 0 otherwise

7 2. Hopf monoids. The Hopf monoid of matroids. M[I ] := vector space with basis the set of matroids on I. The species of matroids M is a Hopf monoid with M[S] M[T ] µ S,T M[I ] m 1 m 2 m 1 m 2 M[I ] S,T M[S] M[T ] m m S m/ S where m 1 m 2 = direct sum of m 1 and m 2, m S = restriction of m to S, m/ S = contraction of S from m. Recall: A matroid on I is a collection B of r-subsets ( bases ) such that: If A, B B and a A B, there exists b B A with A a b B. Prototypical example: I = collection of vectors in V, B = subsets of I which are bases of V

8 2. Hopf monoids. The Hopf monoid of graphs. G[I ] := vector space with basis the set of graphs (with half edges) with vertex set I. The species of graphs G is a Hopf monoid with G[S] G[T ] µ S,T G[I ] g 1 g 2 g 1 g 2 G[I ] S,T G[S] G[T ] g g S g/ S where g 1 g 2 = disjoint union of g 1 and g 2, g S = keep everything incident to S, g/ S = remove everything incident to S. I = {a, b, c, x, y} x a y b c S,T S = {a, b, c} a b c T = {x, y} x y

9 2. Hopf monoids. Antipode. The antipode of a connected Hopf monoid P consists of the maps s I : P[I ] P[I ] given by s I = ( 1) k µ S1,...,S k S1,...,S k. S 1,...,S k k 1 The sum is over all ordered decompositions I = S 1 S k into nonempty disjoint subsets. For each such decomposition, P[I ] S 1,...,S k P[S1 ] P[S k ] µ S 1,...,S k P[I ] General problem. Find a simple formula for the antipode of a Hopf monoid. (Very often there is much cancellation in the definition above.)

10 2. Hopf monoids. Antipode for graphs. I = {a, b, c, x, y} x a y b c S,T b g g S g/ S Ex. For n = 3, we expect 13 terms (ordered Bell number), but a S = {a, b, c} c T = {x, y} x y s I

11 2. Hopf monoids. Characters. Let P be a Hopf monoid. A character ζ consists of maps such that for each I = S T, ζ I : P[I ] k P[S] P[T ] µ S,T P[I ] ζ S ζ T ζ I k k = k and P[ ] ζ =id k.

12 2. Hopf monoids. Polynomial invariants. Let P be a Hopf monoid and ζ a character. Define, for each x P[I ] and n N, χ I (x)(n) := S 1 S n=i (ζ S1 ζ Sn ) S1,...,S n (x). Sum over all ordered decompositions of I into n disjoint subsets. Proposition. 1. χ I (x) is a polynomial function of n. 2. χ I (x)(1) = ζ I (x). ) 3. χ I (x)( 1) = ζ I (s(x). ( reciprocity theorems) The function χ I (x) is a polynomial invariant of the structure x (canonically associated to P and ζ).

13 2. Hopf monoids. Invariants of graphs and matroids. Let ζ I : G[I ] k be ζ I (g) := { 1 if g consists of half-edges only, 0 otherwise. Then χ I (g) = chromatic polynomial of g. (For n N, counts proper colorings of g with [n].) Let ζ I : M[I ] k be ζ I (m) := { 1 if m has a unique basis, 0 otherwise. Then χ I (m) = polynomial defined by Billera-Jia-Reiner (2006). (For n N, counts m-generic functions f : I [n].) Let ζ I : Q[I ] k be ζ I (q) := { 1 if q is an antichain 0 otherwise. Then χ I (q) = (strict) order polynomial of Stanley (1970 or 1971). (For n N, counts order-preserving labelings of q with [n].)

14 3. Generalized permutahedra. Euclidean space R I The standard permutahedron := {functions x : I R}. π I := Convex Hull{bijective functions x : I [n]} R I (where n = I ). I = {a, b, c} π I = 3, 1, 2 3, 2, 1 2, 1, 3 2, 3, 1 1, 2, 3 1, 3, 2 {x a + x b + x c = 6}

15 3. Generalized permutahedra. Postnikov (2005). Postnikov, Reiner, Williams (2007). A., Benedetti, Doker (2008). Move vertices of π I keeping new edges parallel to old ones. Generalized permutahedra

16 3. Generalized permutahedra. Restriction and contraction. Given a polytope P R I and v R I, let P v := face of P where v, is maximum. Given I = S T, let v S,T R I be any vector such that { v i = v j if i, j S or i, j T, v i > v j if i S and j T. Proposition. (AA, 08) Let P be a generalized permutahedron. 1. P vs,t =: P S,T depends only on S and T. 2. There are generalized permutahedra P 1 R S and P 2 R T such that P S,T = P 1 P 2. (Note R I = R S R T.) Define P S := P 1 R S, P/ S := P 2 R T.

17 3. Generalized permutahedra. The Hopf monoid. GP[I ] := vector space with basis the set of generalized permutahedra in R I. Theorem. (AA, 08) The species of generalized permutahedra GP is a Hopf monoid with: GP[S] GP[T ] µ S,T GP[I ] P Q P Q GP[I ] S,T GP[S] GP[T ] P P S P/ S Theorem. (AA, 08) GP is a Hopf monoid. Let ζ I : GP[I ] k be ζ I (P) := { 1 if P is a point, 0 otherwise. Then χ I (P) = polynomial defined by Billera-Jia-Reiner (2006).

18 3. Generalized permutahedra. Graphs and matroids. There are G[I ] GP[I ] and M[I ] GP[I ] and Q[I ] GP[I ]. G graph graphic zonotope Z(G) Z(G) = ij G (e i e j ) = { ij G a ij(e i e j ) : 0 a ij 1}. M matroid matroid polytope P M P M = conv{e i1 + + e ik {i 1,..., i k } B}. Q poset poset polyhedron P Q P Q : x i = 0; i A x i < 0 for order ideals A Q. Proposition. (AA, 08) These are morphisms of Hopf monoids. In addition, the diagram G[I ] ζ I GP[I ] ζ I k commutes, and similarly for M[I ] and Q[I ].

19 3. Generalized permutahedra. The antipode. Theorem. Let P be a generalized permutahedron. s I (P) = ( 1) I Q P( 1) dim Q Q. The sum is over all faces Q of P. Proof. Antipode formula gives a big alternating sum of faces of P: µ S1,...,S k S1,...,S k (P) = P S1,...,S k, with lots of repeated terms. The coefficient of each face is the reduced Euler characteristic of a sphere. Note: This is the best possible formula. No cancellation at all! (One advantage of working over species...)

20 3. Generalized permutahedra. Reciprocity theorems. Corollary. χ I (P)( 1) = ( 1) I #{vertices of P}. Some instances: Graphs: χ I (g) = chromatic polynomial χ I (g)( 1) = ( 1) I #{acyclic orientations of g} (Stanley). Matroids: χ M (g) = BJR matroid polynomial χ I (m)( 1) = ( 1) I #{bases of m} (Billera-Jia-Reiner). Posets: χ Q (q) = strict order polynomial ; Ω(q) = order poly. χ Q (q)( 1) = ( 1) I χ Q (q)( n) = ( 1) I Ω Q (q)(n) (Stanley). Others: polymatroids, hypergraphs,...

21 3. Generalized permutahedra. Graphs and the antipode. s I A polytopal explanation: The graphic zonotope [a, b] + [b, c] + {c}

22 Thank you.