The algebraic and combinatorial structure of generalized...

inverting series hopf monoids antipodes generalized permutahedra GP characters reciprocity future

The algebraic and combinatorial structure ofgeneralized permutahedra

Marcelo Aguiar Federico ArdilaCornell University San Francisco State University

MSRI Summer SchoolJuly 19, 2017

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...or: how I came to appreciateHopf monoids (and algebras)


University of MichiganMarch 17, 2017

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...or: how I came to appreciateHopf monoids (and algebras)


This is work from 2008-2017, closely related to papers by:

Carolina Benedetti, Nantel Bergeron, Lou Billera, Eric Bucher, Harm

Derksen, Alex Fink, Rafael Gonzalez D’Leon, Vladimir Grujic, Joshua

Hallam, Brandon Humpert, Ning Jia, Carly Klivans, John Machacek,

Swapneel Mahajan, Jeremy Martin, Vic Reiner, Bruce Sagan, Tanja

Stojadinovic, Jacob White...

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0. Inverting formal power series: MultiplicationA new take on an old question:

How do we invert a formal power series?

Let A(x) =∑

anxn

n! and B(x) =∑

bnxn

n! be multiplicative inverses.Assume a0 = b0 = 1.

Then B(x) = 1/A(x) is given by

b1 = −a1

b2 = −a2 + 2a21

b3 = −a3 + 6a2a1 − 6a31

b4 = −a4 + 8a3a1 + 6a22 − 36a2a

21 + 24a4

1

...

How to make sense of these numbers?4 / 49


0. Inverting formal power series: Multiplication.Permutahedra:

π1: point, π2: segment, π3: hexagon, π4: truncated octahedron...

For exponential generating functions, B(x) = 1/A(x) is given by

b1 = −a1

b2 = −a2 + 2a21

b3 = −a3 + 6a2a1 − 6a31

b4 = −a4 + 8a3a1 + 6a22 − 36a2a

21 + 24a4

1

Faces of π4: • 1 truncated octahedron π4

• 8 hexagons π3 × π1 and 6 squares π2 × π2

• 36 segments π2 × π1 × π1

• 24 points π1 × π1 × π1 × π15 / 49


0. Inverting formal power series: Composition.

A new take on an old question:

How do we invert a formal power series?

A(x) =∑

an−1xn, B(x) =

∑bn−1x

n: compositional inverses.Assume a0 = b0 = 1.

Then B(x) = A(x)〈−1〉 is given by Lagrange inversion:

b1 = −a1

b2 = −a2 + 2a21

b3 = −a3 + 5a2a1 − 5a31

b4 = −a4 + 6a3a1 + 3a22 − 21a2a

21 + 14a4

1

...

How to make sense of these numbers?6 / 49


0. Inverting formal power series: Composition.Associahedra:

a1: point, a2: segment, a3: pentagon, a4: 3-associahedron...

For ordinary generating functions, B(x) = A(x)〈−1〉 is given by

b1 = −a1

b2 = −a2 + 2a21

b3 = −a3 + 5a2a1 − 5a31

b4 = −a4 + 6a3a1 + 3a22 − 21a2a

21 + 14a4

1

Faces of a4: • 1 3-associahedron a4

• 6 pentagons a3 × a1 and 3 squares a2 × a2

• 21 segments a2 × a1 × a1

• 14 points a1 × a1 × a1 × a17 / 49


Hopf algebras? Hopf monoids? A remark.

Hopf monoids are a bit more abstract than Hopf algebras, butbetter suited for many combinatorial purposes. We have a functor

Hopf monoids −→ Hopf algebras

so there are Hopf algebra analogs of all of our results.

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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 ] = spancombinatorial structures of type P on I

Examples.

G[I ] := spangraphs on vertex set I.Q[I ] := spanposets on I.M[I ] := spanmatroids on ground set I.

