Dendriform structures for restriction-deletion and restriction-contraction matroid Hopf algebras

We endow the set of isomorphic classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform coalgebra structures on matroids and introduce a monomial invariant which satisfy a convolution identity with respect to restriction and deletion.


Introduction
It is widely acknowledged that major recent progress in combinatorics stems from the construction of algebraic structures associated to combinatorial objects, and from the design of algebraic invariants for those objects. For over three decades, numerous Hopf algebras with distinguished bases indexed by families of permutations, words, posets, graphs, tableaux, or variants thereof, have been brought to light (see, for example, [6] and references within). The study of such combinatorial Hopf algebras (a class of free or cofree, connected, finitely graded bialgebras thoroughly characterized by Loday and Ronco [9]) has grown into an active research area; many connections with other mathematical domains and, perhaps more surprisingly, to theoretical physics (see, for example, [14,15] and references therein) have been uncovered and tightened.
Since Schmitt's pioneering work [13], matroids have also been the subject of algebraic structural investigations, though to a much lesser extent than other familiar combinatorial species. In the present paper, we aim to carry the study of Hopf algebras on matroids one step forward by providing an alternative coproduct on matroids and by exploring their dendriform structures.
Let us now outline the paper's contents. After a short reminder of the basic theory of matroids and a review of Schmitt's restriction-contraction Hopf algebra (section 2), we define a new coproduct, relying on two standard operations on matroids, namely restriction and deletion, and show that the set of isomorphic classes of matroids can be endowed with a new commutative and cocommutative Hopf algebra structure, different from the one introduced by Schmitt (section 3). We then prove that Schmitt's coproduct as well as ours can be adequately split into two pieces so as to give rise to two dendriform coalgebras (section 4). To the best of our knowledge, it is the first time that such an analysis has been carried out for matroids. Finally, we define a polynomial invariant of matroids which satisfy an identity which is the restriction-deletion analog of the more classical convolution identity satisfied by the Tutte polynomial for matroids (section 5). Although that polynomial turns out to be a monomial, its definition exemplifies the usefulness of the theory of Hopf algebra characters in our context.

Matroid theory reminder
A particular class of matroids is the graphic matroids (or cyclic matroids), for whom the ground set is the set of edges of the graph and for whom the collection of independent sets is given by the sets of edges which do not contain all the edges of a cycle of the graph.
Let E be an n−element set and let I be the collection of subsets of E with at most r elements, 0 ≤ r ≤ n. The pair (E, I) is a matroid -the uniform matroid U r,n . The smallest (with respect to the cardinal of the edge set) non-graphic matroid is U 2,4 .
The bases of a matroid are the maximal independent sets of the respective matroid. Note that bases have all the same cardinality. By relaxing this condition one then has delta-matroids [1].
Let Let M = (E, I) be a matroid. The rank r(A) of A ⊂ E is given by the following formula: Lemma 2.2 (Lemma 1.3.1 [11]) The rank function r of a matroid M on a set E satisfies the following condition: If X and Y are subsets of E, then Note that loops and coloops correspond, in graph theory, to bridges and self-loops, respectively (see Fig. 1a and 1b).

A restriction-contraction matroid Hopf algebra
In this subsection we recall the restriction-contraction matroid Hopf algebra introduced in [13] (see also [2] for details). Let us first give the following definition: If a matroid N is obtained from a matroid M by any combination of restrictions and contractions or deletions, we call the matroid N a minor of M. We write that a family of matroids is minor-closed if it is closed under formation of minors and direct sums. If M is a minor-closed family of matroids, we denote by M the set of isomorphic classes of matroids belonging to M. Direct sums induce a product on M (see [13] for details). We denote by k( M) the monoid algebra of M over some commutative ring k with unit.
One has In the rest of the paper, we follow [2] and, by a slight abuse of notation, we denote in the same way a matroid and its isomorphic class, since the distinction will be clear from the context (as it is already in Proposition 2.6). This is the same for the restriction-deletion matroid Hopf algebra that we will introduce in the following section.
The empty matroid (or U 0,0 ) is the unit of this Hopf algebra and is denoted by 1.
Example 2.7 (Example 2.4 of [2]) Let M = U k,n be a uniform matroid with rank k. Its coproduct is given by

A restriction-deletion matroid Hopf algebra
Let us define the following restriction-deletion map: Example 3.1 One has 1) If 2k ≤ n, 2) If n < 2k, One has: 1) Let X ′ be a subset of X which is a subset of the ground set E. One has 2) Let X and Y be subsets of E such that X and Y are disjoint, one has Proof. These identities follow directly from the definitions of restriction and deletion for matroids (see previous section). (5) is coassociative.

