Lagrange's Theorem for Hopf Monoids in Species

Aguiar, Marcelo ; Lauve, Aaron

Canadian journal of mathematics, Tome 65 (2013) no. 2, pp. 241-265

Résumé

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 $\mathbf{k}$ 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.

Détail

DOI : 10.4153/CJM-2011-098-9

Mots-clés : 05A15, 05A20, 05E99, 16T05, 16T30, 18D10, 18D35, Hopf monoids, species, graded Hopf algebras, Lagrange's theorem, generating series, Poincaré–Birkhoff–Witt theorem, Hopf kernel, Lie kernel, primitive element, partition, composition, linear order, cyclic order, derangement