h -polynomials of triangulations of flow polytopes Karola M esz - PowerPoint PPT Presentation

h -polynomials of triangulations of flow polytopes Karola M´ esz´ aros (Cornell University)

h -polynomials of triangulations of flow polytopes (and of reduction trees) Karola M´ esz´ aros (Cornell University)

Plan Background on flow polytopes Reduction trees and reduced forms Reduced forms generalize h -polynomials of triangulations Canonical triangulations of flow polytopes Shellings and h -polynomials of reduction trees Nonnegativity results on reduced forms Back to where we started: last words on flow polytopes

Plan Background on flow polytopes Reduction trees and reduced forms Reduced forms generalize h -polynomials of triangulations Canonical triangulations of flow polytopes Shellings and h -polynomials of reduction trees Nonnegativity results on reduced forms Back to where we started: last words on flow polytopes

Plan Background on flow polytopes Reduction trees and reduced forms Reduced forms generalize h -polynomials of triangulations Canonical triangulations of flow polytopes Shellings and h -polynomials of reduction trees Nonnegativity results on reduced forms Back to where we started: last words on flow polytopes

Plan Background on flow polytopes Reduction trees and reduced forms Reduced forms generalize h -polynomials of triangulations Canonical triangulations of flow polytopes Shellings and h -polynomials of reduction trees Nonnegativity results on reduced forms Back to where we started: last words on flow polytopes

Plan Background on flow polytopes Reduction trees and reduced forms Reduced forms generalize h -polynomials of triangulations Canonical triangulations of flow polytopes Shellings and h -polynomials of reduction trees Nonnegativity results on reduced forms Back to where we started: last words on flow polytopes

Plan Background on flow polytopes Reduction trees and reduced forms Reduced forms generalize h -polynomials of triangulations Canonical triangulations of flow polytopes Shellings and h -polynomials of reduction trees Nonnegativity results on reduced forms Back to where we started: last words on flow polytopes

Plan Background on flow polytopes Reduction trees and reduced forms Reduced forms generalize h -polynomials of triangulations Canonical triangulations of flow polytopes Shellings and h -polynomials of reduction trees Nonnegativity results on reduced forms Back to where we started: last words on flow polytopes

Plan Background on flow polytopes Reduction trees and reduced forms Reduced forms generalize h -polynomials of triangulations Canonical triangulations of flow polytopes Shellings and h -polynomials of reduction trees Nonnegativity results on reduced forms Back to where we started: last words on flow polytopes

Flow polytopes F K 5 (1 , 0 , 0 , 0 , − 1)

Flow polytopes F K 5 (1 , 0 , 0 , 0 , − 1) d 1 = a + b + c + d g K 5 c 0 = e + f + g − a i f b 0 = h + i − b − e 0 = j − c − f − h 1 0 0 0 − 1 a j e h 1 4 2 3 5 a, b, c, d, e, f, g, h, i, j ≥ 0

Flow polytopes F K 5 (1 , 0 , 0 , 0 , − 1) d 1 = a + b + c + d g K 5 c 0 = e + f + g − a i f b 0 = h + i − b − e 0 = j − c − f − h 1 0 0 0 − 1 a j e h 1 4 2 3 5 a, b, c, d, e, f, g, h, i, j ≥ 0 For a general graph G on the vertex set [ n ] , with net flow a = (1 , 0 , . . . , 0 , − 1) , the flow polytope of G , denoted F G , is the set of flows f : E ( G ) → R ≥ 0 such that the total flow going in at vertex 1 is one, and there is flow conservation at each of the inner vertices.

Examples of flow polytopes 1 − 1 simplex K 4 1 0 0 − 1

An intriguing theorem Theorem [Postnikov-Stanley]: For a graph G on the vertex set { 1 , 2 . . . , n } we have vol ( F G (1 , 0 , . . . , 0 , − 1)) = K G (0 , d 2 , . . . , d n − 1 , − � n − 1 i =2 d i ) , where d i = ( indegree of i ) − 1 and K G is the Kostant partition function.

Some interesting examples of flow polytopes Theorem [Zeilberger 99]: vol( F K n +1 ) = Cat (1) Cat (2) · · · Cat ( n − 2) .

Some interesting examples of flow polytopes Theorem [Zeilberger 99]: vol( F K n +1 ) = Cat (1) Cat (2) · · · Cat ( n − 2) . F K n +1 is a member of a larger family of polytopes with volumes given by nice product formulas.

Some interesting examples of flow polytopes Theorem [Zeilberger 99]: vol( F K n +1 ) = Cat (1) Cat (2) · · · Cat ( n − 2) . F K n +1 is a member of a larger family of polytopes with volumes given by nice product formulas. � m + n + i +1 � (Think � m + n − 1 1 . ) i = m +1 2 i +1 2 i

Triangulating F G p ≥ q q ≥ p p = q G 0 G 1 G 2 G 3 p = q p q p q q − p p − q →

Triangulating F G p ≥ q q ≥ p p = q G 0 G 1 G 2 G 3 p = q p q p q q − p p − q → Proposition: F G 0 = F G 1 ∪ F G 2 , F G 1 ∩ F G 2 = F G 3 .

