Chromatic problems in polytope Hopf algebras Ra ul Penagui ao - - PowerPoint PPT Presentation

Chromatic problems in polytope Hopf algebras

Ra´ ul Penagui˜ ao

University of Zurich

December 18th, 2017

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 1 / 1

Introduction CF on graphs

The chromatic symmetric function on graphs

A colouring on a graph G is a map f : V (G) → N. It is proper if f(v1) = f(v2) when {v1, v2} ∈ E(G).

Figure: A proper colouring f ∗ of a graph

Set xf =

• v

xf(v). We have xf∗ = x2

1x2 2x4 in the figure.

The chromatic symmetric function (CF) is ΨG(G) =

• f proper

xf.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 2 / 1

Introduction CF on graphs

CF on graphs - The kernel problem

Question (The kernel problem on graphs) Describe all linear relations of the form

• i

aiΨG(Gi) = 0 . Let G = the linear span of all graphs. Equivalent to find kernel of the linear extension of ΨG : G → QSym.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 3 / 1

Introduction CF on graphs

Outline

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 4 / 1

Introduction CF on graphs

Symmetric functions

A weak composition of n is an infinite list α = (α1, . . . , ) of non-negative integers that sum up to n. Write xα =

• i

xαi

i .

Example: β = (3, 1, 2, 1, 0, 0, · · · ) weakly composes 7. We have xβ = x3

1x2x2 3x4.

A homogeneous symmetric function of degree n is a sum of the form f =

• α

aαxα , where the sum runs over weak compositions of n, and reordering α → β preserves the coefficient aα = aβ (i.e. changing xi ↔ xj does not change the sum).

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 5 / 1

Introduction CF on graphs

Symmetric functions

The graded ring of symmetric functions Sym = ⊕n≥0Symn is the span

• f all homogeneous symmetric functions.

Monomial basis of Symn is mλ =

• λ(α)=λ

xα, where the sum runs over weak compositions that, after reordering, generate the partition λ. The chromatic symmetric function on a graph is a symmetric function. Proposition (Monomial formula for graphs) ΨG(G) =

• π

autλ(π) mλ(π) , where the sum runs over all stable set partitions.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 6 / 1

Kernel problem on graphs

Graphs terminology

The edge deletion of a graph: H \ {e}. The edge addition of a graph: G + {e}.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 7 / 1

Kernel problem on graphs

Modular relations

ΨG(G) =

• f proper on G

xf . Proposition (Modular relations - Guay-Paquet, Orellana, Scott, 2013) Let G be a graph that contains an edge e3 and does not contain e1, e2 such that the edges {e1, e2, e3} form a triangle. Then, ΨG(G) − ΨG(G + {e1}) − ΨG(G + {e2}) + ΨG(G + {e1, e2}) = 0 .

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 8 / 1

Kernel problem on graphs

The kernel problem

For G1, G2 isomorphic graphs, we have G1 − G2 ∈ ker ΨG. These are called isomorphism relation. Theorem (RP-2017) The kernel of ΨG is generated by modular relations and isomorphism relations. Let M = modular relations, isomorphism relations ⊆ G. Goal: ker ΨG = M.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 9 / 1

Kernel problem on graphs

Idea of proof - Rewriting graph combinations

ΨG(G) − ΨG(G + {e1}) − ΨG(G + {e2}) + ΨG(G + {e1, e2}) = 0 . Take z =

• i

Giai ∈ G/M in the kernel of ˜ ΨG : G/M → Sym. Goal: show that z = 0. Some of the Gi can be rewritten as graphs with more edges (through modular relation). We call them extendible. The badly behaved graphs {H1, H2, · · · } are not a lot, and {ΨG(H1), ΨG(H2), · · · } is linearly independent. Linear algebra magic. Cash in the theorem.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 10 / 1

Kernel problem on graphs

Idea of proof - Rewriting graph combinations

ΨG(G) − ΨG(G + {e1}) − ΨG(G + {e2}) + ΨG(G + {e1, e2}) = 0 . Proposition (Non-extendible graphs) A graph is non-extendible if and only if any connected component Gc, the complement graph of G, is a complete graph. Consequence: Up to isomorphism, we can identify naturally a partition λ with a non-extendible graph Kc

λ in such a way λ = λ(Gc).

