The antipode of linearized Hopf monoids

Abstract

In this paper, a Hopf monoid is an algebraic structure built on objects in the category of Joyal’s vector species. There are two Fock functors, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒦</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , that map a Hopf monoid <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">H</mml:mi> </mml:math> to graded Hopf algebras <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒦</mml:mi> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , respectively. There is a natural Hopf monoid structure on linear orders <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">L</mml:mi> </mml:math> , and the two Fock functors are related by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒦</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>×</mml:mo> <mml:mi mathvariant="bold">L</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Unlike the functor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , the functor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒦</mml:mi> </mml:math> applied to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">H</mml:mi> </mml:math> may not preserve the antipode of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">H</mml:mi> </mml:math> . In view of the relation between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒦</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , one may consider instead of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">H</mml:mi> </mml:math> the larger Hopf monoid <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="bold">L</mml:mi> <mml:mo>×</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> </mml:math> and study the antipode of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="bold">L</mml:mi> <mml:mo>×</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> </mml:math> . One of the main results in this paper provides a cancellation free and multiplicity free formula for the antipode of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="bold">L</mml:mi> <mml:mo>×</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> </mml:math> . As a consequence, we obtain a new antipode formula for the Hopf algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mi>𝒦</mml:mi> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We explore the case when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">H</mml:mi> </mml:math> is commutative and cocommutative, and obtain new antipode formulas that, although not cancellation free, they can be used to obtain an antipode formula for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> in some cases. We also recover many well-known identities in the literature involving antipodes of certain Hopf algebras. In our study of commutative and cocommutative Hopf monoids, hypergraphs and acyclic orientations play a central role. We obtain polynomials analogous to the chromatic polynomial of a graph, and also identities parallel to Stanley’s ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> )-color theorem. An important consequence of our notion of acyclic orientation of hypergraphs is a geometric interpretation for the antipode formula for hypergraphs. This interpretation, which differs from the recent work of Aguiar and Ardila as the Hopf structures involved are different, appears in subsequent work by the authors.