Chromatic Symmetric Functions with respect to Complete Graphs Sof - PowerPoint PPT Presentation

Chromatic Symmetric Functions with respect to Complete Graphs Sof´ ıa Mart´ ınez Alberga in collaboration with J. Kazdan, L. Kr¨ oll, O. Melnyk, and A. Tenenbaum University of California, Riverside smart040@ucr.edu January 26, 2019

Outline 1 Background 2 Motivation 3 E-positivity

Graph Theory: Definitions Let Γ = ( V, E ) be a graph on n vertices Formally we can define a coloring , of Γ, via a map κ : V → N Proper colorings of Γ are the colorings of vertices for which any two adjacent vertices have different colors. Figure 1: Examples (a) Proper coloring of K 4

Graph Theory: Definitions Let Γ = ( V, E ) be a graph on n vertices Formally we can define a coloring , of Γ, via a map κ : V → N Proper colorings of Γ are the colorings of vertices for which any two adjacent vertices have different colors. Figure 1: Examples (a) Proper coloring of K 4 (b) Proper coloring

Graph Theory: Definitions Let Γ = ( V, E ) be a graph on n vertices Formally we can define a coloring , of Γ, via a map κ : V → N Proper colorings of Γ are the colorings of vertices for which any two adjacent vertices have different colors. Figure 1: Examples (a) Proper coloring of K 4 (b) Proper coloring (c) Not a proper coloring

Graph Theory: Definitions Let Γ = ( V, E ) be a graph on n vertices Formally we can define a coloring , of Γ, via a map κ : V → N Proper colorings of Γ are the colorings of vertices for which any two adjacent vertices have different colors. Figure 1: Examples (a) Proper coloring of K 4 (b) Proper coloring (d) Proper coloring (c) Not a proper coloring

Subgraphs

Subgraphs Let ( i , j ) denote an element of E where i , j ∈ V .

Subgraphs Let ( i , j ) denote an element of E where i , j ∈ V . Given any graph, Γ = ( V, E ), we can define an subgraph , ∆, as the graph with vertex set ˜ V ⊂ V and edge set ˜ E ⊂ E .

Subgraphs Let ( i , j ) denote an element of E where i , j ∈ V . Given any graph, Γ = ( V, E ), we can define an subgraph , ∆, as the graph with vertex set ˜ V ⊂ V and edge set ˜ E ⊂ E . Given any graph, Γ = ( V, E ), we can define an induced subgraph , ∆, as the graph with vertex set ˜ V ⊂ V and edge set � ˜ � V ˜ E = { ( i, j ) | ( i, j ) ∈ ∩ E } 2

Subgraphs Let ( i , j ) denote an element of E where i , j ∈ V . Given any graph, Γ = ( V, E ), we can define an subgraph , ∆, as the graph with vertex set ˜ V ⊂ V and edge set ˜ E ⊂ E . Given any graph, Γ = ( V, E ), we can define an induced subgraph , ∆, as the graph with vertex set ˜ V ⊂ V and edge set � ˜ � V ˜ E = { ( i, j ) | ( i, j ) ∈ ∩ E } 2 A graph, Γ, is said to be free of a subgraph, i.e ∆ - free , if ∆ is not a induced subgraph of Γ.

Subgraph: Example Figure 3: K 4 and C 4 If the original graph is K 4 , then C 4 is only a subgraph of K 4 not an induced subgraph.

Algebra: Background Suppose we have a partition λ = ( λ 1 , λ 2 , ..., λ ℓ ) of some n ∈ N , denoted λ ⊢ n .

Algebra: Background Suppose we have a partition λ = ( λ 1 , λ 2 , ..., λ ℓ ) of some n ∈ N , denoted λ ⊢ n . The monomial symmetric function corresponding to λ � x λ 1 i 1 x λ 2 i 2 x λ 3 i 3 ...x λ ℓ m λ = i ℓ i ∈ I where the sum is over all monomials having exponents λ 1 , λ 2 , ..., λ ℓ .

Algebra: Background Suppose we have a partition λ = ( λ 1 , λ 2 , ..., λ ℓ ) of some n ∈ N , denoted λ ⊢ n . The monomial symmetric function corresponding to λ � x λ 1 i 1 x λ 2 i 2 x λ 3 i 3 ...x λ ℓ m λ = i ℓ i ∈ I where the sum is over all monomials having exponents λ 1 , λ 2 , ..., λ ℓ . Example : Take n = 3

Algebra: Background Suppose we have a partition λ = ( λ 1 , λ 2 , ..., λ ℓ ) of some n ∈ N , denoted λ ⊢ n . The monomial symmetric function corresponding to λ � x λ 1 i 1 x λ 2 i 2 x λ 3 i 3 ...x λ ℓ m λ = i ℓ i ∈ I where the sum is over all monomials having exponents λ 1 , λ 2 , ..., λ ℓ . Example : Take n = 3 λ =(3) , m (3) = x 3 1 + x 3 2 + x 3 3 + x 3 4 + ...

Algebra: Background Suppose we have a partition λ = ( λ 1 , λ 2 , ..., λ ℓ ) of some n ∈ N , denoted λ ⊢ n . The monomial symmetric function corresponding to λ � x λ 1 i 1 x λ 2 i 2 x λ 3 i 3 ...x λ ℓ m λ = i ℓ i ∈ I where the sum is over all monomials having exponents λ 1 , λ 2 , ..., λ ℓ . Example : Take n = 3 λ =(3) , m (3) = x 3 1 + x 3 2 + x 3 3 + x 3 4 + ... λ ′ = (2,1) , m (2 , 1) = x 2 1 x 2 + x 1 x 2 2 + x 2 1 x 3 + x 1 x 2 3 + x 2 2 x 3 + ...

