Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach

Transcription

1 Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach Nguyen Hoang Nghia (in collaboration with Gérard Duchamp, Thomas Krajewski and Adrian Tanasă) Laboratoire d Informatique de Paris Nord, Université Paris 13 arxiv: [math.co] accepted by Discrete Mathematics and Theoretical Computer Science Proceedings 70th Séminaire Lotharingien de Combinatoire March 25, 2013 Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 1 / 18

2 Plan 1 Matroids: The Tutte polynomial and the Hopf algebra 2 Characters of the Hopf algebras of matroids 3 Convolution formula for the Tutte polynomials for matroids 4 Proof of the universality of the Tutte polynomials for matroids Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 2 / 18

3 Matroid theory - some definitions Definition 1.1 A matroid M = (E, I) is a pair (E, I) a finite set: E a collection of subsets of E: I satisfying: Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 3 / 18

4 Matroid theory - some definitions Definition 1.1 A matroid M = (E, I) is a pair (E, I) a finite set: E a collection of subsets of E: I satisfying: I is non-empty, Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 3 / 18

5 Matroid theory - some definitions Definition 1.1 A matroid M = (E, I) is a pair (E, I) a finite set: E a collection of subsets of E: I satisfying: I is non-empty, every subset of every member of I is also in I, Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 3 / 18

6 Matroid theory - some definitions Definition 1.1 A matroid M = (E, I) is a pair (E, I) a finite set: E a collection of subsets of E: I satisfying: I is non-empty, every subset of every member of I is also in I, if X, Y I and X = Y + 1, then x X Y s.t. Y {x} I. Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 3 / 18

7 Matroid theory - some definitions Definition 1.1 A matroid M = (E, I) is a pair (E, I) a finite set: E a collection of subsets of E: I satisfying: I is non-empty, every subset of every member of I is also in I, if X, Y I and X = Y + 1, then x X Y s.t. Y {x} I. The set E : the ground set of the matroid The members of I : the independent sets of the matroid. Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 3 / 18

8 Uniform matroids The uniform matroid U k,n is the matroid on n elements whose independent sets are all the subsets of size k. Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 4 / 18

9 Uniform matroids The uniform matroid U k,n is the matroid on n elements whose independent sets are all the subsets of size k. Example 1.2 Let E = {1}. One hase two uniform matroids U 0,1 = (E, { }); Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 4 / 18

10 Uniform matroids The uniform matroid U k,n is the matroid on n elements whose independent sets are all the subsets of size k. Example 1.2 Let E = {1}. One hase two uniform matroids U 0,1 = (E, { }); U 1,1 = (E, {, {1}}). Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 4 / 18

11 Uniform matroids The uniform matroid U k,n is the matroid on n elements whose independent sets are all the subsets of size k. Example 1.2 Let E = {1}. One hase two uniform matroids U 0,1 = (E, { }); U 1,1 = (E, {, {1}}). Example 1.3 Let E = {1, 2} and I = {, {1}, {2}}. One has the uniform matroid U 1,2. Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 4 / 18

12 Rank function Definition 1.4 Let M = (E, I) be a matroid and A E. The rank function of A: r(a) = max{ B : B I, B A}. (1) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 5 / 18

13 Rank function Definition 1.4 Let M = (E, I) be a matroid and A E. The rank function of A: r(a) = max{ B : B I, B A}. (1) The nullity function of A: n(a) = A r(a). (2) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 5 / 18

14 Loop-coloop Definition 1.5 Let e E. The element e is called a loop if r({e}) = 0. Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 6 / 18

15 Loop-coloop Definition 1.5 Let e E. The element e is called a loop if r({e}) = 0. The element e is called a coloop if r(e {e}) = r(e) 1. Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 6 / 18

16 Two operations [1] Definition 1.6 (Deletion) One sets the collection of subsets that I = {I E T : I I}. (3) Then one has that the pair (E T, I ) is a matroid, called that the deletion of T from M. One denotes that M\ T Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 7 / 18

17 Two operations [2] Definition 1.7 (Contraction) One sets the collection of subsets that I = {I E T : I B T I}, (4) where B T is a maximal independent subset of T. Then one has that the pair (E T, I ) is a matroid, called that the contraction of T from M. One denotes that M/ T Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 8 / 18

18 Tutte polynomial for matroids Definition 1.8 The Tutte polynomial of matroid M: T M (x, y) = A E(x 1) r(e) r(a) (y 1) n(a). (5) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 9 / 18

19 Tutte polynomial for matroids Definition 1.8 The Tutte polynomial of matroid M: T M (x, y) = A E(x 1) r(e) r(a) (y 1) n(a). (5) Example 1.9 T U1,2 (x, y) = x + y. (6) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 9 / 18

20 Tutte polynomial for matroids Definition 1.8 The Tutte polynomial of matroid M: T M (x, y) = A E(x 1) r(e) r(a) (y 1) n(a). (5) Example 1.9 T U1,2 (x, y) = x + y. (6) Theorem 1.10 Deletion-contraction relation: T M (x, y) = T M/e (x, y) + T M\e (x, y). (7) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC 70 9 / 18

21 Hopf algebra on matroids (H. Crapo and W. Schmitt. A free subalgebra of the algebra of matroids. EJC, 26(7), 05.) Coproduct (M) = A E M A M/A. (8) ɛ(m) = { 1, if M = U 0,0, 0 otherwise. (U 1,2 ) = 1 U 1,2 + 2U 1,1 U 0,1 + U 1,2 1. (k( M),, 1,, ɛ) is bialgebra. Moreover, this bialgebra is graded by the cardinal of the ground set, then it is a Hopf algebra. Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

