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BIRS
Workshop
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Combinatorial Hopf Algebras
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28 August — 2 September 2004
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Schedule
Abstracts
Speaker: Marcelo Aguiar (Texas A&M University)
Title:
The smash product of symmetric functions
Abstract:
We introduce a product on the space of symmetric
functions that interpolates between the classical "internal"
and "external" products (which are constructed in terms of tensor
products and induction of representations).
This product is best understood in terms of Hopf algebraic constructions
(the "smash product").
The smash product exists as well on the space of non-commutative
symmetric functions. At this level it interpolates between Solomon's
product (the "descent algebra") and the usual product of non-commutative
symmetric functions (as defined by Thibon et al).
The dual coproduct can be described in terms of the multiplicative
formal group law for alphabets.
The smash product also exists on larger spaces such as the algebra of
Solomon-Tits and the algebra of Malvenuto-Reutenauer.
This is joint work with Walter Ferrer and Walter Moreira.
Return.
Speaker: Pierre Cartier (IHES)
Title:
Hopf algebras, Lie algebras, algebraic groups and their use
in the renormalization in quantum field theory
Abstract:
This talk will present an alternative description
(obtained in collaboration with Marcus Berg) of the Hopf algebraic
methods used by Connes and Kreimer in their reformulation of quantum
field renormalization. Our approach uses extensively Vinberg
algebras (alias pre-Lie algebras) and brings together the
combinatorics of Feynman diagrams and algebro-differential geometry
(in the line of the original approach of Vinberg). There are obvious
connections with the lectures of Frabetti and Chapoton.
Return.
Speaker: Frédéric Chapoton (Lyon)
Title:
pre-Lie algebras, rooted trees and posets
Abstract:
We will recall the relation between free pre-Lie algebras and rooted
trees. This is linked to the Hopf algebra of rooted trees used by Connes
and Kreimer. Somehow this explains many structures of Lie algebras on
all kinds of graphs.
Then we will introduce the posets of pointed partitions and the posets
of hypertrees and explain what links exist or should exist with the
combinatorics of rooted trees.
Return.
Speaker: Richard Ehrenborg (Kentucky)
Title:
Geometric transforms of posets and associated coalgebras
Abstract:
We discuss geometric applications of the coproduct on the algebra
Z<a, b >. Our three examples are the
prism operation, the geometric lattic to zonotope omega
correspondence, and most recently, the Tchebyshev transform. The
similarity of the last two examples suggests a deeper connection which
we explore. We define two general chain maps of the first and second
kind. We prove that the chain map g of the second kind
is a Hopf algebra endomorphism on the quasisymmetric functions. To
understand the chain map of the first kind g, we must introduce an
extension of the quasisymmetric functions. On this algebra g is an
algebra map and also a comodule map on the extended quasisymmetric
functions.
Return.
Speaker: Alessandra Frabetti (Lyon)
Title:
Hopf algebras and renormalization in physics
Abstract:
In Quantum Field Theory, the Green functions are formal series of the
coupling constants, with coefficients computed as divergent integrals
associated to Feynman graphs. The meaningful Green functions can be found
by applying the renormalization group action, which is based on a
complicated combinatorial formula giving the renormalization of each
Feynman graph. Recently, the renormalization of Feynman graphs has been
very efficiently described by A. Connes and D. Kreimer in terms of a
suitable Hopf algebra. We will try to motivate the use of Hopf algebras
in the context of renormalization, recalling briefly the physical problem,
showing the three kinds of Hopf algebras developped in this context on
graphs and trees, and presenting some feed backs of the Hopf algebraic
method of renormalization in physics and in mathematics.
Return.
Speaker: Adriano Garsia (San Diego)
Title:
The Haglund Macdonald statistics and the open problems it creates
Abstract:
In 1990 Garsia-Haiman conjectured that a certain version
H\mu(x;q,t) of the Macdonald integral
forms gave the Frobenius characteristic of certain bigraded Sn modules. This
came to be known as the "n! conjecture". In various attempts at proving this conjecture
a variety of other conjectures were formulated in the decade that passed before
Haiman finally proved the n-factorial conjecture using algebraic-geometrical
methods. Nevertheless the basic combinatorial problems arisen since the
discovery of the Macdonald polynomials are still open. Very recently
Jim Haglund has given a combinatorial interpretation for the coefficients
in the monomial expansion of
H\mu(x;q,t).
This opens up a purely combinatorial
attack to the variety of conjectures surrounding these remarkable
polynomials and their Schur function expansions. In the brief time
of this talk it is impossible to cover all these developments but
a prominent selection will be presented.
Return.
