1. CJM 2012 (vol 65 pp. 241)
||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
As a corollary, we obtain necessary conditions for a given subspecies
$\mathbf k$ of a Hopf monoid $\mathbf h$ to be a Hopf submonoid: the quotient of
any one of the generating series of $\mathbf h$ by the corresponding
generating series of $\mathbf k$ must have nonnegative coefficients. Other
corollaries include a necessary condition for a sequence of
nonnegative integers to be the
dimension sequence 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.
Keywords:Hopf monoids, species, graded Hopf algebras, Lagrange's theorem, generating series, PoincarÃ©-Birkhoff-Witt theorem, Hopf kernel, Lie kernel, primitive element, partition, composition, linear order, cyclic order, derangement
Categories:05A15, 05A20, 05E99, 16T05, 16T30, 18D10, 18D35