Algebra: Background Suppose we have a partition λ = ( λ 1 , λ 2 , ..., λ ℓ ) of some n ∈ N , denoted λ ⊢ n . The monomial symmetric function corresponding to λ � x λ 1 i 1 x λ 2 i 2 x λ 3 i 3 ...x λ ℓ m λ = i ℓ i ∈ I where the sum is over all monomials having exponents λ 1 , λ 2 , ..., λ ℓ . Example : Take n = 3 λ =(3) , m (3) = x 3 1 + x 3 2 + x 3 3 + x 3 4 + ... λ ′ = (2,1) , m (2 , 1) = x 2 1 x 2 + x 1 x 2 2 + x 2 1 x 3 + x 1 x 2 3 + x 2 2 x 3 + ... λ ′′ = (1,1,1) , m (1 , 1 , 1) = x 1 x 2 x 3 + x 1 x 2 x 4 + x 1 x 3 x 4 + ...

Algebra: the Ring of Symmetric Functions The ring of symmetric functions is denoted as Λ and is defined as Λ = C m λ from which we get that the space Λ n has basis { m λ : λ ⊢ n }

Algebra: Another Base Λ n can also be spanned by other bases, one of which is follows, where λ ⊢ n :

Algebra: Another Base Λ n can also be spanned by other bases, one of which is follows, where λ ⊢ n : nth elementary symmetric function � e n = m 1 n = x i 1 x i 2 x i 3 ...x i n i 1 <i 2 <...<i n Example: e 3 = x 1 x 2 x 3 + x 1 x 2 x 4 + x 1 x 3 x 4 + ...

Algebra: Another Base Λ n can also be spanned by other bases, one of which is follows, where λ ⊢ n : nth elementary symmetric function � e n = m 1 n = x i 1 x i 2 x i 3 ...x i n i 1 <i 2 <...<i n Example: e 3 = x 1 x 2 x 3 + x 1 x 2 x 4 + x 1 x 3 x 4 + ... Property: e λ = e λ 1 ...e λ n

Algebra: Another Base Λ n can also be spanned by other bases, one of which is follows, where λ ⊢ n : nth elementary symmetric function � e n = m 1 n = x i 1 x i 2 x i 3 ...x i n i 1 <i 2 <...<i n Example: e 3 = x 1 x 2 x 3 + x 1 x 2 x 4 + x 1 x 3 x 4 + ... Property: e λ = e λ 1 ...e λ n What is E -Postivity? A term used to describe a Chromatic Symmetric Function in e -basis with positive coefficients.

Algebra: Another Base Λ n can also be spanned by other bases, one of which is follows, where λ ⊢ n : nth elementary symmetric function � e n = m 1 n = x i 1 x i 2 x i 3 ...x i n i 1 <i 2 <...<i n Example: e 3 = x 1 x 2 x 3 + x 1 x 2 x 4 + x 1 x 3 x 4 + ... Property: e λ = e λ 1 ...e λ n What is E -Postivity? A term used to describe a Chromatic Symmetric Function in e -basis with positive coefficients. Why E -Positivity? E -Positivity is an invariant which allows for classifying to take place.

Chromatic Symmetric Functions Chromatic Symmetric Function, (CSF) , of Γ : � X Γ = x κ ( v 1 ) x κ ( v 2 ) ...x κ ( v n ) where the sum is over all proper colorings of Γ with colors from the positive integers and v i ∈ V

A Few Examples Figure 4: P 3 and K 4

A Few Examples Figure 4: P 3 and K 4 Example: P 3 , path on three vertices, has either all colors different or the two outer vertices are the same color and the middle vertex is of a different color. So we have: X P 3 = e 2 , 1 + 3 e 3

A Few Examples Figure 4: P 3 and K 4 Example: P 3 , path on three vertices, has either all colors different or the two outer vertices are the same color and the middle vertex is of a different color. So we have: X P 3 = e 2 , 1 + 3 e 3 Example: K n only has colorings where all vertices are of different colors. Then we get: X K n = n ! e n

Outline 1 Background 2 Motivation 3 E-positivity

Some Major Questions in Algebraic Combinatorics Figure 5: Claw and Incomparibility Graph

Some Major Questions in Algebraic Combinatorics Figure 5: Claw and Incomparibility Graph What families of graphs are e -positive?

Some Major Questions in Algebraic Combinatorics Figure 5: Claw and Incomparibility Graph What families of graphs are e -positive? → Claw-free, incomparability graphs

Some Major Questions in Algebraic Combinatorics Figure 5: Claw and Incomparibility Graph What families of graphs are e -positive? → Claw-free, incomparability graphs Can graphs be distinguished by their chromatic symmetric functions?

Some Major Questions in Algebraic Combinatorics Figure 5: Claw and Incomparibility Graph What families of graphs are e -positive? → Claw-free, incomparability graphs Can graphs be distinguished by their chromatic symmetric functions? → Trees : a graph with no cycles

Motivation Classifying graphs by their CSF. Which graph classes are e-positive? Identifying properties encoded in the CSF. Which families of graphs are uniquely determined by there CSF.

Our Research Ways We Proved: e -positivity Explicit Formula Tableaux Method

Our Research Families of graphs we Ways We Proved: proved e -postivity e -positivity for: Explicit Formula Generalized Tableaux Nets Method Horseshoe Crab

Our Research Families of graphs we Ways We Proved: What we proved was proved e -postivity e -positivity uniquely determined for: but CSF: Explicit Formula Generalized Generalized Tableaux Nets Spiders Method Horseshoe Crab