Mathematics – Combinatorics
Scientific paper
2011-05-27
Mathematics
Combinatorics
20 pages
Scientific paper
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.
Aguiar Marcelo
Lauve Aaron
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