Speaker: Robert Grossman (University of Illinois at Chicago)
Title:
Some Hopf Algebra Structures on Trees and their Applications
Abstract:
Let k be a field, R be a commutative k-algebra, and
Der(R) the Lie algebra of derivations of R. In this talk, we describe some Hopf
algebras defined on the vector space spanned by rooted trees. In
particular, we describe some properties of the the Hopf algebra H
formed from rooted trees labeled using derivations D in Der(R). We
also describe a construction which uses a connection on Der(R) to
define a H-module algebra structure on R. We also give some
applications of these ideas to symbolic computation and numerical
algorithms.
Return.
Speaker: Kurusch Ebrahimi-Fard (Bonn)
Title:
Rota-Baxter Algebras and Hopf Algebras
Abstract:
Rota-Baxter algebras, first occurred in the study of Glen Baxter
in probability, and were popularized by the work of Gian-Carlo Rota and
Cartier during the same period that Rota introduced Hopf algebras into
combinatorics. We will present connections between Rota-Baxter algebras
and Hopf algebras in the contexts of perturbative quantum field theory,
combinatorics and operads.
We first briefly show how a non-commutative generalization of
the Spitzer identity for Rota-Baxter algebras naturally gives
the algebraic Birkhoff factorization used by Connes and Kreimer
to describe renormalization in terms of Hopf algebras of Feynman graphs.
We then show that the free Rota-Baxter algebra on a set is a Hopf
algebra when the set is empty and, for a non-empty one, contains the Hopf
algebras of quasi-symmetric functions, shuffle algebras and quasi-shuffle
algebras. This latter connection allows us to use a theorem of Loday
relating free commutative dendriform algebra to shuffle algebra to deduce
that every commutative dendriform algebra is a subalgebra of a Rota-Baxter
algebra. We also characterize coherent ABQR operads which give Hopf algebras
thanks to another theorem of Loday.
Return.
Speaker: Michiel Hazewinkel (Centrum voor Wiskunde en Informatica)
Title:
Hopf algebras of endomorphisms of Hopf algebras
Abstract:
In the last decennia two generalizations of the Hopf algebra of
symmetric functions have appeared and shown themselves important, the
Hopf algebra of noncommutative symmetric functions NSymm and the
Hopf algebra of quasisymmetric functions QSymm. It has also become
clear that it is important to understand the noncommutative versions of such
important structures as Symm,s the Hopf algebra of symmetric functions.
Not least because the right noncommmutative versions are often more
beautiful than the commutaive ones (not all cluttered up with counting
coefficients). NSymm and QSymm are not truly the full
noncommutative
generalizations. One is maximally noncommutative but cocommutative, the
other is maximally non cocommutative but commutative. There is a
common, selfdual generalization, the Hopf algebra of permutations of
Malvenuto, Poirier, and Reutenauer (MPR). This one is, I feel, best
understood as a Hopf algebra of endomorphisms. In any case, this point of
view suggests vast generalizations leading to the Hopf algebras of
endomorphisms and word Hopf algebras with which this paper is
concerned. This point of view also sheds light on the somewhat mysterious
formulas of MPR and on the question where all the extra structure (such
as autoduality) comes from. The paper concludes with a few sections on the
structure of MPR and the question of algebra retractions of the natural
inclusion of Hopf algebras NSymm>--->MPR and coalgebra
sections of the dual natural projection of Hopf algebras
MPR--->>QSymm. Several
of these will be described explicitly.
Return.
Speaker: Florent Hivert and Jean-Christophe
Novelli
(Université de Marne-la-Valée)
Title:
Colored quasi-symmetric function and representation theory
Abstract:
We introduce analogues of the Hopf algebra of Free quasi-symmetric
functions with bases labelled by colored permutations. As an
application, we recover in a simple way the descent algebras
associated with wreath products of cyclic groups and the symmetric group
Sn and the corresponding
generalizations of quasi-symmetric functions, the Poirier
quasi-symmetric algebra and the dual Mantaci-Reutenauer algebra (also
considered by Poirier). We then show that this last algebra (and its
dual) can be interpreted as the Grothendieck rings of the tower of
some algebras, namely the specialization at q=0 of the Ariki-Koike
algebras in the Shoji's presentation.
Return.
Speaker: Ralf Holtkamp (Bochum)
Title:
Operads of primitive elements and their combinatorics
Abstract:
We study P-Hopf algebras with one coassociative cooperation over
different operads P. Important examples are the dendriform operad,
the operad Mag freely generated by a non-commutative non-associative
binary operation, and the operad of Stasheff polytopes. In order to
describe the operads of primitive elements we prove an analog of
the Poincare-Birkhoff-Witt theorem. We determine the generating
series for these operads and show that the dimension of PrimMag(n)
is related to the log-Catalan numbers. A similar theory may be
devloped for (unitary) infinitesimal P-Hopf algebra structures.
Return.
Speaker: Samuel Hsiao (Michigan)
Title:
Canonical characters on quasi-symmetric functions and
bivariate Catalan numbers
Abstract:
Every character on a graded connected Hopf algebra decomposes uniquely
as a product of an even character and an odd character (Aguiar, Bergeron, and
Sottile, Combinatorial Hopf algebra and generalized Dehn-Sommerville
equations, 2003, math.CO/0310016).
