Chromatic symmetric functions on graphs and polytopes 30th FPSAC, Hanover NH, Darmouth College Ra´ ul Penagui˜ ao University of Zurich July 16th, 2018 Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 1 / 23
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 ( v 1 ) � = f ( v 2 ) when { v 1 , v 2 } ∈ E ( G ) . Figure: Example of a proper colouring f of a graph � x f ( v ) . We have x f = x 2 1 x 2 Set x f = 2 x 4 in the figure. v Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 2 / 23
Introduction CF on graphs The chromatic symmetric function on graphs � The chromatic symmetric function (CSF) of G is Ψ G ( G ) = x f . f proper This is a Hopf algebra morphism between G = span { all graphs } and Sym . Example: Figure: The line graph P 2 and the path P 3 Their CSF are � � � x 2 . Ψ G ( P 2 ) = 2 x i x j , Ψ G ( P 3 ) = 6 x i x j x k + i x j 1 ≤ i<j 1 ≤ i<j<k i � = j Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 3 / 23
Introduction CF on graphs Tree conjecture on graphs Evaluating x 1 = · · · = x t = 1 and x i = 0 for i > t we obtain the chromatic polynomial χ G ( t ) . With the CSF , we can compute the number of connected components , compute the degree sequence for trees, etc... , but Figure: Non-isomorphic graphs with the same CSF 1 Conjecture (Tree conjecture - Stanley and Stembridge) Any two non-isomorphic trees T 1 , T 2 have distinct CSF . Think about the chromatic polynomial 1 Rose Orelanna and Scott Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 4 / 23
Introduction CF on graphs CF on graphs - The kernel problem Question (The kernel problem on graphs) Compute generators of ker Ψ G . I.e. describe all linear relations of the form � a i Ψ G ( G i ) = 0 . i Theorem (RP-2017) The space ker Ψ G is spanned by the modular relations and isomorphism relations. Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 5 / 23
Introduction CF on graphs Outline Introduction 1 CF on graphs Kernel problem on graphs 2 CF on polytopes 3 Generalised permutahedra Kernel problem on nestohedra Tree conjecture 4 Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 6 / 23
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 July 16th, 2018 7 / 23
Kernel problem on graphs Modular relations � Ψ G ( G ) = x f . f proper on G Proposition (Modular relations - Guay-Paquet, Orellana, Scott, 2013) Let G be a graph that contains an edge e 3 and does not contain e 1 , e 2 such that the edges { e 1 , e 2 , e 3 } form a triangle. Then, Ψ G ( G ) − Ψ G ( G + { e 1 } ) − Ψ G ( G + { e 2 } ) + Ψ G ( G + { e 1 , e 2 } ) = 0 . Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 8 / 23
Kernel problem on graphs The kernel problem For G 1 , G 2 isomorphic graphs, we have G 1 − G 2 ∈ ker Ψ G . These are called isomorphism relation . Theorem (RP-2017) The kernel of Ψ G is generated by modular relations and isomorphism relations. Let M = � modular relations , isomorphism relations � . Goal: ker Ψ G = M . Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 9 / 23
Kernel problem on graphs Idea of proof - Rewriting graph combinations Condition to be a modular relation: e 3 ∈ G ⇒ G − ( G + { e 1 } ) − ( G + { e 2 } ) + ( G + { e 1 , e 2 } ) ∈ M . � Take z = G i a i in the kernel of Ψ G . i Goal: by working on ker Ψ G / M , show that z ∈ M . Some of the G i can be rewritten as graphs with more edges (through modular relation). We call them extendible . The non-extendible graphs { H 1 , H 2 , · · · } are not a lot, and { Ψ G ( H 1 ) , Ψ G ( H 2 ) , · · · } is linearly independent. Linear algebra ‘magic’ ⇒ a theorem is born. Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 10 / 23
Kernel problem on graphs Idea of proof - Rewriting graph combinations e 3 ∈ G ⇒ G − ( G + { e 1 } ) − ( G + { e 2 } ) + ( G + { e 1 , e 2 } ) ∈ M . Proposition (Non-extendible graphs) A graph is non-extendible if and only if any connected component of G c , the complement graph of G , is a complete graph. Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 11 / 23