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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 T , product and coproduct maps

P[S ]⊗ P[T ]µS,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:

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Hopf monoids. Compatibility axiom.Fix two decompositions S tT = I = S ′tT ′, and let A,B,C ,D be:'

&

$

%

S

T

'

&

$

%S ′ T ′

'

&

$

%

A B

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]

µA,C⊗µB,D

P[S]⊗ P[T ]

µS,T

//

∆A,B⊗∆C,D

OO

P[I ]∆S′,T ′

// P[S ′]⊗ P[T ′]

Think: “merge, then break” = ”break, then merge”

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Examples of Hopf monoids: Graphs.G[I ] := spangraphs (with half edges) on vertex set I.The species of graphs G is a Hopf monoid with

G[S ]⊗ G[T ]µS,T−−−→ G[I ] G[I ]

∆S,T−−−→ G[S ]⊗ G[T ]

g1 ⊗ g2 7−→ g1 t g2 g 7−→ g |S ⊗ g/S

where g1 t g2 = disjoint union of g1 and g2,

g |S = keep everything incident to S ,

g/S = remove everything incident to S .

r rr r

r

@@@

JJ

b

a

c

y

xI = a, b, c , x , y

g

∆S,T7−→ r rrJJ

b

a

c

S = a, b, c

g |S

rr@@

@

y

xT = x , y

g/S

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Examples of Hopf monoids: Posets.

Q[I ] := spanposets on I.

The species of posets Q is a Hopf monoid, under:

Product: disjoint union.

µS ,T : Q[S ]⊗ Q[T ] −→ Q[I ]

q1 ⊗ q2 7−→ q1 t q2

Coproduct: splitting.

∆S ,T : Q[I ] −→Q[S ]⊗ Q[T ]

q 7−→

q|S ⊗ q|T if S is an order ideal of q

0 otherwise

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Examples of Hopf monoids: Matroids.M[I ] := spanmatroids on I.The species of matroids M is a Hopf monoid with

M[S ]⊗M[T ]µS,T−−−→ M[I ] M[I ]

∆S,T−−−→ M[S ]⊗M[T ]

m1 ⊗m2 7−→ m1 ⊕m2 m 7−→ m|S ⊗m/S

wherem1 ⊕m2 = direct sum of m1 and m2,

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

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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 (Faa di Bruno)• simplicial complexes SC• hypergraphs HG• building sets BS

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3. The antipode of a Hopf monoid.

The antipode of a connected Hopf monoid P consists of the maps

sI : P[I ]→ P[I ]

sI =∑

I=S1t···tSkk≥1

(−1)k µS1,...,Sk ∆S1,...,Sk .

summing over all ordered set partitions I = S1 t · · · t Sk . (Si 6= ∅)

Think: groups 7→ inversesHopf monoids 7→ antipodes

(s2 = id)

General problem. Find the simplest possible formulafor the antipode of a Hopf monoid.

(Usually there is much cancellation in the definition above.)

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Examples: The antipode of a graph, matroid, poset.Ex. Takeuchi: sI =

∑I=S1t···tSk

(−1)k µS1,...,Sk ∆S1,...,Sk .

For n = 3, 4 this sum has 13, 73 terms. However,

How do we explain (and predict) the simplification?

• Inclusion-exclusion• Sign-reversing involution• Mobius functions• Euler characteristics

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∑I=S1t···tSk

(−1)k µS1,...,Sk ∆S1,...,Sk .



• Inclusion-exclusion• Sign-reversing involution• Mobius functions

• Euler characteristics

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∑I=S1t···tSk

(−1)k µS1,...,Sk ∆S1,...,Sk .



• Inclusion-exclusion• Sign-reversing involution• Mobius functions• Euler characteristics 17 / 49


Example: The antipode of a graph.Ex. Takeuchi: sI =

∑I=S1t···tSk (−1)k µS1,...,Sk ∆S1,...,Sk .

r r ra b c

sI7−→ − r r ra b c

+ r r ra b c

+ r r ra b c

+ r r ra b c

+ r r r@a b c

− r r ra b c

− r r r@a b c

− r r r@a b c

− r r r@a b c

Theorem. (Aguiar–A., Humpert–Martin, Benedetti–Sagan)

s(g) =∑f flat

(−1)|I |−r(f )a(g/f )f

where a(h) = number of acyclic orientations of h.

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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).

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4. Generalized permutahedra.

