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 ( v 1 ) � = f ( v 2 ) when { v 1 , v 2 } ∈ E ( G ) . Figure: 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 � The chromatic symmetric function (CF) is Ψ G ( G ) = x f . f proper 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 � a i Ψ G ( G i ) = 0 . i Let G = the linear span of all graphs. Equivalent to find kernel of the linear extension of Ψ G : 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 = ( α 1 , . . . , ) of non-negative integers that sum up to n . Write x α = � x α i i . i Example: β = (3 , 1 , 2 , 1 , 0 , 0 , · · · ) weakly composes 7 . We have x β = x 3 1 x 2 x 2 3 x 4 . A homogeneous symmetric function of degree n is a sum of the form a α x α , � f = α where the sum runs over weak compositions of n , and reordering α → β preserves the coefficient a α = a β (i.e. changing x i ↔ x j 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 ≥ 0 Sym n is the span of all homogeneous symmetric functions. � x α , where the sum runs over Monomial basis of Sym n is m λ = λ ( α )= λ 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) � Ψ G ( G ) = aut λ ( π ) m λ ( π ) , 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 � 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 December 18th, 2017 8 / 1

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 � ⊆ G . Goal: ker Ψ G = 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 Ψ G ( G ) − Ψ G ( G + { e 1 } ) − Ψ G ( G + { e 2 } ) + Ψ G ( G + { e 1 , e 2 } ) = 0 . � G i a i ∈ G / M in the kernel of ˜ Take z = Ψ G : G / M → Sym . i Goal: show that z = 0 . Some of the G i can be rewritten as graphs with more edges (through modular relation). We call them extendible . The badly behaved graphs { H 1 , H 2 , · · · } are not a lot, and { Ψ G ( H 1 ) , Ψ G ( H 2 ) , · · · } 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 Ψ G ( G ) − Ψ G ( G + { e 1 } ) − Ψ G ( G + { e 2 } ) + Ψ G ( G + { e 1 , e 2 } ) = 0 . Proposition (Non-extendible graphs) A graph is non-extendible if and only if any connected component G c , 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 K c λ in such a way λ = λ ( G c ) . Possible to show: the set { Ψ G ( K c λ ) } λ is linearly independent. � K c z = λ a λ ∈ ker Ψ G , λ 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 � K c z = λ a λ ∈ ker Ψ G , λ Apply Ψ G to get � Ψ G ( K c 0 = λ ) a λ ⇒ a λ = 0 . λ 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 a α x α , � f = α where the sum runs over weak compositions of n , and the coefficients respect a α = a β whenever β is obtained from α by changing the order of the zeroes . Monomial basis of QSym n : x β , � M α = α ( β )= α 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 �→ f ( b ) b ∈ B has a minimum in a unique basis B ∈ B . The chromatic quasisymmetric function on matroids is then defined as � Ψ Mat ( M ) = x f . f is M-generic 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 � Ψ Pos ( P ) = x f . f non-decreasing Theorem (F´ eray, 2014) The kernel of Ψ Pos 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 ≥ 0 H n 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 ( G 1 , G 2 ) is a graph with vertices V ( G 1 ) ⊔ V ( G 2 ) , and edges E ( G 1 ) ⊔ E ( G 2 ) , with some relabelling. Graph comultiplication ∆ G is a linear combination of graphs � ∆ G = G | S ⊗ G | T . S ⊔ T = V ( G ) 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 : η 0 ( M α ) = 1 [ ∃ n ≥ 0 α = ( n )] . Theorem (Aguiar, Bergeron and Sottile, 2006) For a combinatorial Hopf algebra ( H , η ) there is a unique Hopf algebra morphism Ψ H 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 , η α is the composition: H ∆ ( l − 1) → H ⊗ l η ⊗ l → K ⊗ l ∼ → H ⊗ l π α − − − − − − − = K . For a ∈ H n , the unique Hopf algebra morphism is � Ψ H ( a ) = η α ( a ) M α , α 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 Ψ G . For the poset Hopf algebra Pos , if we choose the character η ( P ) = 1 [ P is an anti-chain ] , we obtain Ψ Pos . For the matroid Hopf algebra Mat , if we choose the character η ( M ) = 1 [ M has a unique basis ] , we obtain Ψ Mat . Ra´ ul Penagui˜ ao (University of Zurich) Kernel problems December 18th, 2017 20 / 1