Kernel problem on graphs Idea of proof - Linear algebra magic So, always working on ker Ψ G / M , we car rewrite: � K c z = λ a λ ∈ ker Ψ G , λ ∈P n Apply Ψ G to get � Ψ G ( K c 0 = λ ) a λ ⇒ a λ = 0 . λ ∈P n Possible to show: the set { Ψ G ( K c λ ) } λ ∈P n is linearly independent. So z = 0 , as desired. Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 12 / 23
CF on polytopes Generalised permutahedra Polytopes Fix a dimension n . A polytope is a bounded set of the form q = { x ∈ R n | Ax ≤ b } . Given a colouring f : [ n ] → N of the coordinates , the face q f is n � q f = arg min x i f ( i ) . x ∈ q i =1 Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 13 / 23
CF on polytopes Generalised permutahedra Polytopes: Examples Simplexes and its dilations: Consider J ⊆ [ n ] non empty. λ s J = conv { λe i | i ∈ J } . Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 14 / 23
CF on polytopes Generalised permutahedra The permutahedron and its generalisations The n order permutahedron: per = conv { ( σ (1) , . . . , σ ( n )) | σ ∈ S n } . Is ( n − 1) -dimensional. Figure: The 4 -permutahedron 2 2https://en.wikipedia.org/wiki/Permutohedron Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 15 / 23
CF on polytopes Generalised permutahedra The permutahedron and its generalisations A generalised permutahedron is a polytope q of the form M � M � − M , q = a J s J b J s J J � = ∅ J � = ∅ A nestohedron is only the positive part: M � q = a J s J . J � = ∅ Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 16 / 23
CF on polytopes Generalised permutahedra Chromatic function and zonotopes We define the chromatic quasisymmetric function (CF) as � Ψ GP ( q ) = x f . q f =pt Given a graph G , its zonotope is defined as M � Z ( G ) = s e . e ∈ E ( G ) These are all Hopf algebra morphisms from the Hopf algebra GP = span { generalised permutahedra in R n , n ≥ 0 } . Also, Ψ G = Ψ GP ◦ Z . Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 17 / 23
CF on polytopes Generalised permutahedra Some relations in nestohedra Proposition (Modular relations on nestohedra) Consider a nestohedron q , { B j | j ∈ T } a family of subsets on { 1 , · · · n } and { a j | j ∈ T } some positive scalars. Suppose “some magic” M � T ⊆ J ( − 1) # T Ψ GP � = 0 . q + M a j s B j happens. Then, j ∈ T Additionally, there are also the so called simple relations - describe that we only care about which coefficients are positive, not how big they are. Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 18 / 23
CF on polytopes Generalised permutahedra Some relations on nestohedra - Example An example of a modular relation: Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 19 / 23
CF on polytopes Kernel problem on nestohedra K c π parallel and conclusion of proof Theorem (RP 2017) The modular relations, the isomorphism relations and the simple relations span the kernel of the restriction of Ψ GP to the nestohedra. Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 20 / 23
Tree conjecture Tree conjecture on graphs The following: � � q # monochromatic edges in f of colour i χ ′ ( G ) = x f i f i is a graph invariant, where the sum runs over all colourings. If we consider the projection of this invariant modulo the relations q i ( q i − 1) 2 = 0 , then the modular relations are in ker χ ′ . We obtain ker Ψ G = ker χ ′ . Conjecture (Tree conjecture - χ ′ formulation) Any two non-isomorphic trees T 1 , T 2 have distinct χ ′ . Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 21 / 23
Conclusion Further questions From nestohedra to generalised permutahedra? The image of the CF on graphs Ψ G is spanned by { Ψ G ( K c λ ) } λ , which forms a basis of im Ψ G . Combinatorial meaning of the coefficients? Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 22 / 23
Conclusion Thank you Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems July 16th, 2018 23 / 23
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