21 research outputs found
Renormalization group-like proof of the universality of the Tutte polynomial for matroids
In this paper we give a new proof of the universality of the Tutte polynomial
for matroids. This proof uses appropriate characters of Hopf algebra of
matroids, algebra introduced by Schmitt (1994). We show that these Hopf algebra
characters are solutions of some differential equations which are of the same
type as the differential equations used to describe the renormalization group
flow in quantum field theory. This approach allows us to also prove, in a
different way, a matroid Tutte polynomial convolution formula published by
Kook, Reiner and Stanton (1999). This FPSAC contribution is an extended
abstract.Comment: 12 pages, 3 figures, conference proceedings, 25th International
Conference on Formal Power Series and Algebraic Combinatorics, Paris, France,
June 201
Renormalization group-like proof of the universality of the Tutte polynomial for matroids
In this paper we give a new proof of the universality of the Tutte polynomial for matroids. This proof uses appropriate characters of Hopf algebra of matroids, algebra introduced by Schmitt (1994). We show that these Hopf algebra characters are solutions of some differential equations which are of the same type as the differential equations used to describe the renormalization group flow in quantum field theory. This approach allows us to also prove, in a different way, a matroid Tutte polynomial convolution formula published by Kook, Reiner and Stanton (1999). This FPSAC contribution is an extended abstract
Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach
Using a quantum field theory renormalization group-like differential
equation, we give a new proof of the recipe theorem for the Tutte polynomial
for matroids. The solution of such an equation is in fact given by some
appropriate characters of the Hopf algebra of isomorphic classes of matroids,
characters which are then related to the Tutte polynomial for matroids. This
Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte
polynomial convolution formula appearing in W. Kook {\it et. al., J. Comb.
Series} {\bf B 76} (1999).Comment: 14 pages, 3 figure
Wilsonian renormalization, differential equations and Hopf algebras
In this paper, we present an algebraic formalism inspired by Butcher's
B-series in numerical analysis and the Connes-Kreimer approach to perturbative
renormalization. We first define power series of non linear operators and
propose several applications, among which the perturbative solution of a fixed
point equation using the non linear geometric series. Then, following
Polchinski, we show how perturbative renormalization works for a non linear
perturbation of a linear differential equation that governs the flow of
effective actions. Then, we define a general Hopf algebra of Feynman diagrams
adapted to iterations of background field effective action computations. As a
simple combinatorial illustration, we show how these techniques can be used to
recover the universality of the Tutte polynomial and its relation to the
-state Potts model. As a more sophisticated example, we use ordered diagrams
with decorations and external structures to solve the Polchinski's exact
renormalization group equation. Finally, we work out an analogous construction
for the Schwinger-Dyson equations, which yields a bijection between planar
diagrams and a certain class of decorated rooted trees.Comment: 42 pages, 26 figures in PDF format, extended version of a talk given
at the conference "Combinatorics and physics" held at Max Planck Institut
fuer Mathematik in Bonn in march 2007, some misprints correcte
Recipe theorems for polynomial invariants on ribbon graphs with half-edges
We provide recipe theorems for the Bollob\`as and Riordan polynomial
defined on classes of ribbon graphs with half-edges introduced in
arXiv:1310.3708[math.GT]. We also define a generalized transition polynomial
on this new category of ribbon graphs and establish a relationship between
and .Comment: 24 pages, 14 figure
Generalization of the Bollob\'as-Riordan polynomial for tensor graphs
Tensor models are used nowadays for implementing a fundamental theory of
quantum gravity. We define here a polynomial encoding the
supplementary topological information. This polynomial is a natural
generalization of the Bollob\'as-Riordan polynomial (used to characterize
matrix graphs) and is different of the Gur\uau polynomial, (R. Gur\uau,
"Topological Graph Polynomials in Colored Group Field Theory", Annales Henri
Poincare {\bf 11}, 565-584 (2010)) defined for a particular class of tensor
graphs, the colorable ones. The polynomial is defined for both
colorable and non-colorable graphs and it is proved to satisfy the
contraction/deletion relation. A non-trivial example of a non-colorable graphs
is analyzed.Comment: 22 pages, 20 figure
Universal Tutte characters via combinatorial coalgebras
The Tutte polynomial is the most general invariant of matroids and graphs
that can be computed recursively by deleting and contracting edges. We
generalize this invariant to any class of combinatorial objects with deletion
and contraction operations, associating to each such class a universal Tutte
character by a functorial procedure. We show that these invariants satisfy a
universal property and convolution formulae similar to the Tutte polynomial.
With this machinery we recover classical invariants for delta-matroids, matroid
perspectives, relative and colored matroids, generalized permutohedra, and
arithmetic matroids, and produce some new convolution formulae. Our principal
tools are combinatorial coalgebras and their convolution algebras. Our results
generalize in an intrinsic way the recent results of
Krajewski--Moffatt--Tanasa.Comment: Accepted version, 51p
Hopf algebras and Tutte polynomials
By considering Tutte polynomials of Hopf algebras, we show how a Tutte
polynomial can be canonically associated with combinatorial objects that have
some notions of deletion and contraction. We show that several graph
polynomials from the literature arise from this framework. These polynomials
include the classical Tutte polynomial of graphs and matroids, Las Vergnas'
Tutte polynomial of the morphism of matroids and his Tutte polynomial for
embedded graphs, Bollobas and Riordan's ribbon graph polynomial, the Krushkal
polynomial, and the Penrose polynomial.
We show that our Tutte polynomials of Hopf algebras share common properties
with the classical Tutte polynomial, including deletion-contraction
definitions, universality properties, convolution formulas, and duality
relations. New results for graph polynomials from the literature are then
obtained as examples of the general results.
Our results offer a framework for the study of the Tutte polynomial and its
analogues in other settings, offering the means to determine the properties and
connections between a wide class of polynomial invariants.Comment: v2: change of title and some reorderin