The algebraic and combinatorial structure of generalized permutahedra

Transcription

1 The algebraic and combinatorial structure of generalized permutahedra Marcelo Aguiar Cornell University Federico Ardila San Francisco State University MSRI Summer School July 19, / 49

2 ...or: how I came to appreciate Hopf monoids (and algebras) Marcelo Aguiar Cornell University Federico Ardila San Francisco State University University of Michigan March 17, / 49

3 ...or: how I came to appreciate Hopf monoids (and algebras) Marcelo Aguiar Cornell University Federico Ardila San Francisco State University This is work from , closely related to papers by: Carolina Benedetti, Nantel Bergeron, Lou Billera, Eric Bucher, Harm Derksen, Alex Fink, Rafael González D León, Vladimir Grujić, Joshua Hallam, Brandon Humpert, Ning Jia, Carly Klivans, John Machacek, Swapneel Mahajan, Jeremy Martin, Vic Reiner, Bruce Sagan, Tanja Stojadinović, Jacob White... 3 / 49

4 0. Inverting formal power series: Multiplication A new take on an old question: How do we invert a formal power series? Let A(x) = x a n n n! and B(x) = x b n n n! be multiplicative inverses. Assume a 0 = b 0 = 1. Then B(x) = 1/A(x) is given by b 1 = a 1 b 2 = a 2 + 2a1 2 b 3 = a 3 + 6a 2 a 1 6a1 3 b 4 = a 4 + 8a 3 a 1 + 6a2 2 36a 2 a a1 4. How to make sense of these numbers? 4 / 49

5 0. Inverting formal power series: Multiplication. Permutahedra: π 1 : point, π 2 : segment, π 3 : hexagon, π 4 : truncated octahedron... For exponential generating functions, B(x) = 1/A(x) is given by b 1 = a 1 b 2 = a 2 + 2a 2 1 b 3 = a 3 + 6a 2 a 1 6a 3 1 b 4 = a 4 + 8a 3 a 1 + 6a a 2 a a 4 1 Faces of π 4 : 1 truncated octahedron π 4 8 hexagons π 3 π 1 and 6 squares π 2 π 2 36 segments π 2 π 1 π 1 24 points π 1 π 1 π 1 π 1 5 / 49

6 0. Inverting formal power series: Composition. A new take on an old question: How do we invert a formal power series? A(x) = a n 1 x n, B(x) = b n 1 x n : compositional inverses. Assume a 0 = b 0 = 1. Then B(x) = A(x) 1 is given by Lagrange inversion: b 1 = a 1 b 2 = a 2 + 2a 2 1 b 3 = a 3 + 5a 2 a 1 5a 3 1 b 4 = a 4 + 6a 3 a 1 + 3a a 2 a a 4 1. How to make sense of these numbers? 6 / 49

7 0. Inverting formal power series: Composition. Associahedra: a 1 : point, a 2 : segment, a 3 : pentagon, a 4 : 3-associahedron... For ordinary generating functions, B(x) = A(x) 1 is given by b 1 = a 1 b 2 = a 2 + 2a 2 1 b 3 = a 3 + 5a 2 a 1 5a 3 1 b 4 = a 4 + 6a 3 a 1 + 3a a 2 a a 4 1 Faces of a 4 : 1 3-associahedron a 4 6 pentagons a 3 a 1 and 3 squares a 2 a 2 21 segments a 2 a 1 a 1 14 points a 1 a 1 a 1 a 1 7 / 49

8 Hopf algebras? Hopf monoids? A remark. Hopf monoids are a bit more abstract than Hopf algebras, but better suited for many combinatorial purposes. We have a functor Hopf monoids Hopf algebras so there are Hopf algebra analogs of all of our results. 8 / 49

9 1. Species and Hopf monoids. (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], consistent with composition. (Relabeling.) Think: ways of putting a certain combinatorial structure on set I. P[I ] = span{combinatorial structures of type P on I } Examples. G[I ] := span{graphs on vertex set I }. Q[I ] := span{posets on I }. M[I ] := span{matroids on ground set I }. 9 / 49

10 Hopf monoids. (Joni-Rota, Coalgebras and bialgebras in combinatorics.) (Aguiar-Mahajan, Monoidal functors, species, and Hopf algebras.) A 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 ]. Think: We have rules for merging and breaking our structures. These maps should satisfy various axioms, including the following: 10 / 49

