Lagrange's Theorem for Hopf Monoids in Species

TL;DR

This paper extends Lagrange's theorem to Hopf monoids in species, providing algebraic conditions for subspecies to be Hopf submonoids and characterizing dimension sequences through polynomial and linear inequalities.

Contribution

It proves Lagrange's theorem for Hopf monoids in species and derives necessary conditions for subspecies to be Hopf submonoids, including inequalities on generating series and dimension sequences.

Findings

Necessary conditions for subspecies to be Hopf submonoids involving generating series quotients.

Dimension sequences of Hopf monoids must satisfy certain polynomial inequalities.

Set-theoretic Hopf monoids have dimension sequences with nonnegative binomial transforms.

Abstract

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 coefficients. Other corollaries include a necessary condition for a sequence of nonnegative integers to be the sequence of dimensions of a Hopf monoid in the form of certain polynomial inequalities, and of a set-theoretic Hopf monoid in the form of certain linear inequalities. The latter express that the binomial transform of the sequence must be nonnegative.