Euclidean space RI := functions x : I → R.The standard permutahedron is

πI := Convex Hullbijective functions x : I → [n] ⊆ RI

(where n = |I |).

rr

rrr

r"""

"""b

bb

bbb

3, 2, 1

3, 1, 2

1, 3, 2

1, 2, 3

2, 3, 1

2, 1, 3I = a, b, c

πI =

xa + xb + xc = 6

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The standard permutahedron.

The standard permutahedron πI for |I | = 4:

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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.

rr

rrr

r"""

"""b

bb

bbb

Generalized permutahedra:

rr

rr

rr

"""

""

bbbb

bbb r

rrr

r

r"

""

bbb

bb

bbb

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Generalized permutahedra in 3-D.

The permutahedron π4. A generalized permutahedron.

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Generalized permutahedra. Product.

Key Lemma. If P and Q are generalized permutahedra in RS andRT , respectively, and I = S t T , then

P × Q

is a generalized permutahedron in RI .

Define the multiplication of two generalized permutahedra to be

P ⊗ Q := P × Q

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Generalized permutahedra. Restriction and contraction.Given a polytope P ⊆ RI and v ∈ RI , let

Pv := face of P where 〈v ,−〉 is maximum.

Given I = S t T , let eS ,T ∈ RI have coordinates:es = 1 for s ∈ S

et = 0 for t ∈ T

Key Lemma. If P is a generalized permutahedron and I = S t T ,

PeS,T = P1 × P2.

for generalized permutahedra P1 ⊆ RS and P2 ⊆ RT .

Define the restriction and contraction

P|S := P1 ⊆ RS , P/S := P2 ⊆ RT .

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Generalized permutahedra: Restriction and contraction.Given I = S t T , the vector eS,T ∈ RI has es = 1, et = 0. Then

PeS,T = P|S × P/S

Two examples:

= X

= X

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5. The Hopf monoid of generalized permutahedra.

GP[I ] := span generalized permutahedra in RI.

Define a product and coproduct:

GP[S ]⊗ GP[T ]µS,T−−−→ GP[I ] GP[I ]

∆S,T−−−→ GP[S ]⊗ GP[T ]

P ⊗ Q 7−→ P × Q P 7−→ P|S ⊗ P/S

Theorem. (Aguiar–A.)GP is a Hopf monoid.

(Derksen–Fink 10 proved a very similar result.)

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Generalized permutahedra: Posets, graphs, matroids.

There are generalized permutahedra associated to:

G graph → graphic zonotope Z (G )Z (G ) =

∑ij∈G (ei − ej)

M matroid → matroid polytope PM

PM = convei1 + · · ·+ eik | i1, . . . , ik is a basis of M.

P poset → poset cone CP

CP : coneei − ej : i < j in P.

Proposition. (Aguiar–A.) These give inclusions of Hopf monoids:

G → GP, M → GP, P → GP

.

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The antipode of GP.Theorem. (Aguiar–A.) Let P be a generalized permutahedron.

s(P) = (−1)|I |∑Q≤P

(−1)dimQ Q.

The sum is over all faces Q of P.

Proof. Takeuchi:

s(P) =∑

I=S1t···tSk

(−1)k PS1,...,Sk

where PS1,...,Sk is the maximum face of P in direction S1| · · · |Sk .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!)

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The antipodes of GP, Q, G, M.

Theorem. (Aguiar–A.) Let P be a generalized permutahedron.

s(P) = (−1)|I |∑Q≤P

(−1)dimQ 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)

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Example: The antipode of graphs.

The (post-cancellation) antipode of a graph:

r r ra b c

sI7−→ − r r r+ r r r + r r r + r r r + r r r@− r r r − r r r@ − r r r@ − r r r@

Geometric explanation: The faces of the graphic zonotope!

rr

rr

bb

bb

b

bb

bb

b

r r ra b c

r r rr r r

r r rr r r@r r r@

r r rr r r@

r r r@

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Many antipode formulas.

objects polytopes Hopf structure antipode

set partitions permutahedra Joni-Rota Joni-Rotapaths associahedra Joni-Rota, new Haiman-Schmitt, newgraphs graphic zonotopes Schmitt new, Humpert-Martinmatroids matroid polytopes Schmitt newposets braid cones Schmitt newsubmodular fns gen. permutahedra Derksen-Fink newhypergraphs hg-polytopes new newsimplicial cxes new: sc-polytopes Benedetti et al Benedetti et albuilding sets nestohedra new, Grujic et al newweaves (graphs) graph associahedra new new

Lots of interesting algebra and combinatorics.