11 Hopf monoids. Compatibility axiom. Fix two decompositions S T = I = S T, and let A, B, C, D be: S A B S T T Then this diagram must commute: C D 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 ] Think: merge, then break = break, then merge 11 / 49

12 Examples of Hopf monoids: Graphs. G[I ] := span{graphs (with half edges) on vertex set I }. The species of graphs G is a Hopf monoid with G[S] G[T ] µ S,T G[I ] G[I ] S,T G[S] G[T ] g 1 g 2 g 1 g 2 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 12 / 49

13 Examples of Hopf monoids: Posets. Q[I ] := span{posets on I }. The species of posets Q is a Hopf monoid, under: Product: disjoint union. Coproduct: splitting. µ S,T : Q[S] Q[T ] Q[I ] q 1 q 2 q 1 q 2 S,T : Q[I ] Q[S] Q[T ] { q S q T if S is an order ideal of q q 0 otherwise 13 / 49

14 Examples of Hopf monoids: Matroids. M[I ] := span{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 is b B A with A a b B. Prototypical example: I = collection of vectors in V, B = subsets which are bases of V 14 / 49

15 Other Hopf monoids. There are many other Hopf monoids of interest in combinatorics. A few of them: graphs G posets P matroids M set partitions Π (symmetric functions) paths A (Faá di Bruno) simplicial complexes SC hypergraphs HG building sets BS 15 / 49

16 3. The antipode of a Hopf monoid. The antipode of a connected Hopf monoid P consists of the maps s I = I =S 1 S k k 1 s I : P[I ] P[I ] ( 1) k µ S1,...,S k S1,...,S k. summing over all ordered set partitions I = S 1 S k. (S i ) Think: groups inverses Hopf monoids antipodes (s 2 = id) 16 / 49

17 3. The antipode of a Hopf monoid. The antipode of a connected Hopf monoid P consists of the maps s I = I =S 1 S k k 1 s I : P[I ] P[I ] ( 1) k µ S1,...,S k S1,...,S k. summing over all ordered set partitions I = S 1 S k. (S i ) Think: groups inverses Hopf monoids antipodes (s 2 = id) General problem. Find the simplest possible formula for the antipode of a Hopf monoid. (Usually there is much cancellation in the definition above.) 16 / 49

18 Examples: The antipode of a graph, matroid, poset. Ex. Takeuchi: s I = ( 1) k µ S1,...,S k S1,...,S k. I =S 1 S k For n = 3, 4 this sum has 13, 73 terms. However, How do we explain (and predict) the simplification? 17 / 49

19 Examples: The antipode of a graph, matroid, poset. Ex. Takeuchi: s I = ( 1) k µ S1,...,S k S1,...,S k. I =S 1 S k For n = 3, 4 this sum has 13, 73 terms. However, How do we explain (and predict) the simplification? Inclusion-exclusion Sign-reversing involution Möbius functions 17 / 49

20 Examples: The antipode of a graph, matroid, poset. Ex. Takeuchi: s I = ( 1) k µ S1,...,S k S1,...,S k. I =S 1 S k For n = 3, 4 this sum has 13, 73 terms. However, How do we explain (and predict) the simplification? Inclusion-exclusion Sign-reversing involution Möbius functions Euler characteristics 17 / 49

21 Example: The antipode of a graph. Ex. Takeuchi: s I = I =S 1 S k ( 1) k µ S1,...,S k S1,...,S k. a b c s I a b c + a b c + a b c + a b c + a b c a b c a b c a b c a b c Theorem. (Aguiar A., Humpert Martin, Benedetti Sagan) s(g) = ( 1) I r(f ) a(g/f )f f flat where a(h) = number of acyclic orientations of h. 18 / 49

22 Other examples of antipodes. There are many other Hopf monoids of interest. Only some of their antipodes were known. graphs G: Humpert Martin 10, Benedetti Sagan 15 posets P: Schmitt, 94 matroids M:? set partitions / symm fns. Π: Aguiar Mahajan 10 paths A:? simplicial complexes SC: Benedetti Hallam Michalak 16 hypergraphs HG:? building sets BS:? Goal 1. a unified approach to compute these and other antipodes. (We do this). 19 / 49

23 4. Generalized permutahedra. Euclidean space R I := {functions x : I R}. The standard permutahedron is 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} 20 / 49

24 The standard permutahedron. The standard permutahedron π I for I = 4: 21 / 49

25 Generalized permutahedra. Edmonds (70), Postnikov (05), Postnikov Reiner Williams (07),... Equivalent formulations: Move facets of standard permutahedron without passing vertices. Move vertices while preserving edge directions. Change polytope while coarsening the normal fan. Generalized permutahedra: 22 / 49

26 Generalized permutahedra in 3-D. The permutahedron π 4. A generalized permutahedron. 23 / 49

27 Generalized permutahedra. Product. Key Lemma. If P and Q are generalized permutahedra in R S and R T, respectively, and I = S T, then P Q is a generalized permutahedron in R I. Define the multiplication of two generalized permutahedra to be P Q := P Q 24 / 49

28 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 e S,T R I have coordinates: { e s = 1 for s S e t = 0 for t T Key Lemma. If P is a generalized permutahedron and I = S T, P es,t = P 1 P 2. for generalized permutahedra P 1 R S and P 2 R T. Define the restriction and contraction P S := P 1 R S, P/ S := P 2 R T. 25 / 49

29 Generalized permutahedra: Restriction and contraction. Given I = S T, the vector e S,T R I has e s = 1, e t = 0. Then P es,t = P S P/ S Two examples: = X = X 26 / 49

30 5. The Hopf monoid of generalized permutahedra. GP[I ] := span {generalized permutahedra in R I }. Define a product and coproduct: 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. (Aguiar A.) GP is a Hopf monoid. (Derksen Fink 10 proved a very similar result.) 27 / 49

31 Generalized permutahedra: Posets, graphs, matroids. There are generalized permutahedra associated to: G graph graphic zonotope Z(G) Z(G) = ij G (e i e j ) M matroid matroid polytope P M P M = conv{e i1 + + e ik {i 1,..., i k } is a basis of M}. P poset poset cone C P C P : cone{e i e j : i < j in P}. Proposition. (Aguiar A.) These give inclusions of Hopf monoids:. G GP, M GP, P GP 28 / 49

32 The antipode of GP. Theorem. (Aguiar A.) Let P be a generalized permutahedron. s(p) = ( 1) I Q P( 1) dim Q Q. The sum is over all faces Q of P. 29 / 49

33 The antipode of GP. Theorem. (Aguiar A.) Let P be a generalized permutahedron. s(p) = ( 1) I Q P( 1) dim Q Q. The sum is over all faces Q of P. Proof. Takeuchi: s(p) = I =S 1 S k ( 1) k P S1,...,S k where P S1,...,S k is the maximum face of P in direction S 1 S k. Coeff. of a face Q: huge sum of 1s and 1s. How to simplify it? It is the reduced Euler characteristic of a sphere! This is the best possible formula. No cancellation or grouping! (One advantage of working over species!) 29 / 49

34 The antipodes of GP, Q, G, M. Theorem. (Aguiar A.) Let P be a generalized permutahedron. s(p) = ( 1) I Q P( 1) dim Q Q. The sum is over all faces Q of P. As a consequence, we get Corollary. Best possible formulas for the antipodes of: posets (Schmitt, 94) graphs (new also Humpert Martin, 12) matroids (new) 30 / 49

35 Example: The antipode of graphs. The (post-cancellation) antipode of a graph: a b c s I Geometric explanation: The faces of the graphic zonotope! a b c 31 / 49

36 Many antipode formulas. objects polytopes Hopf structure antipode set partitions permutahedra Joni-Rota Joni-Rota paths associahedra Joni-Rota, new Haiman-Schmitt, new graphs graphic zonotopes Schmitt new, Humpert-Martin matroids matroid polytopes Schmitt new posets braid cones Schmitt new submodular fns gen. permutahedra Derksen-Fink new hypergraphs hg-polytopes new new simplicial cxes new: sc-polytopes Benedetti et al Benedetti et al building sets nestohedra new, Grujić et al new weaves (graphs) graph associahedra new new Lots of interesting algebra and combinatorics. 32 / 49

37 6. Characters of Hopf monoids. Let P be a Hopf monoid. A character ζ consists of maps ζ I : P[I ] k which are multiplicative: for each I = S T, s P[S], t P[T ]: ζ S (s)ζ T (t) = ζ I (s t). Think: character = multiplicative function on our objects 33 / 49

