Universal Tutte characters via combinatorial coalgebras

Abstract

This work discusses the extraction of meaningful invariants of combinatorial objects from coalgebra or bialgebra structures. The Tutte polynomial is an invariant of graphs well known for the formula which computes it recursively by deleting and contracting edges, and for its universality with respect to similar recurrence. We generalize this to all classes 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. We also produce some new invariants along with new convolution formulae.obtain several convolution formulae for different classes of combinatorial objects. Some of these formulae are well known, while others are (to the best of our knowledge) new: see for instance Propositions 5.12, 5.13 and 5.14 for the classical Tutte polynomial, Proposition 6.17 for the Las Vergnas polynomial, Proposition 6.20 for the Bollobás-Riordan polynomial and Theorem 10.9 for the arithmetic Tutte polynomial.We have chosen to exemplify our results with mostly matroid-like combinatorial structures, which are arguably simpler to deal with than topological examples. Our formalism, and in particular the mechanism of twist maps, should help uncover new invariants in the latter class and produce interesting convolution formulae. This will be the subject of a subsequent article.Layout. The structure of this paper is as follows. In Section 2 we give the definitions of minors systems and comonoids in set species. Section 3 contains the main results, including the definitions and statements of our universal invariants and formulae. Section 4 presents a number of further results which are less essential for our main developments.The remainder of the paper, Sections 5 through 10, comprises a sequence of applications of our theory to numerous individual minors systems. We work out Grothendieck groups, universal Tutte characters, and in many cases universal convolution formulae, and point out how these specialize to invariants and formulae present in the literature. Section 5 covers the minors system of matroids, of which all the subsequent sections are in one way or another generalizations, together with the minors system of graphs. Of the following sections, Section 6 is on delta-matroids, matroid perspectives, and their ilk is called on in Section 7 on relative matroids, but there are no (or at most incidental) dependences between these and Section 8 on polymatroids and generalised permutohedra, Section 9 on colored matroids, or Section 10 on arithmetic matroids, nor among the latter three, so the reader should have no trouble taking these in any order. The last four sections are also new by comparison with [37].Notation. We fix a commutative ring with unit K...

Delta-matroids as subsystems of sequences of Higgs lifts

“…. , k}, we have the full Higgs lift delta-matroid of the pair (Q, L); they were studied by Tardos [30], who called them generalized matroids, and more recently in [18], where they are called saturated delta-matroids. If k and all elements of K are even, we have the even Higgs lift delta-matroid of the pair (Q, L).…”

Logarithmic concavity for morphisms of matroids

“…In general, the homogeneous multivariate Tutte polynomial satisfies the deletion-contraction relation Remark 16. The homogeneous multivariate Tutte polynomials are the reduced multivariate Tutte characters for the minor system of flag matroids with constituents in the sense of [DFM18]. See [KMT18] for an equivalent theory of canonical Tutte polynomials of minor systems.…”

Hopf algebras for matroids over hyperfields

“…In fact, many combinatorial objects posses notions of "deletion" and "contraction" and hence one can associate Hopf algeras (or bialgebras in the disconnected case). In [10], C. Dupont, A. Fink, and L. Moci associate a universal Tutte character to such combinatorial objects specializing to Tutte polynomials in the case of matroids and graphs (among others) generalizing the work [14] of T. Krajewski Hyperfields were first introduced by M. Krasner in his work [15] on an approximation of a local field of positive characteristic by using local fields of characteristic zero. Krasner's motivation was to impose, for a given multiplicative subgroup G of a commutative ring A, "ring-like" structure on the set of equivalence classes A/G (G acts on A by left multiplication).…”

“…By appealing to the above results, we define Hopf algebras for matroids over hyperfields in §5. Finally, in §6, we explain how our current work can be thought in views of matroids over fuzzy rings as in Dress-Wenzel theory [9], matroids over partial hyperfields [6], and universal Tutte characters [10] as well as Tutte polynomials of Hopf algebras [14].…”

“…The term a minors system is used in [10] to define universal Tutte characters. The following is an easy consequence of §3.…”