Possible to show: the set {ΨG(Kc

λ)}λ is linearly independent.

z =

• λ

Kc

λaλ ∈ ker ΨG ,

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 11 / 1

Kernel problem on graphs

Idea of proof - Rewriting graph combinations

So z =

• λ

Kc

λaλ ∈ ker ΨG ,

Apply ΨG to get 0 =

• λ

ΨG(Kc

λ)aλ ⇒ aλ = 0 .

So z = 0, as desired.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 12 / 1

CF on other objects

Quasisymmetric functions

A homogeneous quasisymmetric function of degree n is a sum of the form f =

• α

aαxα , where the sum runs over weak compositions of n, and the coefficients respect aα = aβ whenever β is obtained from α by changing the order

• f the zeroes.

Monomial basis of QSymn: Mα =

• α(β)=α

xβ , where the sum runs over weak compositions that, after deleting zeroes, generate the (strong) composition α.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 13 / 1

CF on other objects

CF on matroids

Let M = (I, B) be a matroid, for I finite set and B ⊆ P(I) a set of bases. A colouring f of M is a map f : I → N. It is called M-generic if B →

• b∈B

f(b) has a minimum in a unique basis B ∈ B. The chromatic quasisymmetric function on matroids is then defined as ΨMat(M) =

• f is M-generic

xf .

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 14 / 1

CF on other objects

CF on posets

For a poset, a colouring f : P → N is called non-decreasing if a ≤ b ⇒ f(a) ≤ f(b). The chromatic quasisymmetric function on posets is then defined as ΨPos(P) =

• f non-decreasing

xf . Theorem (F´ eray, 2014) The kernel of ΨPos is generated by the cyclic inclusion exclusion relations and isomorphism relations.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 15 / 1

CF on other objects

(Graded) Hopf algebras

Given a field K, a graded Hopf algebra is a linear space H = ⊕n≥0Hn with graded operations µ and ∆. Operation µ : H ⊗ H → H is a multiplication and says how to merge two objects together. Operation ∆ : H → H ⊗ H is a comultiplication and says how to split an object into two.

Figure: The coproduct determines how objects decompose

Some extra conditions for compatibility and an antipode s : H → H.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 16 / 1

CF on other objects

(Graded) Hopf algebras

Examples: The one dimensional vector space K. Sym and QSym. The vector space spanned freely by graphs G. Hopf algebra structure on graphs: The multiplication µ(G1, G2) is a graph with vertices V (G1) ⊔ V (G2), and edges E(G1) ⊔ E(G2), with some relabelling. Graph comultiplication ∆G is a linear combination of graphs ∆G =

• S⊔T=V (G)

G|S ⊗ G|T .

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 17 / 1

CF on other objects

CF in combinatorial Hopf algebras

Character: a linear map η : H → K, preserves multiplication and unit. On graphs: η(G) = 1[G has no edges]. On QSym: η0(Mα) = 1[∃n≥0α = (n)]. Theorem (Aguiar, Bergeron and Sottile, 2006) For a combinatorial Hopf algebra (H, η) there is a unique Hopf algebra morphism ΨH that makes the diagram commute:

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 18 / 1

CF on other objects

CF in combinatorial Hopf algebras

For a composition α of size l, ηα is the composition: H ∆(l−1) − − − − → H⊗l πα − → H⊗l η⊗l − − → K⊗l ∼ = K . For a ∈ Hn, the unique Hopf algebra morphism is ΨH(a) =

• α

ηα(a)Mα , where the sum runs over compositions of n.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 19 / 1

CF on other objects

CF in combinatorial Hopf algebras

For the graph Hopf algebra G, if we choose the character η(G) = 1[G has no edges], we obtain ΨG. For the poset Hopf algebra Pos, if we choose the character η(P) = 1[P is an anti-chain], we obtain ΨPos. For the matroid Hopf algebra Mat, if we choose the character η(M) = 1[M has a unique basis], we obtain ΨMat.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 20 / 1

CF on polytopes Generalised permutahedra

Polytopes and fans

A polytope is a bounded set of the form q = {x ∈ Rn|Ax ≤ b}. Functional f : {1, · · · , n} → R. Colouring f : {1, · · · , n} → N. → Linear optimisation problem min

(xi)n

i=1

n

i=1 f(i)xi

→ Solution qf is called a face. This partitions the colourings into cones, for each face. This partition is called the normal fan of a polytope.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 21 / 1

CF on polytopes Generalised permutahedra

The permutahedron and its generalisations

The n order permutahedron is per = conv{(σ(1), . . . , σ(n))|σ ∈ Sn}. Is (n − 1)-dimentional.