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6. Characters of Hopf monoids.

Let P be a Hopf monoid. A character ζ consists of maps

ζI : P[I ]→ k

which are multiplicative: for each I = S t T , s ∈ P[S ], t ∈ P[T ]:

ζS(s)ζT (t) = ζI (s · t).

Think: character = multiplicative function on our objects

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Group of characters, inversion.

The group of characters of a Hopf monoid P:

Operation: For p ∈ P[I ]

ζ1 ∗ ζ2(p) =∑

I=StTζ1(p|S)ζ2(p/S)

Identity:

u(p) =

1 if p = 1 ∈ P[∅]0 otherwise.

Inverses:ζ−1 = ζ s

• This group is hard to describe in general.• To understand it, we must understand the antipode.

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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: permutationsfaces: ordered set partitions(Schoute, 1911)

2. The Hopf monoid of associahedra.

vertices: binary parenthesizations of a wordfaces: arbitrary parenthesizations(Tamari 51, Stasheff-Milnor 63,Haiman-Lee 89, Loday 02,...)

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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:


vertices: permutationsfaces: ordered set partitions(Schoute, 1911)


vertices: binary parenthesizations of a wordfaces: arbitrary parenthesizations(Tamari 51, Stasheff-Milnor 63,Haiman-Lee 89, Loday 02,...)

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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 thegroup of exponential generating functions 1 + a1x + a2

x2

2! + · · ·under multiplication.

Sketch of Proof.char. of Π ↔ seq. 1, a1, a2, . . . ↔ egf A(x) = 1 + a1x + a2

x2

2! + · · ·

Group operation:

characters: a ∗ b(p) =∑

I=StT a(p|S)b(p/S)

sequences: cn =∑(n

k

)akbn−k

power series: C (x) = A(x)B(x)

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Group of characters, reciprocity for permutahedra.

Proposition. The group of characters of Π is the multiplicative groupof exponential generating functions 1 + a1x + a2

x2

2! + · · ·.

inversion of egfs ←→ antipode of permutahedra

For exponential generating functions, B(x) = 1/A(x) is given byb1 = −a1

b2 = −a2 + 2a21

b3 = −a3 + 6a2a1 − 6a31

b4 = −a4 + 8a3a1 + 6a22 − 36a2a

21 + 24a4

1...

These numbers come from theantipode of the permutahedron:s(π4) = −π4 + 8π3π1 + 6π2

2 − 36π2π21 + 24π4

1

(1 perm., 8 hexagons and 6 squares, 36 segments, 24 points.)37 / 49


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 thegroup of ordinary generating functions x + a1x

2 + a2x3 + · · ·

under composition.

Sketch of Proof.char. of Π ↔ seq 1, a1, a2, . . . ↔ ogf A(x) = x + a1x

2 + a2x3 + · · ·

Group operation:

characters: a ∗ b(p) =∑

I=StT a(p|S)b(p/S)

sequences: cn =∑

ak−1bi1bi2 · · · bikpower series: C (x) = A(B(x))

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Group of characters, reciprocity for associahedra.

Proposition. The group of characters of Π is the compositional groupof generating functions x + a1x

2 + a2x3 + · · ·.

compositional inversion of gfs ←→ antipode of associahedra

For ordinary generating functions, B(x) = A(x)〈−1〉 is given byb1 = −a1

b2 = −a2 + 2a21

b3 = −a3 + 5a2a1 − 5a31

b4 = −a4 + 6a3a1 + 3a22 − 21a2a

21 + 14a4

1...These numbers come from theantipode of the associahedron:s(a4) = −a4 + 6a3a1 + 3a2

2 − 21a2a21 + 14a4

1

(1 assoc., 6 pentagons and 3 squares, 21 segments, 14 points.)39 / 49


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 [Lagrangeinversion] formula which explicitly involves theparenthesizings, but it would be interesting to find onewhich involves the topological structure of theassociahedron.” (Loday, 2005)

s(a4) = −a4 + 6a3a1 + 3a22 − 21a2a

21 + 14a4

1 (for Loday’s a4!)