22 Two infinitesimal characters Let us define two linear forms. δ loop (M) = δ coloop (M) = { 1 K if M = U 0,1, 0 K otherwise. { 1 K if M = U 1,1, 0 K otherwise. (9) (10) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

23 Two infinitesimal characters Let us define two linear forms. δ loop (M) = δ coloop (M) = { 1 K if M = U 0,1, 0 K otherwise. { 1 K if M = U 1,1, 0 K otherwise. (9) (10) One has δ loop (M 1 M 2 ) = δ loop (M 1 )ɛ(m 2 ) + ɛ(m 1 )δ loop (M 2 ). δ coloop (M 1 M 2 ) = δ coloop (M 1 )ɛ(m 2 ) + ɛ(m 1 )δ coloop (M 2 ). Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

24 Two infinitesimal characters Let us define two linear forms. δ loop (M) = δ coloop (M) = { 1 K if M = U 0,1, 0 K otherwise. { 1 K if M = U 1,1, 0 K otherwise. (9) (10) One has δ loop (M 1 M 2 ) = δ loop (M 1 )ɛ(m 2 ) + ɛ(m 1 )δ loop (M 2 ). δ coloop (M 1 M 2 ) = δ coloop (M 1 )ɛ(m 2 ) + ɛ(m 1 )δ coloop (M 2 ). Theorem 2.1 exp {aδ coloop + bδ loop } is a Hopf algebra character. exp {aδ coloop + bδ loop }(M) = a r(m) b n(m). (11) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

25 A mapping α Let us define α(x, y, s, M) := exp s{δ coloop + (y 1)δ loop } exp s{(x 1)δ coloop + δ loop }(M). (12) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

26 A mapping α Let us define α(x, y, s, M) := exp s{δ coloop + (y 1)δ loop } exp s{(x 1)δ coloop + δ loop }(M). (12) Proposition 3.1 α is a Hopf algebra character. Moreover, one has α(x, y, s, M) = s E T M (x, y). (13) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

27 A convolution formula for Tutte polynomials The character α can be rewritten: α(x, y, s, M) = exp (s(δ coloop + (y 1)δ loop )) exp (s( δ coloop + δ loop )) exp (s(δ coloop δ loop )) exp (s((x 1)δ coloop + δ loop ))(14). Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

28 A convolution formula for Tutte polynomials The character α can be rewritten: α(x, y, s, M) = exp (s(δ coloop + (y 1)δ loop )) exp (s( δ coloop + δ loop )) exp (s(δ coloop δ loop )) exp (s((x 1)δ coloop + δ loop ))(14). Corollary 3.2 (Theorem 1 of [KRS99]) The Tutte polynomial satisfies T M (x, y) = A E T M A (0, y)t M/A (x, 0). (15) W. Kook, V. Reiner, and D. Stanton. A Convolution Formula for the Tutte Polynomial. Journal of Combinatorial Series (99) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

29 Differential equation of character α Proposition 4.1 The character α is the solution of the differential equation: dα ds (M) = (xα δ coloop + yδ loop α + [δ coloop, α] [δ loop, α] )(M). (16) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

30 We take a four-variable matroid polynomial Q M (x, y, a, b) which has the following properties: a multiplicative law if e is a coloop, then if e is a loop, then Q M1 M 2 (x, y, a, b) = Q M1 (x, y, a, b)q M2 (x, y, a, b), (17) if e is a nonseparating point, then Q M (x, y, a, b) = xq M\e (x, y, a, b), (18) Q M (x, y, a, b) = yq M/e (x, y, a, b), (19) Q M (x, y, a, b) = aq M\e (x, y, a, b) + bq M/e (x, y, a, b). (20) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

31 A mapping β β(x, y, a, b, s, M) := s E Q M (x, y, a, b). (21) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

32 A mapping β β(x, y, a, b, s, M) := s E Q M (x, y, a, b). (21) Lemma 4.2 The mapping β is a matroid Hopf algebra character. Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

33 A mapping β β(x, y, a, b, s, M) := s E Q M (x, y, a, b). (21) Lemma 4.2 The mapping β is a matroid Hopf algebra character. Proposition 4.3 The character β satisfies the following differential equation: dβ ds (M) = (xβ δ coloop + yδ loop β + b[δ coloop, β] a[δ loop, β] ) (M). (22) Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

34 A mapping β β(x, y, a, b, s, M) := s E Q M (x, y, a, b). (21) Lemma 4.2 The mapping β is a matroid Hopf algebra character. Proposition 4.3 The character β satisfies the following differential equation: dβ ds (M) = (xβ δ coloop + yδ loop β + b[δ coloop, β] a[δ loop, β] ) (M). (22) Sketch of the proof: Using the definitions of the infitesimal character δ loop and δ coloop and the conditions of the polynomial Q M (x, y, a, b), one can get the result. Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

35 Main theorem From the propositions 3.1, 4.1 and 4.3, one gets the result Theorem 4.4 Q(x, y, a, b, M) = a n(m) b r(m) T M ( x b, y ). (23) a Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18

36 Thank you for your attention! Nguyen Hoang-Nghia (LIPN, Université Paris 13) Recipe theorem for the Tutte polynomial for matroids, renormalization group-like Ellwangen, approach SLC / 18