Figure: The 4-permutahedron1

1https://en.wikipedia.org/wiki/Permutohedron Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 22 / 1

CF on polytopes Generalised permutahedra

The permutahedron and its generalisations

For a cone C of per,

• f∈C

xf is a quasisymmetric function. A generalised permutahedra q is a polytope which fan coarsens the normal fan of the permutahedron (i.e. results from merging cones from the n-permutahedron).Define the CF: ΨGP(q) =

• qf= point

xf .

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 23 / 1

CF on polytopes Generalised permutahedra

Minkowsky sum

A +M B = {a + b|a ∈ A, b ∈ B} . C := A −M B if A = C +M B. C may not exist but if exists it is unique (only for polytopes).

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 24 / 1

CF on polytopes Generalised permutahedra

Minkowsky sum

Examples of generalised permutahedra: The J-simplex, for J ⊆ {1, · · · , n}: sJ = conv{ej|j ∈ J} and its dilations. The permutahedron per = conv{(σ(1), . . . , σ(n))|σ ∈ Sn} . The permutahedron is also given as per =

M i≤j

s{i,j} .

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 25 / 1

CF on polytopes Generalised permutahedra

Generalised permutahedra and nestohedra

A generalised permutahedron is a polytope q of the form q =    

M J=∅ aJ>0

aJsJ     −M    

M J=∅ aJ<0

|aJ|sJ     , A nestohedron is only the positive part: q =

M J=∅ aJ>0

aJsJ .

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 26 / 1

CF on polytopes Generalised permutahedra

Zonotopes and other embedings

Given a graph G, its zonotope is defined as Z(G) =

M e∈E(G)

se . This is a Hopf algebra morphism, so ΨG = ΨGP ◦ Z . There is also a Hopf algebra embedding Z : Mat → GP. ΨMat = ΨGP ◦ Z .

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 27 / 1

CF on polytopes Generalised permutahedra

Faces of nestohedra

For a colouring f, note that qf =

M J=∅ aJ>0

(aJsJ)f = point ⇔ ∀J:aJ>0 (sJ)f = point , This allows us to establish a parallel for the modular relation on graphs: Proposition (Modular relations on nestohedra) Consider a nestohedron q, {Bj|j ∈ T} a family of subsets on {1, · · · n} and {aj|j ∈ T} some positive scalars. Suppose “some magic”

• happens. Then,
• T⊆J (−1)#T ΨGP

 q +M

M j∈T

ajsBj   = 0 .

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 28 / 1

CF on polytopes Kernel problem on nestohedra

Kc

π parallel and conclusion of proof

The nestohedra that are not extendable are exactly pf =

• J:(sJ)f= point

aJsJ , for positive aJ. Up to isomorphism there is only one such pα for each composition α of

• n. Also, {ΨGP(pα)}α are linearly independent.

Theorem (RP 2017) The modular relations and the isomorphism relations span the kernel

• f the restriction of ΨGP to the nestohedra.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 29 / 1

Tree conjecture

Tree conjecture on graphs

Figure: Non-isomorphic graphs with the same CSF

Conjecture (Tree conjecture -Stanley and Stembridge) Any two non-isomorphic trees T1, T2 have distinct CSF .

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 30 / 1

Tree conjecture

Tree conjecture on graphs

This is a graph invariant: χ′(G) =

• f

xf

• i

q# monochromatic edges in f of colour i

i

where the sum runs over all colourings. The modular relations and isomorphism relations are in ker χ′. So ker ΨG ⊆ χ′ . Conjecture (Tree conjecture -Stanley and Stembridge) Any two non-isomorphic trees T1, T2 have distinct χ′.

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 31 / 1

Conclusion

Further questions

From nestohedra to generalised permutahedra? Modular relations on matroids? The image of the CF on graphs ΨG is spanned by {ΨG(Kc

λ)}λ,

which forms a basis of im ΨG. Combinatorial meaning of the coefficients?

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 32 / 1

Conclusion

Thank you

Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 33 / 1