38 Group of characters, inversion. The group of characters of a Hopf monoid P: Operation: For p P[I ] ζ 1 ζ 2 (p) = ζ 1 (p S )ζ 2 (p/ S ) I =S T Identity: Inverses: u(p) = { 1 if p = 1 P[ ] 0 otherwise. ζ 1 = ζ s This group is hard to describe in general. To understand it, we must understand the antipode. 34 / 49

39 Group of characters, reciprocity. The group of characters of a Hopf monoid P: operation: convolution (via coproduct ) This is hard to describe in general. inverse: antipode 35 / 49

40 Group of characters, reciprocity. The group of characters of a Hopf monoid P: operation: convolution (via coproduct ) inverse: antipode This is hard to describe in general. Let s study two special cases: 1. The Hopf monoid of permutahedra. vertices: permutations faces: ordered set partitions (Schoute, 1911) 35 / 49

41 Group of characters, reciprocity. The group of characters of a Hopf monoid P: operation: convolution (via coproduct ) inverse: antipode This is hard to describe in general. Let s study two special cases: 1. The Hopf monoid of permutahedra. vertices: permutations faces: ordered set partitions (Schoute, 1911) 2. The Hopf monoid of associahedra. vertices: binary parenthesizations of a word faces: arbitrary parenthesizations (Tamari 51, Stasheff-Milnor 63, Haiman-Lee 89, Loday 02,...) 35 / 49

42 Group of characters, reciprocity for permutahedra. 1. Let Π = submonoid of GP generated by permutahedra. Proposition. The group of characters of Π is isomorphic to the x group of exponential generating functions 1 + a 1 x + a 2 2 2! + under multiplication. Sketch of Proof. char. of Π seq. 1, a 1, a 2,... egf A(x) = 1 + a 1 x + a 2 x 2 2! + Group operation: characters: a b(p) = I =S T a(p S)b(p/ S ) sequences: c n = ( n k) ak b n k power series: C(x) = A(x)B(x) 36 / 49

43 Group of characters, reciprocity for permutahedra. Proposition. The group of characters of Π is the multiplicative group of exponential generating functions 1 + a 1 x + a 2 x 2 2! +. inversion of egfs antipode of permutahedra For exponential generating functions, B(x) = 1/A(x) is given by b 1 = a 1 b 2 = a 2 + 2a 2 1 b 3 = a 3 + 6a 2 a 1 6a 3 1 b 4 = a 4 + 8a 3 a 1 + 6a a 2a a4 1. These numbers come from the antipode of the permutahedron: s(π 4 ) = π 4 + 8π 3 π 1 + 6π π 2π π4 1 (1 perm., 8 hexagons and 6 squares, 36 segments, 24 points.) 37 / 49

44 Group of characters, reciprocity for associahedra. 2. Let A = submonoid of GP generated by Loday s associahedra. Proposition. The group of characters of A is isomorphic to the group of ordinary generating functions x + a 1 x 2 + a 2 x 3 + under composition. Sketch of Proof. char. of Π seq 1, a 1, a 2,... ogf A(x) = x + a 1 x 2 + a 2 x 3 + Group operation: characters: a b(p) = I =S T a(p S)b(p/ S ) sequences: c n = a k 1 b i1 b i2 b ik power series: C(x) = A(B(x)) 38 / 49

45 Group of characters, reciprocity for associahedra. Proposition. The group of characters of Π is the compositional group of generating functions x + a 1 x 2 + a 2 x 3 +. compositional inversion of gfs antipode of associahedra For ordinary generating functions, B(x) = A(x) 1 is given by b 1 = a 1 b 2 = a 2 + 2a 2 1 b 3 = a 3 + 5a 2 a 1 5a 3 1 b 4 = a 4 + 6a 3 a 1 + 3a a 2a a4 1. These numbers come from the antipode of the associahedron: s(a 4 ) = a 4 + 6a 3 a 1 + 3a a 2a a4 1 (1 assoc., 6 pentagons and 3 squares, 21 segments, 14 points.) 39 / 49

46 Group of characters, reciprocity for associahedra. This reformulation of the Lagrange inversion formula for B(x) = A(x) 1 may be seen as an answer to Loday s question: There exists a short operadic proof of the [Lagrange inversion] formula which explicitly involves the parenthesizings, but it would be interesting to find one which involves the topological structure of the associahedron. (Loday, 2005) s(a 4 ) = a 4 + 6a 3 a 1 + 3a a 2a a4 1 (for Loday s a 4!) 40 / 49

