Lagrange's Theorem for Hopf Monoids in Species

Marcelo Aguiar, Aaron Lauve
2012 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
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
more » ... tive 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.
doi:10.4153/cjm-2011-098-9 fatcat:hbfxmrgkujdvffacs5ysztrqai