The Arithmetic of Coxeter Permutahedra Federico Ardila San - PowerPoint PPT Presentation

The Arithmetic of Coxeter Permutahedra Federico Ardila San Francisco State University Universidad de Los Andes Matthias Beck San Francisco State University Freie Universit¨ at Berlin Jodi McWhirter Washington University St. Louis

The Menu 3/22/2020 https://upload.wikimedia.org/wikipedia/commons/3/3e/Permutohedron.svg (4,1,2,3) Ehrhart (quasi-)polynomials (3,1,2,4) ◮ (4,2,1,3) (3,2,1,4) (4,1,3,2) Zonotopes ◮ (2,1,3,4) (4,3,1,2) (2,3,1,4) (3,1,4,2) (4,2,3,1) (4,3,2,1) (2,1,4,3) Coxeter permutahedra ◮ (3,4,1,2) (1,2,3,4) (1,3,2,4) (2,4,1,3) (3,2,4,1) Signed graphs ◮ (1,2,4,3) (3,4,2,1) (1,4,2,3) (2,3,4,1) Tree generating functions ◮ (1,3,4,2) (2,4,3,1) (1,4,3,2) The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter https://upload.wikimedia.org/wikipedia/commons/3/3e/Permutohedron.svg 1/1

Measuring Polytopes Rational polytope — convex hull of finitely many points in Q d — solution set of a system of linear (in-)equalities with integer coefficients Goal: measuring... � � 1 � P ∩ 1 � t Z d � volume vol( P ) = lim ◮ � � t d t →∞ � � � P ∩ Z d � discrete volume ◮ � � � � P ∩ 1 � for t ∈ Z > 0 � t Z d � � � t P ∩ Z d � Ehrhart function ehr P ( t ) := � = � � The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Discrete Volumes & Ehrhart Quasipolynomials Rational polytope — convex hull of finitely many points in Q d q ( t ) = c d ( t ) t d + · · · + c 0 ( t ) is a quasipolynomial if c 0 ( t ) , . . . , c d ( t ) are periodic functions; the lcm of their periods is the period of q ( t ) . Theorem (Ehrhart 1962) For any rational polytope P ⊂ R d , � is a quasipolynomial in t whose period � � t P ∩ Z d � ehr P ( t ) := divides the lcm of the denominators of the vertex coordinates of P . Example P = [ − 1 2 , 1 2 ] 2 The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Why care about... Ehrhart (Quasi-)Polynomials Linear systems are everywhere, and so polyhedra are everywhere. ◮ In applications, the volume of the polytope represented by a linear ◮ system measures some fundamental data of this system (“average”). Polytopes are basic geometric objects, yet even for these basic objects ◮ volume computation is hard and there remain many open problems. Many discrete problems in various mathematical areas are linear ◮ problems, thus they ask for the discrete volume of a polytope in disguise. Much discrete geometry can be modeled using polynomials ◮ and, conversely, many combinatorial polynomials can be modeled geometrically. The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Zonotopes Zonotope — Minkowski sum of line segments Z = � n j =1 [ a j , b j ] Shephard (1974) Decomposition of Z into translates of half-open parallele- pipeds spanned by the linearly indepen- dent subsets of { b j − a j : 1 ≤ j ≤ n } . Stanley (1991) For a finite set of vectors U ⊂ Z d , let Z ( U ) := � u ∈ U [ 0 , u ] Then � vol( W ) t | W | ehr Z ( U ) ( t ) = W ⊆ U lin. indep. where | W | denotes the number of vectors in W and vol( W ) is the relative volume of the parallelepiped generated by W . The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Lie Combinatorics Finite crystallographic root systems := {± ( e i − e j ) : 1 ≤ i < j ≤ n } A n − 1 := {± ( e i − e j ) , ± ( e i + e j ) : 1 ≤ i < j ≤ n } ∪ {± e i : 1 ≤ i ≤ n } B n {± ( e i − e j ) , ± ( e i + e j ) : 1 ≤ i < j ≤ n } ∪ {± 2 e i : 1 ≤ i ≤ n } C n := := {± ( e i − e j ) , ± ( e i + e j ) : 1 ≤ i < j ≤ n } D n . . . and E 6 , E 7 , E 8 , F 4 , G 2 . Positive roots are obtained by choosing the plus sign in each ± above. Standard Coxeter permutahedron of the finite root system Φ � − α 2 , α � � Π(Φ) := = conv { w · ρ : w ∈ W } 2 α ∈ Φ + where ρ := 1 � α ∈ Φ + α and W is the Weyl group of Φ 2 The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Lie Combinatorics Finite crystallographic root systems := {± ( e i − e j ) : 1 ≤ i < j ≤ n } A n − 1 {± ( e i − e j ) , ± ( e i + e j ) : 1 ≤ i < j ≤ n } ∪ {± e i : 1 ≤ i ≤ n } B n := := {± ( e i − e j ) , ± ( e i + e j ) : 1 ≤ i < j ≤ n } ∪ {± 2 e i : 1 ≤ i ≤ n } C n := {± ( e i − e j ) , ± ( e i + e j ) : 1 ≤ i < j ≤ n } D n . . . and E 6 , E 7 , E 8 , F 4 , G 2 . Positive roots are obtained by choosing the plus sign in each ± above. Standard Coxeter permutahedron of the finite root system Φ � − α 2 , α � � Π(Φ) := 2 α ∈ Φ + � Integral Coxeter permutahedron Π Z (Φ) := [0 , α ] α ∈ Φ + The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Standard Coxeter Permutahedra = {± ( e i − e j ) : 1 ≤ i < j ≤ n } A n − 1 {± ( e i − e j ) , ± ( e i + e j ) : 1 ≤ i < j ≤ n } ∪ {± e i : 1 ≤ i ≤ n } B n = {± ( e i − e j ) , ± ( e i + e j ) : 1 ≤ i < j ≤ n } ∪ {± 2 e i : 1 ≤ i ≤ n } C n = = {± ( e i − e j ) , ± ( e i + e j ) : 1 ≤ i < j ≤ n } D n conv { permutations of 1 Π( A n − 1 ) = 2 ( − n + 1 , − n + 3 , . . . , n − 3 , n − 1) } conv { signed permutations of 1 2 (1 , 3 , . . . , 2 n − 1) } Π( B n ) = conv { signed permutations of (1 , 2 , . . . , n ) } Π( C n ) = Π( D n ) = conv { evenly signed permutations of (0 , 1 , . . . , n − 1) } The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Why care about... Coxeter Permutahedra Many questions about permutations can be answered looking at the ◮ geometry of the permutahedron Fundamental objects in the representation theory of semisimple Lie ◮ algebras Connections to optimization (Ardila–Castillo–Eur–Postnikov 2020) ◮ Zonotopes with natural connections to tree enumeration ◮ The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Signed Graphs A signed graph G = ( V, E, σ ) comes with a signature σ : E ∗ → {±} A simple cycle is balanced if its product of signs is + . A signed graph is balanced if it contains no half edges and all of its simple cycles are balanced. An all-negative signed graph is balanced if and only if it is bipartite. A signed graph is balanced if and only if it has no half edges and can be switched to an all-positive signed graph. The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Signed Graphs and Root Systems Zaslavsky Encoding (1981) of a subset S ⊆ Φ + into the signed graph G S with a positive edge ij for each e i − e j ∈ S ◮ a negative edge ij for each e i + e j ∈ S ◮ a halfedge at j for each e j ∈ S ◮ a negative loop at j for each 2 e j ∈ S ◮ Linear independent subsets of Φ + correspond precisely to signed pseudo- forests which consist of signed trees plus possibly a single halfedge (halfedge-tree) ◮ | Φ G | = n − tc( G ) a single loop (loop-tree) ◮ vol(Φ G ) = 2 pc( G )+lc( G ) a single unbalanced cycle (pseudotree) ◮ The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Why care about... Signed Graphs Earliest appearance in social psychology (Heider 1946, Cartwright– ◮ Harary 1956) “The enemy of my enemy is my friend” Type-B analogues of graphs, natural from the viewpoint of incidence ◮ matrices Applications to ◮ Knot theory (positive/negative crossings) ◮ Biology (perturbed large-scale biological networks ◮ Chemistry (M¨ obius systems) ◮ Physics (spin glasses—mixed Ising model) ◮ Computer science (correlation clustering) ◮ The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Integral Coxeter Permutahedra � Fix Φ ∈ { A n , B n , C n , D n : n ≥ 2 } and consider Π Z (Φ) = [0 , α ] α ∈ Φ + Linear independent subsets of Φ + correspond precisely to signed pseudo- forests which consist of signed trees plus possibly a single halfedge (halfedge-tree) ◮ | Φ G | = n − tc( G ) a single loop (loop-tree) ◮ vol(Φ G ) = 2 pc( G )+lc( G ) a single unbalanced cycle (pseudotree) ◮ Ardila–Castillo–Henley (2015) Let F (Φ) be the set of Φ -forests. Then 2 pc( G )+lc( G ) t n − tc( G ) . � ehr Π Z (Φ) ( t ) = G ∈F (Φ) The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter

Almost Integral Zonotopes Lemma Let U ⊂ Z d be a finite set and v ∈ Q d . Then � χ W ( t ) vol( W ) t | W | ehr v + Z ( U ) ( t ) = W ⊆ U lin. indep. if ( t v + span( W )) ∩ Z d � = ∅ , � 1 where χ W ( t ) := 0 otherwise. Ardlia–M B–McWhirter Fix Φ ∈ { A n : n ≥ 2 even } ∪ { B n : n ≥ 1 } . Let F (Φ) be the set of Φ -forests and E (Φ) ⊆ F (Φ) be the set of Φ -forests such that every tree component has an even number of vertices. Then  � 2 pc( G ) t n − tc( G ) if t is even,     G ∈F (Φ) ehr Π(Φ) ( t ) = � 2 pc( G ) t n − tc( G ) if t is odd.     G ∈E (Φ) The Arithmetic of Coxeter Permutahedra Federico Ardila, Matthias Beck & Jodi McWhirter