Project. (A.–Benedetti–Gonzalez.)Compute the group of characters andreciprocity rules for other interestingsubmonoids of GP.

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Group of characters, reciprocity for associahedra.Two (of many) interesting enumerative consequences:


Recall: ζ(g) is 1 if g has no edges and 0 otherwise.Proposition. (Aguiar–A.)(−1)nζ−1(Kn) = Dn (# of derangements)(Conjectured by Humpert + Martin 12)

Key idea:The polytope of Kn is the permutahedron πn, so

we can compute in the group of characters of Π.


Proposition. (Aguiar–A.)The number of face parallelism classes ofLoday’s associahedron an is Catalan # Cn.Counted by dimension: Narayana numbers.

This follows from the antipode of an.

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Group of characters, reciprocity for associahedra.Two (of many) interesting enumerative consequences:


Recall: ζ(g) is 1 if g has no edges and 0 otherwise.Proposition. (Aguiar–A.)(−1)nζ−1(Kn) = Dn (# of derangements)(Conjectured by Humpert + Martin 12)

Key idea:The polytope of Kn is the permutahedron πn, so

we can compute in the group of characters of Π.


Proposition. (Aguiar–A.)The number of face parallelism classes ofLoday’s associahedron an is Catalan # Cn.Counted by dimension: Narayana numbers.

This follows from the antipode of an. 41 / 49


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) :=∑

S1t···tSn=I

(ζS1 ⊗ · · · ⊗ ζSn) ∆S1,...,Sn(p),

summing over all weak ordered set partitions I = S1 t · · · t Sn(where Si 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.

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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) :=∑

S1t···tSn=I

(ζS1 ⊗ · · · ⊗ ζSn) ∆S1,...,Sn(p),

summing over all weak ordered set partitions I = S1 t · · · t Sn(where Si 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


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:ζ(q) :=

1 if q is an antichain,

0 otherwise.

χ(q) = strict order polynomial of q (Stanley, 1970).For n ∈ N, it counts order-preserving labelings of q with [n].

• Matroids:ζ(m) :=

1 if m has a unique basis,

0 otherwise.

χ(m) = BJR polynomial of m. (Billera-Jia-Reiner, 2006).For n ∈ N, it counts m-generic functions f : I → [n].

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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:ζ(q) :=

1 if q is an antichain,

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:ζ(m) :=

1 if m has a unique basis,

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.

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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:

ζI (P) :=

1 if P is a point,

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]

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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 ζI (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 polynomialχ(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 polynomialχ(m)(−1) = (−1)|I |#bases of m (Billera-Jia-Reiner).

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8. A current direction: the polytope algebra.

• In the polytope algebra:The antipode of P is

s(P) = (−1)|I |∑Q≤P

(−1)dimQ Q = −Po

where Po = rel. interior of P. An involution! (Ehrhart,McMullen)

• Brion’s theorem: P =∑

v vertex

conev (P)

Theorem. (Aguiar – A.) There is a Brion morphism B : GP→ P

P 7→∑

v vertex

posetv (P).

This restricts to B: W (weaves) −→ Connes-Kreimer (rooted trees).Several interesting algebraic and combinatorial consequences.

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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 GPW of Coxeter permutahedra πW ,apply them to Coxeter associahedra, Coxeter matroids,... Thisrequires extending Hopf monoids to type W . (Aguiar–Mahajan)

• Extend to deformations of any simple polytope. This requiresextending Hopf monoids to any arrangement. (Aguiar–Mahajan)

• Study the connection with polyhedral valuations on generalizedpermutahedra. (Derksen–Fink)

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¡Muchas gracias!

The preprints are coming soon!49 / 49

The algebraic and combinatorial structure of generalized...

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