Abstract view
Lagrange's Theorem for Hopf Monoids in Species


Published:20120419
Printed: Apr 2013
Marcelo Aguiar,
Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA
Aaron Lauve,
Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, 60660, USA
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
$\mathbf k$ 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 settheoretic 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, PoincarĂ©BirkhoffWitt theorem, Hopf kernel, Lie kernel, primitive element, partition, composition, linear order, cyclic order, derangement
Hopf monoids, species, graded Hopf algebras, Lagrange's theorem, generating series, PoincarĂ©BirkhoffWitt theorem, Hopf kernel, Lie kernel, primitive element, partition, composition, linear order, cyclic order, derangement

MSC Classifications: 
05A15, 05A20, 05E99, 16T05, 16T30, 18D10, 18D35 show english descriptions
Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] Combinatorial inequalities None of the above, but in this section Hopf algebras and their applications [See also 16S40, 57T05] Connections with combinatorics Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] Structured objects in a category (group objects, etc.)
05A15  Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A20  Combinatorial inequalities 05E99  None of the above, but in this section 16T05  Hopf algebras and their applications [See also 16S40, 57T05] 16T30  Connections with combinatorics 18D10  Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 18D35  Structured objects in a category (group objects, etc.)
