h-polynomials of triangulations of flow polytopes (Cornell University)

Transcription

1 h-polynomials of triangulations of flow polytopes Karola Mészáros (Cornell University)

2 h-polynomials of triangulations of flow polytopes (and of reduction trees) Karola Mészáros (Cornell University)

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 F K5 (1, 0, 0, 0, 1) Flow polytopes

12 Flow polytopes F K5 (1, 0, 0, 0, 1) 1 = a + b + c + d 0 = e + f + g a 0 = h + i b e 0 = j c f h a, b, c, d, e, f, g, h, i, j 0 K 5 d c b f a e h j g i

13 Flow polytopes F K5 (1, 0, 0, 0, 1) 1 = a + b + c + d 0 = e + f + g a 0 = h + i b e 0 = j c f h a, b, c, d, e, f, g, h, i, j 0 K 5 d c b f a e h j g i 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.

14 Examples of flow polytopes 1 1 simplex K

15 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.

16 Some interesting examples of flow polytopes Theorem [Zeilberger 99]: vol(f Kn+1 ) =Cat(1)Cat(2) Cat(n 2).

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

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

19 Triangulating F G q p p q p = q G 0 G 1 G 2 G 3 p q p=q p q q p p q

20 Triangulating F G q p p q p = q p G 0 G 1 G 2 G 3 p q p=q q q p p q Proposition: F G0 = F G1 F G2, F G1 F G2 = F G3.

21 Triangulating F G q p p q p = q p G 0 G 1 G 2 G 3 p q p=q q q p p q Proposition: F G0 = F G1 F G2, F G1 F G2 = F G3. F G1 or F G2 could be empty.

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

23 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

24 Reduction tree T (G) 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.

25 Reduction tree T (G) Lemma. If the leaves are labeled by graphs H 1,, H k then the flow polytopes F H1,..., F Hk are simplices.

26 Reduction tree T (G) Lemma. The normalized volume of F G is equal to the number of leaves in a reduction tree T (G).

27 Reductions in variables q p p q 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 k i j k i j k x ij x jk x jk x ik + x ik x ij + βx ik

28 Reduced form x 12 x 23 x

29 Reduced form x 12 x 23 x x 13 x 23 x 24 x 12 x 14 x 34 x 14 x 24 x x 12 x 13 x 14 x 13 x 14 x 24

30 Reduced form x 12 x 23 x x 13 x 23 x 24 x 12 x 14 x 34 x 14 x 24 x x 12 x 13 x 14 x 13 x 14 x 24 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)

31 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

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

33 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.

34 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)

35 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)

36 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.

37 Why triangulation?

38 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.

39 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 2 e 1 e 3

40 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.

41 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?

42 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

43 Yes, triangulation! Theorem. (M, 2014) There is a way to play the game and obtain a shellable triangulation of the flow polytope F G.

44 Yes, triangulation! Theorem. (M, 2014) There is a way to play the game and obtain a shellable triangulation of the flow polytope F G. A triangulation is said to be shellable, if we can order the top dimensional simplices F 1,..., F k, so that F i, 1 < i, attaches to the preceeding simplices F 1,..., F i 1, on a union of its facets (at least one of them).

45 Yes, triangulation! Theorem. (M, 2014) There is a way to play the game and obtain a shellable triangulation of the flow polytope F G. The key is to use a special reduction order. Namely, do the reductions from left to right and always on the topmost edges.

46 Yes, triangulation! Theorem. (M, 2014) There is a way to play the game and obtain a shellable triangulation of the flow polytope F G. The key is to use a special reduction order. Namely, do the reductions from left to right and always on the topmost edges. We call this special order O.

47 Reduction tree R O G

48 Reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

49 Shelling T O Let F 1,..., F l be the full-dimensional leaves of RG O depth-first search order. ordered by Theorem. (M, 2014) F F1,..., F Fl is a shelling order of the triangulation T O of F G.

50 Shelling T O Let F 1,..., F l be the full-dimensional leaves of RG O depth-first search order. ordered by Theorem. (M, 2014) F F1,..., F Fl is a shelling order of the triangulation T O of F G. The idea of proof is weak embeddability of reduction trees.