47 Group of characters, reciprocity for associahedra. This reformulation of the Lagrange inversion formula for B(x) = A(x) 1 may be seen as an answer to Loday s question: There exists a short operadic proof of the [Lagrange inversion] formula which explicitly involves the parenthesizings, but it would be interesting to find one which involves the topological structure of the associahedron. (Loday, 2005) s(a 4 ) = a 4 + 6a 3 a 1 + 3a a 2a a4 1 (for Loday s a 4!) Project. (A. Benedetti González.) Compute the group of characters and reciprocity rules for other interesting submonoids of GP. 40 / 49

48 Group of characters, reciprocity for associahedra. Two (of many) interesting enumerative consequences: 1. The Hopf monoid of permutahedra. Recall: ζ(g) is 1 if g has no edges and 0 otherwise. Proposition. (Aguiar A.) ( 1) n ζ 1 (K n ) = D n (# of derangements) (Conjectured by Humpert + Martin 12) Key idea: The polytope of K n is the permutahedron π n, so we can compute in the group of characters of Π. 41 / 49

49 Group of characters, reciprocity for associahedra. Two (of many) interesting enumerative consequences: 1. The Hopf monoid of permutahedra. Recall: ζ(g) is 1 if g has no edges and 0 otherwise. Proposition. (Aguiar A.) ( 1) n ζ 1 (K n ) = D n (# of derangements) (Conjectured by Humpert + Martin 12) Key idea: The polytope of K n is the permutahedron π n, so we can compute in the group of characters of Π. 2. The Hopf monoid of associahedra. Proposition. (Aguiar A.) The number of face parallelism classes of Loday s associahedron a n is Catalan # C n. Counted by dimension: Narayana numbers. This follows from the antipode of a n. 41 / 49

50 7. Polynomial invariants from characters. Each character gives a polynomial invariant. Let P be a Hopf monoid and ζ a character. Define, for each p P[I ] and n N, χ(p)(n) := (ζ S1 ζ Sn ) S1,...,S n (p), S 1 S n=i summing over all weak ordered set partitions I = S 1 S n (where S i could be empty). 42 / 49

51 7. Polynomial invariants from characters. Each character gives a polynomial invariant. Let P be a Hopf monoid and ζ a character. Define, for each p P[I ] and n N, χ(p)(n) := (ζ S1 ζ Sn ) S1,...,S n (p), S 1 S n=i summing over all weak ordered set partitions I = S 1 S n (where S i could be empty). Proposition. 1. χ(p)(n) is a polynomial function of n. ( ) 2. χ(p)( n) = χ s(p) (n). (antipode reciprocity thms) χ(p): a polynomial invariant of the structure p. 42 / 49

52 Examples: Invariants of posets, graphs, matroids. Graphs: ζ(g) := { 1 if g has no edges, 0 otherwise. χ(g) = chromatic polynomial of g. (Birkhoff, 1912). For n N, it counts proper colorings of g with [n]. 43 / 49

53 Examples: Invariants of posets, graphs, matroids. Graphs: ζ(g) := { 1 if g has no edges, 0 otherwise. χ(g) = chromatic polynomial of g. (Birkhoff, 1912). For n N, it counts proper colorings of g with [n]. { Posets: 1 if q is an antichain, ζ(q) := 0 otherwise. χ(q) = strict order polynomial of q (Stanley, 1970). For n N, it counts order-preserving labelings of q with [n]. 43 / 49

54 Examples: Invariants of posets, graphs, matroids. Graphs: ζ(g) := { 1 if g has no edges, 0 otherwise. χ(g) = chromatic polynomial of g. (Birkhoff, 1912). For n N, it counts proper colorings of g with [n]. { Posets: 1 if q is an antichain, ζ(q) := 0 otherwise. χ(q) = strict order polynomial of q (Stanley, 1970). For n N, it counts order-preserving labelings of q with [n]. { Matroids: 1 if m has a unique basis, ζ(m) := 0 otherwise. χ(m) = BJR polynomial of m. (Billera-Jia-Reiner, 2006). For n N, it counts m-generic functions f : I [n]. 43 / 49

