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Lagrange's Theorem for Hopf Monoids in Species

Published online by Cambridge University Press:  20 November 2018

Marcelo Aguiar  and
Aaron Lauve
Marcelo Aguiar
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA, e-mail: maguiar@math.tamu.edu
Aaron Lauve
Affiliation:
Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, 60660, USA, e-mail: lauve@math.luc.edu
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Abstract

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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

05A1505A2005E9916T0516T3018D1018D35Hopf monoidsspeciesgraded Hopf algebrasLagrange's theoremgenerating seriesPoincaré–Birkhoff–Witt theoremHopf kernelLie kernelprimitive elementpartitioncompositionlinear ordercyclic orderderangement
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 2 , 01 April 2013 , pp. 241 - 265
Copyright
Copyright © Canadian Mathematical Society 2013

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