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LAGRANGE'S THEOREM FOR HOPF MONOIDS IN SPECIES MARCELO AGUIAR AND AARON LAUVE
 

Summary: LAGRANGE'S THEOREM FOR HOPF MONOIDS IN SPECIES
MARCELO AGUIAR AND AARON LAUVE
Abstract. Following Radford's proof of Lagrange's theorem for pointed Hopf alge-
bras, 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 k of a
Hopf monoid h to be a Hopf submonoid: the quotient of any one of the generating
series of h by the corresponding generating series of k must have nonnegative coeffi-
cients. Other corollaries include a necessary condition for a sequence of nonnegative
integers to be the sequence of dimensions of a Hopf monoid in the form of certain
polynomial inequalities, and of a set-theoretic Hopf monoid in the form of certain lin-
ear inequalities. The latter express that the binomial transform of the sequence must
be nonnegative.
Introduction
Lagrange's theorem states that for any subgroup K of a group H, H = KQ as (left)
K-sets, where Q = H/K. In particular, if H is finite, then |K| divides |H|. Passing to
group algebras over a field k, we have that kH = kK kQ as (left) kK-modules, or
that kH is free as a kK-module. Kaplansky [7] conjectured that the same statement
holds for Hopf algebras--group algebras being principal examples. It turns out that the
result does not not in general, as shown by Oberst and Schneider [16, Proposition 10]
and [14, Example 3.5.2]. On the other hand, the result does hold for certain large

  

Source: Aguiar, Marcelo - Department of Mathematics, Texas A&M University

 

Collections: Mathematics