Strong forms of linearization for Hopf monoids in species

A vector species is a functor from the category of finite sets with bijections to vector spaces; informally, one can view this as a sequence of $S_n$-modules. A Hopf monoid (in the category of vector species) consists of a vector species with unit, counit, product, and coproduct morphisms satisfying several compatibility conditions, analogous to a graded Hopf algebra. We say that a Hopf monoid is strongly linearized if it has a "basis" preserved by its product and coproduct in a certain sense. We prove several equivalent characterizations of this property, and show that any strongly linearized Hopf monoid which is commutative and cocommutative possesses four bases which one can view as analogues of the classical bases of the algebra of symmetric functions. There are natural functors which turn Hopf monoids into graded Hopf algebras, and applying these functors to strongly linearized Hopf monoids produces several notable families of Hopf algebras. For example, in this way we give a simple unified construction of the Hopf algebras of superclass functions attached to the maximal unipotent subgroups of three families of classical Chevalley groups.Comment: 35 pages; v2: corrected some typos, fixed attribution for Theorem 5.4.4; v3: some corrections, slight revisions, added references; v4: updated references, numbering of results modified to conform with published version, final versio

Strong forms of self-duality for Hopf monoids in species

A vector species is a functor from the category of finite sets with bijections to vector spaces (over a fixed field); informally, one can view this as a sequence of $S_n$-modules. A Hopf monoid (in the category of vector species) consists of a vector species with unit, counit, product, and coproduct morphisms satisfying several compatibility conditions, analogous to a graded Hopf algebra. A vector species has a basis if and only if it is given by a sequence of $S_n$-modules which are permutation representations. We say that a Hopf monoid is freely self-dual if it is connected and finite-dimensional, and if it has a basis in which the structure constants of its product and coproduct coincide. Such Hopf monoids are self-dual in the usual sense, and we show that they are furthermore both commutative and cocommutative. We prove more specific classification theorems for freely self-dual Hopf monoids whose products (respectively, coproducts) are linearized in the sense that they preserve the basis; we call such Hopf monoids strongly self-dual (respectively, linearly self-dual). In particular, we show that every strongly self-dual Hopf monoid has a basis isomorphic to some species of block-labeled set partitions, on which the product acts as the disjoint union. In turn, every linearly self-dual Hopf monoid has a basis isomorphic to the species of maps to a fixed set, on which the coproduct acts as restriction. It follows that every linearly self-dual Hopf monoid is strongly self-dual. Our final results concern connected Hopf monoids which are finite-dimensional, commutative, and cocommutative. We prove that such a Hopf monoid has a basis in which its product and coproduct are both linearized if and only if it is strongly self-dual with respect to a basis equipped with a certain partial order, generalizing the refinement partial order on set partitions.Comment: 42 pages; v2: a few typographical errors corrected and references updated; v3: discussion in Sections 3.1 and 3.2 slightly revised, Theorem A corrected to include hypothesis about ambient field, final versio