51 Weak embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

52 Weak embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

53 Weak embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

54 Weak embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

55 Weak embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

56 Weak embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

57 Weak embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

58 Weak embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

59 Leaves of R O G F 2 F 4 F 3 F 5 (F 3 F 5 ) (F 4 F 5 ) F 1 F 1 F 2 F 5 F 6 F 6 F 2 F 2 F 3 F 3 F 4 F 4 F 5 F 5

60 Leaves of R O G Theorem. (M, 2014) Let F 1,..., F l be the full-dimensional leaves of R O G depth-first search order. Let ordered by {Q i 1,..., Q i f(i) } = {F i F j 1 j < i, E(F i F j ) = E(F i ) 1}. Then l i=1 f(i) j=1 (F i + Q i j ) is the formal sum of the set of the leaves of RG O, where the product of graphs is their intersection. If f(i) = 0 we define f(i) j=1 (F i + Q i j ) = F i.

61 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

62 Shellable reduction trees The idea of proof for the previous theorem is to define a notion alike shellability for reduction trees.

63 Shellable reduction trees The idea of proof for the previous theorem is to define a notion alike shellability for reduction trees. Given a full dimensional leaf L of R G, H is a preceeding facet of L if

64 Shellable reduction trees The idea of proof for the previous theorem is to define a notion alike shellability for reduction trees. Given a full dimensional leaf L of R G, H is a preceeding facet of L if 1. H is a leaf before L in R G in depth-first search order

65 Shellable reduction trees The idea of proof for the previous theorem is to define a notion alike shellability for reduction trees. Given a full dimensional leaf L of R G, H is a preceeding facet of L if 1. H is a leaf before L in R G in depth-first search order 2. E(H) E(L) and E(H) = E(L) 1

66 Shellable reduction trees The idea of proof for the previous theorem is to define a notion alike shellability for reduction trees. Given a full dimensional leaf L of R G, H is a preceeding facet of L if 1. H is a leaf before L in R G in depth-first search order 2. E(H) E(L) and E(H) = E(L) 1 3.

67 Strong embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

68 Strong embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

69 Strong embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

70 Strong embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

71 Strong embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

72 Strong embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

73 Strong embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

74 Strong embeddable reduction tree R O G F 1 F 6 F 2 F 3 F 4 F 5

75 h-polynomials of reduction trees

76 h-polynomials of reduction trees Define the h-polynomial of a reduction tree R G as h(r G, β) = i=0 s iβ i, where s i is the number of full dimensional leaves L of R G with exactly i preceeding facets.

77 h-polynomials of reduction trees Define the h-polynomial of a reduction tree R G as h(r G, β) = i=0 s iβ i, where s i is the number of full dimensional leaves L of R G with exactly i preceeding facets. All above can also be defined for partial reduction trees R p G, or alternatively reduction trees in other algebras.

78 h-polynomials of reduction trees Define the h-polynomial of a reduction tree R G as h(r G, β) = i=0 s iβ i, where s i is the number of full dimensional leaves L of R G with exactly i preceeding facets. All above can also be defined for partial reduction trees R p G, or alternatively reduction trees in other algebras. Theorem. (M, 2014) For strong embeddable R p G we have Q R p G (β 1) = h(rp G, β)

79 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

80 Reduced forms are shifted h-polynomials Theorem. (M, 2014) For strong embeddable R p G we have Q R p G (b 1) = h(rp G, b)

81 Reduced forms are shifted h-polynomials Theorem. (M, 2014) For strong embeddable R p G we have Q R p G (b 1) = h(rp G, b) Corollary. (M, 2014) For strong embeddable R p G the coefficients of Q R p (b 1) are nonnegative. G

82 Reduced forms are shifted h-polynomials Theorem. (M, 2014) For strong embeddable R p G we have Q R p G (b 1) = h(rp G, b) Corollary. (M, 2014) For strong embeddable R p G the coefficients of Q R p (b 1) are nonnegative. G Generalizations of the above theorem and corollary can be used to address a nonnegativity conjecture of Kirillov.

83

84 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

85 If a triangulation is shellable... Recall that the motivation for the definitions of weak and strong embeddability was the shellable triangulation T O obtained from R O G.

86 If a triangulation is shellable......one wonders if it is regular.

87 If a triangulation is shellable......one wonders if it is regular. Question: Is T O regular?

88 If a triangulation is shellable......one wonders if it is regular. Question: Is T O regular? (I am not sure about that, but...)

89 If a triangulation is shellable......one wonders if it is regular. Question: Is T O regular? (I am not sure about that, but... I know something else)

90 The something else Theorem. (M, 2014) There are ways to play the game and obtain regular and flag triangulations of the flow polytope F G.

91 The something else Theorem. (M, 2014) There are ways to play the game and obtain regular and flag triangulations of the flow polytope F G. This result builds on work of Danilov-Karzanov-Koshevoy.

92 Happy birthday, Richard!