Triangulating F G p ≥ q q ≥ p p = q G 0 G 1 G 2 G 3 p = q p q p q q − p p − q → Proposition: F G 0 = F G 1 ∪ F G 2 , F G 1 ∩ F G 2 = F G 3 . F G 1 or F G 2 could be empty.

� G = G with s and t G 1 2 s t Purpose: we can simply do the reductions on G and at the end arrive to a triangulation of F � G .

Plan Background on flow polytopes Reduction trees and reduced forms Reduced forms generalize h -polynomials of triangulations Canonical triangulations of flow polytopes Shellings and h -polynomials of reduction trees Nonnegativity results on reduced forms Back to where we started: last words on flow polytopes

Reduction tree T ( G ) 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 A reduction tree of G = ([4] , { (1 , 2) , (2 , 3) , (3 , 4) } ) with five leaves. The edges on which the reductions are performed are in bold.

Reduction tree T ( G ) 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Lemma. If the leaves are labeled by graphs H 1 , , H k then the flow polytopes F � H 1 , . . . , F � H k are simplices.

Reduction tree T ( G ) 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Lemma. The normalized volume of F � G is equal to the number of leaves in a reduction tree T ( G ) .

Reductions in variables p ≥ q q ≥ p p = q G 0 G 1 G 2 G 3 p = q p q p q q − p p − q → i j k i j i j k i j k k x ij x jk → x jk x ik + x ik x ij + βx ik

Reduced form x 12 x 23 x 34 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Reduced form x 12 x 23 x 34 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 x 13 x 23 x 24 x 12 x 14 x 34 x 14 x 24 x 34 1 2 3 4 1 2 3 4 x 13 x 14 x 24 x 12 x 13 x 14

Reduced form x 12 x 23 x 34 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 x 13 x 23 x 24 x 12 x 14 x 34 x 14 x 24 x 34 1 2 3 4 1 2 3 4 x 13 x 14 x 24 x 12 x 13 x 14 x 12 x 13 x 14 + x 13 x 14 x 24 x 13 x 23 x 24 x 12 x 14 x 34 x 14 x 24 x 34 + + + ( β = 0)

Plan Background on flow polytopes Reduction trees and reduced forms Reduced forms generalize h -polynomials of triangulations Canonical triangulations of flow polytopes Shellings and h -polynomials of reduction trees Nonnegativity results on reduced forms Back to where we started: last words on flow polytopes

Reduced form Denote by Q G ( β, x ) the reduced form of the monomial � ( i,j ) ∈ E ( G ) x ij .

Reduced form Denote by Q G ( β, x ) the reduced form of the monomial � ( i,j ) ∈ E ( G ) x ij . Let Q G ( β ) denote the reduced form when all x ij = 1 .

Reduced form Denote by Q G ( β, x ) the reduced form of the monomial � ( i,j ) ∈ E ( G ) x ij . Let Q G ( β ) denote the reduced form when all x ij = 1 . Theorem. (M, 2014) Q G ( β − 1) = h ( T , β )

Reduced form Denote by Q G ( β, x ) the reduced form of the monomial � ( i,j ) ∈ E ( G ) x ij . Let Q G ( β ) denote the reduced form when all x ij = 1 . Theorem. (M, 2014) Q G ( β − 1) = h ( T , β ) (where T is a “triangulation” of F ˜ G obtained via the game)

Reduced form Denote by Q G ( β, x ) the reduced form of the monomial � ( i,j ) ∈ E ( G ) x ij . Let Q G ( β ) denote the reduced form when all x ij = 1 . Theorem. (M, 2014) Q G ( β − 1) = h ( T , β ) (where T is a “triangulation” of F ˜ G obtained via the game) In particular the coefficients of Q G ( β − 1) are nonnegative.

Why “triangulation”?

Why “triangulation”? If we just play the game in any way we like, we might not get a triangulation in the sense of a simplicial complex.

Why “triangulation”? If we just play the game in any way we like, we might not get a triangulation in the sense of a simplicial complex. e 3 − e 2 − e 1 e 1 e 2 − e 3

Why “triangulation”? If we just play the game in any way we like, we might not get a triangulation in the sense of a simplicial complex. Nevertheless, the notions of f -vectors and h -vectors still make sense.

Why “triangulation”? If we just play the game in any way we like, we might not get a triangulation in the sense of a simplicial complex. Nevertheless, the notions of f -vectors and h -vectors still make sense. Still, we wonder: Is there a way to play the game and get a triangulation?