47 research outputs found
Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n
The aim of this work is to study the quotient ring R_n of the ring
Q[x_1,...,x_n] over the ideal J_n generated by non-constant homogeneous
quasi-symmetric functions. We prove here that the dimension of R_n is given by
C_n, the n-th Catalan number. This is also the dimension of the space SH_n of
super-covariant polynomials, that is defined as the orthogonal complement of
J_n with respect to a given scalar product. We construct a basis for R_n whose
elements are naturally indexed by Dyck paths. This allows us to understand the
Hilbert series of SH_n in terms of number of Dyck paths with a given number of
factors.Comment: LaTeX, 3 figures, 12 page
Coloured peak algebras and Hopf algebras
For a finite abelian group, we study the properties of general
equivalence relations on G_n=G^n\rtimes \SG_n, the wreath product of with
the symmetric group \SG_n, also known as the -coloured symmetric group. We
show that under certain conditions, some equivalence relations give rise to
subalgebras of \k G_n as well as graded connected Hopf subalgebras of
\bigoplus_{n\ge o} \k G_n. In particular we construct a -coloured peak
subalgebra of the Mantaci-Reutenauer algebra (or -coloured descent algebra).
We show that the direct sum of the -coloured peak algebras is a Hopf
algebra. We also have similar results for a -colouring of the Loday-Ronco
Hopf algebras of planar binary trees. For many of the equivalence relations
under study, we obtain a functor from the category of finite abelian groups to
the category of graded connected Hopf algebras. We end our investigation by
describing a Hopf endomorphism of the -coloured descent Hopf algebra whose
image is the -coloured peak Hopf algebra. We outline a theory of
combinatorial -coloured Hopf algebra for which the -coloured
quasi-symmetric Hopf algebra and the graded dual to the -coloured peak Hopf
algebra are central objects.Comment: 26 pages latex2
The Algebra of Binary Search Trees
We introduce a monoid structure on the set of binary search trees, by a
process very similar to the construction of the plactic monoid, the
Robinson-Schensted insertion being replaced by the binary search tree
insertion. This leads to a new construction of the algebra of Planar Binary
Trees of Loday-Ronco, defining it in the same way as Non-Commutative Symmetric
Functions and Free Symmetric Functions. We briefly explain how the main known
properties of the Loday-Ronco algebra can be described and proved with this
combinatorial point of view, and then discuss it from a representation
theoretical point of view, which in turns leads to new combinatorial properties
of binary trees.Comment: 49 page
Geometric combinatorial algebras: cyclohedron and simplex
In this paper we report on results of our investigation into the algebraic
structure supported by the combinatorial geometry of the cyclohedron. Our new
graded algebra structures lie between two well known Hopf algebras: the
Malvenuto-Reutenauer algebra of permutations and the Loday-Ronco algebra of
binary trees. Connecting algebra maps arise from a new generalization of the
Tonks projection from the permutohedron to the associahedron, which we discover
via the viewpoint of the graph associahedra of Carr and Devadoss. At the same
time that viewpoint allows exciting geometrical insights into the
multiplicative structure of the algebras involved. Extending the Tonks
projection also reveals a new graded algebra structure on the simplices.
Finally this latter is extended to a new graded Hopf algebra (one-sided) with
basis all the faces of the simplices.Comment: 23 figures, new expanded section about Hopf algebra of simplices,
with journal correction
Lattice congruences of the weak order
We study the congruence lattice of the poset of regions of a hyperplane
arrangement, with particular emphasis on the weak order on a finite Coxeter
group. Our starting point is a theorem from a previous paper which gives a
geometric description of the poset of join-irreducibles of the congruence
lattice of the poset of regions in terms of certain polyhedral decompositions
of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let
\eta_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show
that the fibers of \eta_K constitute the smallest lattice congruence with
1\equiv s for every s\in(S-K). We give an algorithm for determining the
congruence lattice of the weak order for any finite Coxeter group and for a
finite Coxeter group of type A or B we define a directed graph on subsets or
signed subsets such that the transitive closure of the directed graph is the
poset of join-irreducibles of the congruence lattice of the weak order.Comment: 26 pages, 4 figure
Combinatorial Hopf algebras and Towers of Algebras
Bergeron and Li have introduced a set of axioms which guarantee that the
Grothendieck groups of a tower of algebras can be
endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap,
and independently Lam and Shimozono constructed dual graded graphs from
primitive elements in Hopf algebras. In this paper we apply the composition of
these constructions to towers of algebras. We show that if a tower
gives rise to graded dual Hopf algebras then we must
have where .Comment: 7 page
Combinatorial Markov chains on linear extensions
We consider generalizations of Schuetzenberger's promotion operator on the
set L of linear extensions of a finite poset of size n. This gives rise to a
strongly connected graph on L. By assigning weights to the edges of the graph
in two different ways, we study two Markov chains, both of which are
irreducible. The stationary state of one gives rise to the uniform
distribution, whereas the weights of the stationary state of the other has a
nice product formula. This generalizes results by Hendricks on the Tsetlin
library, which corresponds to the case when the poset is the anti-chain and
hence L=S_n is the full symmetric group. We also provide explicit eigenvalues
of the transition matrix in general when the poset is a rooted forest. This is
shown by proving that the associated monoid is R-trivial and then using
Steinberg's extension of Brown's theory for Markov chains on left regular bands
to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in
terms of discrete time Markov chain
Associahedra via spines
An associahedron is a polytope whose vertices correspond to triangulations of
a convex polygon and whose edges correspond to flips between them. Using
labeled polygons, C. Hohlweg and C. Lange constructed various realizations of
the associahedron with relevant properties related to the symmetric group and
the classical permutahedron. We introduce the spine of a triangulation as its
dual tree together with a labeling and an orientation. This notion extends the
classical understanding of the associahedron via binary trees, introduces a new
perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's
original approach, and sheds light upon the combinatorial and geometric
properties of the resulting realizations of the associahedron. It also leads to
noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction
Duality between quasi-symmetric functions and the Solomon descent algebra
AbstractThe ring QSym of quasi-symmetric functions is naturally the dual of the Solomon descent algebra. The product and the two coproducts of the first (extending those of the symmetric functions) correspond to a coproduct and two products of the second, which are defined by restriction from the symmetric group algebra. A consequence is that QSym is a free commutative algebra