Abstract
We endow the set of isomorphism 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.
Highlights
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
The study of such combinatorial Hopf algebras has grown into an active research area; many connections with other mathematical domains and, perhaps more surprisingly, to theoretical physics have been uncovered and tightened
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
Summary
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. 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. That polynomial turns out to be a monomial, its definition exemplifies the usefulness of the theory of Hopf algebra characters in our context
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