Hopf monoids, permutohedral cones, and generalized retarded functions

Supercharacters and Hopf Monoids

Before the workshop, we suggested that the following reading list [1, 2, 3, 4, 5, 6] would be a good refreshment for all of us. At the start of the workshop, we identified the following problems that were worth considering given the recent progress in the area. 1. Higman’s conjecture. 2. Compute the antipode of the monoid scf(U) in the supercharacter basis. 3. Explicitly describe the multiplica...

Hopf Monoids in Varieties

We show that entropic varieties provide the canonical setting for a generaliziation of Hopf algebra theory. In particular we show that all naturally occurring functors in this context have the expected adjoints, when generalized to this level. In particular, universal measuring comonoids exist over every entropic variety, as do generalized group algebras, and these carry a canonical Hopf struct...

Lagrange’s Theorem for Hopf Monoids in Species

Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras, we prove Lagrange’s theorem for Hopf monoids in the category of connected species. As a corollary, we obtain necessary conditions for a given subspecies k of a Hopf monoid h to be a Hopf submonoid: the quotient of any one of the generating series of h by the corresponding generating series of k must have nonnegative coef...

Monoids and L-functions

Hopf monoids from class functionson unitriangular matrices

We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal’s category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatori...