The Grothendieck ring of a Tower of algebras is equipped with a graded
Hopf algebra structure. We have many examples of this phenomena where
we represent interesting graded Hopf algebras. Here we provide a
representation theoretical interpretation of the peak algebra and its
graded dual as Grothendieck rings of the tower of Hecke-Clifford
algebras at *q*=0.

The parabolic Kazhdan-Lusztig and *R*-polynomials were introduced by
Deodhar in 1987, and are defined for any two elements of any quotient
of any Coxeter group. They include the classical Kazhdan-Lusztig and
*R*-polynomials, which correspond to elements of the groups
themselves, and have applications to representation theory and
geometry.

Hermitian symmetric pairs are quotients of Weyl groups with
particularly nice geometric properties. Combinatorially, these are
quotients that, as posets under Bruhat order, are distributive
lattices. It is therefore natural to wonder if the parabolic
Kazhdan-Lusztig and *R*-polynomials, which are usually very difficult
to compute, also have particularly nice properties in the Hermitian
symmetric case. The purpose of this talk is to show that this is
indeed the case.

More precisely, we show that for Hermitian symmetric pairs the
*R*-polynomials are always given by a nice product formula, and that
the Kazhdan-Lusztig ones are always a monomial. In the case of
classical Hermitian symmetric pairs we show how one can compute
combinatorially all the exponents appearing in a given polynomial from
the representation of the elements of these quotients as (possibly
signed) permutations and (possibly shifted) partitions. In particular,
for the Kazhdan-Lusztig ones, this involves a generalization of the
concept of Dyck partition (introduced in [F. Brenti, Pacific J. Math.,
**207**(2002), 257-286] for the computation of these polynomials in
type *A*) to shifted partitions. We also show that the polynomials are
combinatorial invariants. Namely that they depend only on the interval
determined by the defining elements under Bruhat order as an abstract
poset, and not on the elements themselves, or on the particular
Hermitian symmetric pair.

The Catalan numbers and their *q*-analogues involving Narayana numbers
have natural counterparts for an arbitrary finite Coxeter group.
These numbers come up in a variety of combinatorial contexts, some of
which will be surveyed in the talk.

A polynomial *P*(*x*) = *P*(*x*_{1},*x*_{2},...,*x*_{n}) is said to be *S*_{n}
*m*-Quasi-Invariant if (1-*s*_{ij}) *P*(*x*) is divisible by
(*x*_{i}-*x*_{j})^{2m+1} for all 1 ≤ i < *j* ≤ *n*. An analogous definition
can be made for any reflection group where the *x*_{i}-*x*_{j} are replaced
by the linear forms giving the equations of the reflecting
hyperplanes. The space *QI*_{m}(*G*) of *m*-Quasi-Invariants of a given
reflection group *G* has truly remarkable properties yielding
*m*-analogues of many of the constructs that occur in the relation
between the polynomial ring *Q*[*X*_{n}] and the ring of *G*-invariants.
One of the fundamental constructs in the Theory of
*m*-Quasi-Invariants is the so-called Baker-Akhiezer function Y_{m}(*x*,*y*) which may be viewed as the *m*-analogue of *e*^{(x,y)}. This
talk is a brief introduction to the theory and the problems arising in
the study of Y_{m}(*x*,*y*).

We give a combinatorial proof of Harer and Zagier's formula for the
disjoint cycle distribution of a long cycle multiplied by an
involution with no fixed points, in the symmetric group on a set of
even cardinality. The main result is a direct bijection for a set
*B*_{p,k}, the enumeration of which is equivalent to the
Harer-Zagier formula. The elements of *B*_{p,k} are of
the form (m,p), where m is a pairing on {1,...,2*p*},
p is a partition into *k* blocks of the same set, and a certain
relation holds between m and p. (The set partitions p
that can appear in *B*_{p,k} are called
"shift-symmetric", for reasons that are explained in the talk.) The
direct bijection for *B*_{p,k} identifies it with a set of
objects of the form (r,*t*), where r is a pairing on a
2(*p*-*k*+1)-subset of {1,...,2*p*} (a "partial pairing"), and *t*
is an ordered tree with *k* vertices. If we specialize to the extreme
case when *p*=*k*-1, then r is empty, and our bijection reduces to
a well-known tree bijection.

This is joint work with Alexandru Nica.

The Temperley-Lieb algebra TL_{n} is an instance of an algebra whose
generators may be represented by planar diagrams of great simplicity.
We show that an enrichment of TL_{n} may be used to provide a very
natural proof of non-singularity of the matrix of chromatic joins.
This matrix was introduced by Tutte in the course of his long
preoccupation with the algebraic approach proposed by Birkhoff-Lewis
in the 1940s to the Four Colour Problem, and he gave the first proof
of its non-singularity. TL_{n} can be regarded as a subalgebra of
the partition algebra, and there are a number of interesting
consequences of this view. The first is that there is an orthogonal
projection from the partition algebra to TL_{n} that has a very
natural interpretation in terms of the Birkhoff-Lewis approach. The
second is a nice connexion with the diagrammatics of topological
quantum field theory and with Schur-Weyl duality. The latter comments
relate to work in progress and the details and further implications
are yet to be completed, so I shall touch upon them only briefly.

