182 research outputs found
Decomposition spaces in combinatorics
A decomposition space (also called unital 2-Segal space) is a simplicial object satisfying an exactness condition weaker than the Segal condition: just as the Segal condition expresses (up to homotopy) composition, the new condition expresses decomposition. It is a general framework for incidence (co)algebras. In the present contribution, after establishing a formula for the section coefficients, we survey a large supply of examples, emphasising the notion's firm roots in classical combinatorics. The first batch of examples, similar to binomial posets, serves to illustrate two key points: (1) the incidence algebra in question is realised directly from a decomposition space, without a reduction step, and reductions are often given by CULF functors; (2) at the objective level, the convolution algebra is a monoidal structure of species. Specifically, we encounter the usual Cauchy product of species, the shuffle product of L-species, the Dirichlet product of arithmetic species, the Joyal-Street external product of q-species and the Morrison `Cauchy' product of q-species, and in each case a power series representation results from taking cardinality. The external product of q-species exemplifies the fact that Waldhausen's S-construction on an abelian category is a decomposition space, yielding Hall algebras. The next class of examples includes Schmitt's chromatic Hopf algebra, the Fa\`a di Bruno bialgebra, the Butcher-Connes-Kreimer Hopf algebra of trees and several variations from operad theory. Similar structures on posets and directed graphs exemplify a general construction of decomposition spaces from directed restriction species. We finish by computing the M\Preprin
Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras
We study the representation theory of three towers of algebras which are
related to the symmetric groups and their Hecke algebras. The first one is
constructed as the algebras generated simultaneously by the elementary
transpositions and the elementary sorting operators acting on permutations. The
two others are the monoid algebras of nondecreasing functions and nondecreasing
parking functions. For these three towers, we describe the structure of simple
and indecomposable projective modules, together with the Cartan map. The
Grothendieck algebras and coalgebras given respectively by the induction
product and the restriction coproduct are also given explicitly. This yields
some new interpretations of the classical bases of quasi-symmetric and
noncommutative symmetric functions as well as some new bases.Comment: 12 pages. Presented at FPSAC'06 San-Diego, June 2006 (minor
explanation improvements w.r.t. the previous version
Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
A combinatorial Hopf algebra is a graded connected Hopf algebra over a field
equipped with a character (multiplicative linear functional) . We show that the terminal object in the category of combinatorial Hopf
algebras is the algebra of quasi-symmetric functions; this explains the
ubiquity of quasi-symmetric functions as generating functions in combinatorics.
We illustrate this with several examples. We prove that every character
decomposes uniquely as a product of an even character and an odd character.
Correspondingly, every combinatorial Hopf algebra possesses two
canonical Hopf subalgebras on which the character is even
(respectively, odd). The odd subalgebra is defined by certain canonical
relations which we call the generalized Dehn-Sommerville relations. We show
that, for , the generalized Dehn-Sommerville relations are the
Bayer-Billera relations and the odd subalgebra is the peak Hopf algebra of
Stembridge. We prove that is the product (in the categorical sense) of
its even and odd Hopf subalgebras. We also calculate the odd subalgebras of
various related combinatorial Hopf algebras: the Malvenuto-Reutenauer Hopf
algebra of permutations, the Loday-Ronco Hopf algebra of planar binary trees,
the Hopf algebras of symmetric functions and of non-commutative symmetric
functions.Comment: 34 page
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