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
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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.)