We obtain explicit formulas for the even and odd parts of the universal
character on the Hopf algebra of quasi-symmetric functions. They can be
described in terms of Legendre's beta function evaluated at half-integers, or
in terms of {\em bivariate Catalan numbers:}
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C(m,n) =
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(2m)!(2m)!
m!(m+n)! n!
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Properties of characters and of quasi-symmetric functions are then used to
derive several interesting identities among bivariate Catalan numbers and in
particular among Catalan numbers and central binomial coefficients.
Return.
Speaker: Jean-Louis Loday (CNRS, Strasbourg)
Title:
Generalized bialgebras
Abstract:
The Poincaré-Birkhoff-Witt theorem and the Milnor-Moore
theorem unravel the structure of connected cocommutative bialgebras.
First, we announce an analogue for connected (not necessarily
cocommutative) bialgebras. In both cases three types of algebras
(i.e. of operads) are involved: (Com, As, Lie) in the classical case
and (As, 2as, B\infty) in the non-cocommutative case. Second, we
examine several other similar cases of triples of operads (C,A,P)
having the same kind of properties (C for coalgebra, A for algebra,
P for primitive). Most of the recently discovered Hopf algebras on
trees (Grossman-Larson, Connes-Kreimer, Brouder-Frabetti, Loday-Ronco,
Holtkamp, Goncharov) fit into this framework.
Return.
Speaker: Nathan Reading (Michigan)
Title:
Lattice congruences and Hopf algebras
Abstract:
We describe a method of constructing sub Hopf algebras of the Malvenuto
Reutenauer Hopf algebra of permutations (MR). The starting point is a
family { Thn }, where each Thn is a lattice
congruence of the weak order on Sn. Under certain conditions, this
family specifies a sub Hopf algebra of MR. The Hopf algebras thus
obtained are indexed by order ideals in an infinite poset, and can be
described by a pattern avoidance condition on permutations. Among the
Hopf algebras arising from this construction are the Hopf algebras of
planar binary trees and of non-commutative symmetric functions.
Return.
Speaker: Victor Reiner (Minnesota)
Title:
The weak order on tableaux
Abstract:
This talk will discuss a partial order on the standard Young tableaux
with $n$ cells, analogous to the weak Bruhat order permutations, or to the
Tamari order on binary trees. It was first introduced by A. Melnikov in
connection with the geometry of orbital varieties. We will explain its
connection to the Poirier-Reutenauer Hopf algebra on tableaux, and discuss
some of its good and bad properties.
This is joint work with Muge Taskin.
Return.
Speaker: William Schmitt (George Washington University)
Title:
A cofree Hopf algebra of matroids
Abstract:
We introduce a
noncommutative binary operation on matroids, called the free product,
and discuss some of its properties. In particular, free product is
characterized by a certain universal property, is associative, and
respects matroid duality. We characterize matroids that are
irreducible with respect to free product and show that, up to
isomorphism, every matroid factors uniquely as a free product of such
matroids. We use these results to prove an inequality involving the
numbers of nonisomorphic matroids on n elements which was conjectured
by Welsh, and to show that the Hopf algebra of matroids with
restriction-contraction coproduct is cofree.
Return.
Speaker: Manfred Schocker (Oxford)
Title:
A decomposition of the Tits algebra of the symmetric group
Abstract:
Tits' notion of projection operator yields the structure of a
semigroup on the set of faces F of a Coxeter complex (or, more
generally, of an arbitrary hyperplane arrangement over a real vector
space). The natural action of the underlying Coxeter group W
preserves this semigroup structure and the ring of invariants of W
in the integral semigroup ring of F is isomorphic to the descent
algebra of W.
I shall analyse the module structure of the ``Tits algebra'' kF over
an arbitrary field k in the case where W is the symmetric group.
This includes a construction of primitive idempotents and projective
indecomposable modules as well as a description of the Cartan
invariants, of the quiver and of the module structure of the Loewy
layers. I shall also discuss the impact of these results on the
descent algebra.
The approach is combinatorial in nature (rather than geometric) and
uses a coproduct and a second product on the direct sum of the Tits
algebras of various symmetric groups, generalising well-known concepts
of the theory of descent algebras.
Return.
Speaker: John Stembridge (Michigan)
Title:
P-partitions and quasi-symmetric functions
Abstract:
This will be a survey talk in which we will (1) review
the combinatorial and algebraic properties of Schur S-functions
and Q-functions as motivation for (2) the theory of P-partitions
(both ordinary and enriched) and (3) their associated (Hopf) algebras
of quasi-symmmetric functions. If time permits, we will discuss our
original motivation for developing the theory of enriched P-partitions:
heap expansions for stable Schubert polynomials.
Return.