This is joint work with Sabin Cautis.

We will present a noncommutative version of the classical theory of invariants and coinvariants of the symmetric group, as well as a noncommutative version of a theorem of Chevalley that says that the ring of polynomials can be written as the tensor product of its invariants times its coinvariants. A brief review of the clasical theory of invariants and coinvariants of the symmetric group will be included in this talk.

This is joint work with Bruce Sagan, Nantel Bergeron, Christophe Reutenauer, and Michael Zabrocki.

We define a symmetric function to be *Schur nonnegative* (SNN) if
it is equal to a nonnegative linear combination of Schur functions.
We define a polynomial *p*(*x*_{1,1},...,*x*_{n,n}) in *n*^{2} variables
to be *Schur nonnegative* if for every Jacobi-Trudi matrix *A* = (*a*_{i,j}), the symmetric function *p*(*a*_{1,1},...,*a*_{n,n}) is
SNN. Using the Kazhdan-Lusztig basis for **C**[*S*_{n}], we will
show that certain differences of products of matrix minors are SNN
polynomials. From these we will deduce that certain differences of
products of Schur functions are SNN symmetric functions.

It is well-known that the number of complete matchings (or partitions
into *n* blocks of size 2) of the set {1,2,...,2*n*} with no
crossings or with no nestings is the Catalan number *C*_{n}. We discuss
how the theory of oscillating tableaux, which originally arose in the
representation theory of the symplectic and orthogonal groups and the
Brauer algebra, can be used to obtain further results on crossings and
nestings of matchings. In particular, if *f*_{n}(*i*,*j*) is the number of
complete matchings on {1,2,...,2*n*} with at most *i* mutually
crossing edges and with at most *j* mutually nested edges, then
*f*_{n}(*i*,*j*) = *f*_{n}(*j*,*i*). Moreover, the Gessel-Zeilberger reflection
principle allows the determination of the generating function
*F*_{i}(*x*) =
∑_{n≥0} *f*_{n}(*i*,∞)
[(*x*^{2n})/((2*n*)!)]. Similar
results hold when matchings are replaced with set partitions, using a
variation of oscillating tableaux called *vacillating tableaux*.
This is joint work with William Y. C. Chen, Eva Y. P. Deng, Rosena
R. X. Du, and Catherine H. Yan.

For each root system, there is a Macdonald kernel-it is the
symmetric bilinear form relative to which the Macdonald polynomials
are orthogonal. It may also be viewed as a virtual character with
coefficients that are formal power series in *q* and *t*. Various
specializations of it are connected to classical work of Kostant and
Chevalley. For example, when *q*=0, the graded multiplicities of the
irreducible characters are (up to normalization) polynomials in *t*
with nonnegative coefficients. Aside from the type *A* case, the
combinatorics of these polynomials still remain rather mysterious.

In this talk, I will describe a general method for decomposing the
Macdonald kernel into irreducible characters, and some of the
consequences of this decomposition (*e.g.*, explicit formulas,
some proved, some conjectured). The most interesting result relates
the *q*,*t*-multiplicities in the Macdonald kernel with a bi-graded
generalization of the co-invariant algebra of the associated Weyl
group.

This generalizes work of A. Broer (1995), Y. Bazlov (2001), and J. Stembridge (Ph.D. thesis, 1985), and both proves and generalizes a conjecture of M. Reeder (1997).

In the Hopf algebra of noncommutative symmetric functions the set of ribbon Schur functions forms a basis, however, in the Hopf algebra of symmetric functions the commutative image of the ribbon Schur functions only forms a spanning set. This raises the question: what are the relations that arise amongst the commutative ribbon Schur functions? In particular, when are two equal?

In this talk we define an equivalence relation on integer compositions such that two commutative ribbon Schur functions, indexed by compositions, are equal if and only if their indices are equivalent in this sense. As an application we derive identities on certain Littlewood-Richardson coefficients.

The problem of enumerating classes of permutation which avoid certain patterns is not new to algebraic combinatorics but very few general techniques exist for attacking this problem. I will present a proof that permutations which avoid any given pattern and have a fixed number of descents are regular and show how this can be used to generalize the standard notion of pattern avoidance and containment. This will also give a procedure for determining the generating function for patterns which avoid a pattern and have a fixed number of descents.

This is joint work with Andrew Rechnitzer and Murray Elder.