550 research outputs found
On the Hadamard product of Hopf monoids
Combinatorial structures which compose and decompose give rise to Hopf
monoids in Joyal's category of species. The Hadamard product of two Hopf
monoids is another Hopf monoid. We prove two main results regarding freeness of
Hadamard products. The first one states that if one factor is connected and the
other is free as a monoid, their Hadamard product is free (and connected). The
second provides an explicit basis for the Hadamard product when both factors
are free.
The first main result is obtained by showing the existence of a one-parameter
deformation of the comonoid structure and appealing to a rigidity result of
Loday and Ronco which applies when the parameter is set to zero. To obtain the
second result, we introduce an operation on species which is intertwined by the
free monoid functor with the Hadamard product. As an application of the first
result, we deduce that the dimension sequence of a connected Hopf monoid
satisfies the following condition: except for the first, all coefficients of
the reciprocal of its generating function are nonpositive
Hopf monoids from class functions on unitriangular matrices
We build, from the collection of all groups of unitriangular matrices, Hopf
monoids in Joyal's category of species. Such structure is carried by the
collection of class function spaces on those groups, and also by the collection
of superclass function spaces, in the sense of Diaconis and Isaacs.
Superclasses of unitriangular matrices admit a simple description from which we
deduce a combinatorial model for the Hopf monoid of superclass functions, in
terms of the Hadamard product of the Hopf monoids of linear orders and of set
partitions. This implies a recent result relating the Hopf algebra of
superclass functions on unitriangular matrices to symmetric functions in
noncommuting variables. We determine the algebraic structure of the Hopf
monoid: it is a free monoid in species, with the canonical Hopf structure. As
an application, we derive certain estimates on the number of conjugacy classes
of unitriangular matrices.Comment: Final Version, 32 pages, accepted in "Algebra and Number Theory
On Operadic Actions on Spaces of Knots and 2-Links
In the present work, we realize the space of string 2-links as
a free algebra over a colored operad denoted (for "Swiss-Cheese
for links"). This result extends works of Burke and Koytcheff about the
quotient of by its center and is compatible with Budney's
freeness theorem for long knots. From an algebraic point of view, our main
result refines Blaire, Burke and Koytcheff's theorem on the monoid of isotopy
classes of string links. Topologically, it expresses the homotopy type of the
isotopy class of a string 2-link in terms of the homotopy types of the classes
of its prime factors.Comment: Comments are welcom
Classification of Reductive Monoid Spaces Over an Arbitrary Field
In this semi-expository paper we review the notion of a spherical space. In
particular we present some recent results of Wedhorn on the classification of
spherical spaces over arbitrary fields. As an application, we introduce and
classify reductive monoid spaces over an arbitrary field.Comment: This is the final versio
Varieties of Languages in a Category
Eilenberg's variety theorem, a centerpiece of algebraic automata theory,
establishes a bijective correspondence between varieties of languages and
pseudovarieties of monoids. In the present paper this result is generalized to
an abstract pair of algebraic categories: we introduce varieties of languages
in a category C, and prove that they correspond to pseudovarieties of monoids
in a closed monoidal category D, provided that C and D are dual on the level of
finite objects. By suitable choices of these categories our result uniformly
covers Eilenberg's theorem and three variants due to Pin, Polak and Reutenauer,
respectively, and yields new Eilenberg-type correspondences
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