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# Lagrange's Theorem for Hopf Monoids in Species

Published:2012-04-19
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
 Format: LaTeX MathJax PDF

## 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 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, Poincaré-Birkhoff-Witt theorem, Hopf kernel, Lie kernel, primitive element, partition, composition, linear order, cyclic order, derangement
 MSC Classifications: 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.)

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