55 Examples: Invariants of posets, graphs, matroids. Graphs: ζ(g) := { 1 if g has no edges, 0 otherwise. χ(g) = chromatic polynomial of g. (Birkhoff, 1912). For n N, it counts proper colorings of g with [n]. For n, it counts... { Posets: 1 if q is an antichain, ζ(q) := 0 otherwise. χ(q) = strict order polynomial of q (Stanley, 1970). For n N it counts order-preserving n-labelings of q. For n it counts... { Matroids: 1 if m has a unique basis, ζ(m) := 0 otherwise. χ(m) = BJR polynomial of m. (Billera-Jia-Reiner, 2006). For n N, it counts m-generic functions f : I [n]. For n it counts / 49

56 Examples: Invariants of posets, graphs, matroids. Graphs: ζ(g) := { 1 if g has no edges, 0 otherwise. χ(g) = chromatic polynomial of g. (Birkhoff, 1912). For n N, it counts proper colorings of g with [n]. For n, it counts... { Posets: 1 if q is an antichain, ζ(q) := 0 otherwise. χ(q) = strict order polynomial of q (Stanley, 1970). For n N it counts order-preserving n-labelings of q. For n it counts... { Matroids: 1 if m has a unique basis, ζ(m) := 0 otherwise. χ(m) = BJR polynomial of m. (Billera-Jia-Reiner, 2006). For n N, it counts m-generic functions f : I [n]. For n it counts... Goal 2. A unified approach to these and other results. 44 / 49

57 Generalized permutahedra: Posets, graphs, matroids. We have Hopf submonoids: P GP, G GP, M GP. Theorem. (Aguiar A., Billera Jia Reiner) The characters: ζ(q) (posets), ζ(g) (graphs), ζ(m) (matroids) are shadows of the same character on GP: { 1 if P is a point, ζ I (P) := 0 otherwise. The (strict order)/(chromatic)/(bjr matroid) polynomials: χ(q)(n) (posets), χ(g)(n) (graphs), χ(m)(n) (matroids) are shadows of the same polynomial on GP: χ(p)(n) := # of P-generic functions f : I [n] 45 / 49

58 Reciprocity. For any character ζ and associated polynomial χ(p) on GP, ( ) ( ) χ(p)( 1) = ζ s(p) χ(p)( n) = χ s(p) (n). Corollary. For ζ I (P) = 1 (if P is a point) or 0 (otherwise), χ(p)( 1) = ( 1) I #{vertices of P}. This gives a unified explanation of: Graphs: χ(g) = chromatic polynomial χ(g)( 1) = ( 1) I #{acyclic orientations of g} (Stanley). Posets: χ(q) = strict order polynomial ; Ω(q) = order poly. χ(q)( n) = ( 1) I Ω(q)(n) (Stanley). Matroids: χ(m) = BJR matroid polynomial χ(m)( 1) = ( 1) I #{bases of m} (Billera-Jia-Reiner). 46 / 49

59 8. A current direction: the polytope algebra. In the polytope algebra: The antipode of P is s(p) = ( 1) I Q P ( 1) dim Q Q = P o where P o = rel. interior of P. An involution! (Ehrhart,McMullen) 47 / 49

60 8. A current direction: the polytope algebra. In the polytope algebra: The antipode of P is s(p) = ( 1) I Q P ( 1) dim Q Q = P o where P o = rel. interior of P. An involution! (Ehrhart,McMullen) Brion s theorem: P = cone v (P) v vertex Theorem. (Aguiar A.) There is a Brion morphism B : GP P P v vertex poset v (P). This restricts to B: W (weaves) Connes-Kreimer (rooted trees). Several interesting algebraic and combinatorial consequences. 47 / 49

61 Current and future directions. For GP and its submonoids P: Describe the Lie algebra P(P ) of primitive elements of P, which determines P via Cartier Milnor Moore. Describe the Brion map B : P(P ) P(GP ) of Lie algebras. Extend this to deformations GP W of Coxeter permutahedra π W, apply them to Coxeter associahedra, Coxeter matroids,... This requires extending Hopf monoids to type W. (Aguiar Mahajan) Extend to deformations of any simple polytope. This requires extending Hopf monoids to any arrangement. (Aguiar Mahajan) Study the connection with polyhedral valuations on generalized permutahedra. (Derksen Fink) 48 / 49

62 Muchas gracias! The preprints are coming soon! 49 / 49