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Algebraic Combinatorics
Org: Nantel Bergeron (York), Christophe Hohlweg (Fields Institute) and Michael Zabrocki (York)
[PDF]
- MARCELO AGUIAR, Texas A&M University
Hopf monoids in Species and associated Hopf algebras
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The recent literature has seen the emergence and profusion of certain
Hopf algebras of a combinatorial nature. We propose a general
framework for the construction and study of these Hopf algebras. We
study the tensor category of species and relate it to the tensor
category of graded vector spaces by means of bilax tensor functors.
Constructions of graded Hopf algebras from Hopf monoids in species are
derived. We use the geometry and combinatorics of the Coxeter complex
of type A to construct Hopf monoids. The corresponding Hopf algebras
include those of symmetric functions, quasisymmetric functions,
noncommutative symmetric functions, other Hopf algebras of prominence
in the recent literature, and new ones. The categorical approach
yields uniform deformations and higher-dimensional generalizations of
all these objects.
- FRANÇOIS BERGERON, UQAM, Dépt. Math, C.P. 8888, Succ. Centre-Ville,
Montréal, H3C 3P8
Harmonics for Steenrod Operators
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We present joint work with A. Garsia and N. Wallach concerning the
space of harmonic polynomials for deformations of Steenrod Operators.
On the side this gives rise to the study of a new variety of skewed
versions of coinvariant spaces for reflection groups.
- PHILIPPE CHOQUETTE, York University, 4700 Keele St., Toronto, Ontario, M3J 1P3
The ring of r-quasisymmetric polynomials
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We present some results about the ring of r-quasisymmetric
polynomials. We discuss the quotients of these rings and morphism
between them.
- IAN GOULDEN, University of Waterloo, Waterloo
Combinatorics and intersection number conjectures
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The purpose of this talk is to explain the combinatorial portion of a
long-standing collaboration with David Jackson and Ravi Vakil. This
is concerned with intersection numbers for moduli spaces and a
combinatorialization by Vakil, an algebraic geometer. In particular,
we describe Kazarian and Lando's recent proof of Witten's Conjecture,
and our recent proof of the lg-Conjecture, as well as our
recent proof of various infinite cases of Faber's conjecture. Our
combinatorial methods feature generating series for various classes of
labelled rooted trees.
- AARON LAUVE, UQAM, LaCIM, C.P. 8888, Succ. Centre-Ville, Montréal,
H3C 3P8
Noncommutative invariants and coinvariants of the symmetric
group
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The algebras NCSymn and Symn (n Î N+) are defined to be the Sn-invariants
inside Q áAn ñ (resp. Q[Xn]),
the polynomial functions on a noncommutative alphabet An (resp. commutative, Xn) of cardinality n. The abelianization (ai® xi) realizes Symn as a quotient of
NCSymn. Here, we view it as a subspace. Some surprising
identities on the ordinary generating function for the Bell numbers
appear as an immediate corollary. In case n=¥, we obtain new
information on the (Hopf) algebraic structure of NCSymn.
Time permitting, we outline similar results for Hivert's
r-QSymn algebras (r,n Î N+ È{¥})
and their noncommutative analogues.
Joint work with F. Bergeron.
- ROSA ORELLANA, Dartmouth
Hopf algebra of uniform block permutations
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In this talk I will present the Hopf algebra of uniform block
permutations and show that it is self-dual, free, and cofree. These
results are closely related to the fact that uniform block
permutations form a factorizable inverse monoid.
This Hopf algebra contains the Hopf algebra of permutations of
Malvenuto and Reutenauer and the Hopf algebra of symmetric functions
in non-commuting variables of Gebhard, Rosas, and Sagan.
- MUGE TASKIN, York University, 2025 TEL Building, North York, ON, M3J 1P3
The properties of four partial orders on standard Young
tableaux
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Standard Young tableaux have been well knowm with their connection
with the representation theory of symmetric group and special linear
algebra sln. In this talk we will focus on the following
four partial orders which are induced from this connection: weak, KL,
geometric and chain orders.
After recalling their definitions and some of their crucial
properties, we will discuss three main results about these orders.
The first one is related to the product in a Hopf algebra of tableaux
defined by Poirier and Reutenauer. The second one is about the
homotopy type of their proper parts. The last one addresses two of
these orders which can be defined on the skew tableaux having fixed
inner boundary, and similarly analyzes their homotopy type and
Möbius function.
The talk will further include some preliminary results about domino
tableaux, which are also related to the representation theory of
symplectic algebra sp2n and orthogonal algebra
so2n+1.
- HUGH THOMAS, Department of Mathematics and Statistics, University of New
Brunswick, Fredericton, NB
A combinatorial rule for (co)minuscule Schubert calculus
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I will discuss a root system uniform, concise combinatorial rule for
Schubert calculus in minuscule and cominuscule flag manifolds G/P.
(The latter are also known as compact Hermitian symmetric spaces.) We
connect this geometry to work of Proctor in poset combinatorics,
thereby generalizing Schutzenberger's jeu de taquin formulation of the
Littlewood-Richardson rule for computing intersection numbers of
Grassmannian Schubert varieties. I will explain the rule, give some
background, and, time permitting, give some idea of the proof,
including the notion, which we introduce, of cominuscule recursion,
which is a general technique which relating the structure constants
for different Lie types.
This talk presents joint work with Alex Yong, and is based on the
preprint (with the same title) math.AG/0608276.
- STEPHANIE VAN WILLIGENBURG, University of British Columbia
A combinatorial classification of skew Schur functions
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We present a single operation for constructing skew diagrams whose
corresponding skew Schur functions are equal. This combinatorial
operation naturally generalises and unifies all results of this type
to date. Moreover, our operation suggests a closely-related condition
that we conjecture is necessary and sufficient for skew diagrams to
yield equal skew Schur functions.
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