Proposition 3.3 The coproduct in
Proof. Let M = (E, I) be a matroid. Let us calculate the left hand side (LHS) and right hand side (RHS) of the coassociativity identity. One has and Equations (10) and (11) lead to the conclusion.
Let us explicitly check the coassociativity of ∆ (II) on U 2,4 .
Proposition 3.5 k( M) is a cocommutative coalgebra with coproduct ∆ (II) and counit ǫ given by Proof. The proof follows directly from the definition (14).
Lemma 3.6 (Proposition 4.2.23 [11]) Let M 1 and M 2 be two matroids. Let A 1 and A 2 be subset of E 1 and E 2 , respectively. One then has 1) 2) Proof. One can check these identities directly from the definitions of direct sum, restriction and deletion for matroids (see again the previous section).
Proposition 3.7 Let M 1 and M 2 be two matroids. One has Proof. Lemma 3.6 leads to: which further leads to: Thus, one gets Proposition 3.8 The triplet (k( M), ⊕, ∆ (II) ) is a commutative and cocommutative bialgebra.
Proof. The claim follows directly from Proposition 3.7 and the results above.
The main result of this section is: The triplet (k( M), ⊕, ∆ (II) ) is a commutative and cocommutative Hopf algebra. The antipode S of this Hopf algebra is given by Proof. The bialgebra is graded by the cardinal of the ground set of matroids. Moreover, M is connected, i. e. M 0 = k1. This leads to the conclusion.
Let us end this section with the following example: Example 3.10 One has:

Definition 4.1 (Definition 2 of [5])
A dendriform coalgebra is a family (C, ∆ ≺ , ∆ ≻ ) such that: 1. C is a k-vector space and one has: 2. For all a ∈ C, one has: If C is a coalgebra, one defines

The restriction-deletion case
Let us now define two maps on M + by: Proof. Let M be the matroid (E, I). Let us first prove identity (24). Its LHS writes: On the other hand, the RHS of identity (24) can be rewritten as follows: (32) From Lemma 2.2, one has: Setting X = B and X ∪ Y = A in equation (32) one now gets that identity (24) holds.
Let us now prove identity (25). Its LHS writes: The RHS of identity (25) writes: As above, we can now set X = B and X ∪ Y = A in the previous equation. One then concludes that identity (25) also holds. Finally, identity (26) holds because of the coassociativity of ∆ ≻ ) to be a codendriform bialgebra [5] write: where ∆ (II) In our case, one gets the identity ∆ (II) which is different of the identity (36) above.

The restriction-contraction case
As in the case of the restriction-deletion coproduct analyzed in the previous subsection, one can use the coproduct ∆ (I) to define a restriction-contraction dendriform coalgebra structure. One defines two maps on M + by: One has ∆ (I) One has Proof. The proof of Proposition 4.2 applies for this case as well.
This dendriform coalgebra is not a codendriform bialgebra for the same reasons as in the case of the dendriform restriction-deletion matroid coalgebra of the previous subsection.

A matroid polynomial
In this section we use an appropriate character of the restriction-deletion matroid Hopf algebra in order to define a certain matroid polynomial.  Note that the matroid of Example 5.2 is a graphic matroid, see Fig. 2. One has, as already noted above l(E) = 3 and c(E) = 1. Let M = (E, I) be a matroid. Let us define the following polynomial: Remark 5.5 Note that the definition above mimics the definition of the Tutte polynomial, where the role of the rank and of the nullity are played by the parameters c and l, respectively.
Example 5.6 One has: As in [3], we now define: and It is easy to check that these maps are infinitesimal characters of the restriction-deletion matroid Hopf algebra.
Following [3] again, we define the map: Using the definition of a Hopf algebra character one can directly check that the map (48) defined above is a character.
Let us now show the relation between the map α and the polynomial in (44).

Lemma 5.7 One has
Proof. The proof of Lemma 4.1 of [3] for the restriction-contraction matroid Hopf algebra applies for the restriction-deletion matroid Hopf algebra as well (where one takes again into consideration that the role of the rank and of the nullity are played by the parameters c and l).
Proof. The proof of Proposition 4.3 of [3] for the restriction-contraction matroid Hopf algebra applies for the restriction-deletion matroid Hopf algebra as well (where one takes again into consideration that the role of the rank and of the nullity are played by the parameters c and l). One further has: Corollary 5.9 Let M 1 and M 2 be two matroids. One has P M 1 ⊕M 2 (x, y) = P M 1 (x, y)P M 2 (x, y).
One has: Proof. The proof of Corollary 4.5 of [3] applies for the polynomial P .

Remark 5.11
The identity above is the analog of a convolution identity for the Tutte polynomial proved initially in [4] and [7]. The difference comes from replacing the matroid contraction, in the Tutte polynomial case, with the matroid deletion, in the last factor of the identity.
Note that P M (x, y) = P M/e (x, y) + P M \e (x, y) where e is neither a loop nor a coloop.
Let us now give the recursive relations satisfied by the polynomial P M (x, y).
Similarly, if e is not a loop, one then has P M (x, y) = xP M \e (x, y).
Corollary 5.13 One has P M (x, y) = x c(E) y l(E) .