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Algebraic Combinatorics / Combinatoire algébrique
Org: François Bergeron, Riccardo Biagioli, Peter McNamara and/et Christophe Reutenauer (UQAM)
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- NANTEL BERGERON, York University
Hopf algebra and representation: the Hecke-Clifford case
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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.
- FRANCESCO BRENTI, Università di Roma II, Via della Ricerca Scientifica 1,
00133 Roma, Italy
Combinatorics of parabolic Kazhdan-Lusztig and
R-polynomials for Hermitian symmetric pairs
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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.
- SERGEY FOMIN, University of Michigan, Ann Arbor, MI 48109, USA
Catalan combinatorics of arbitrary type
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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.
- IAN GOULDEN, University of Waterloo, Waterloo, Ontario N2L 3G1
A direct bijection for the Harer-Zagier formula
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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
Bp,k, the enumeration of which is equivalent to the
Harer-Zagier formula. The elements of Bp,k are of
the form (m,p), where m is a pairing on {1,...,2p},
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 Bp,k are called
"shift-symmetric", for reasons that are explained in the talk.) The
direct bijection for Bp,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,...,2p} (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.
- FREDERICO INCITTI, KTH, Stockholm
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- DAVID JACKSON, University of Waterloo, Waterloo, Ontario
Combinatorial aspects of the Temperley-Lieb algebra
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The Temperley-Lieb algebra TLn is an instance of an algebra whose
generators may be represented by planar diagrams of great simplicity.
We show that an enrichment of TLn 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. TLn 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 TLn 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.
- MERCEDES ROSAS, York University
Noncommutative invariants and coinvariants of the symmetric group
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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.
- MARK SKANDERA, Dartmouth College
Schur nonnegative polynomials
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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(x1,1,...,xn,n) in n2 variables
to be Schur nonnegative if for every Jacobi-Trudi matrix A = (ai,j), the symmetric function p(a1,1,...,an,n) is
SNN. Using the Kazhdan-Lusztig basis for C[Sn], 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.
- RICHARD STANLEY, M.I.T., Cambridge, MA 02139, USA
Crossings, nestings, oscillating tableaux, and vacillating
tableaux
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It is well-known that the number of complete matchings (or partitions
into n blocks of size 2) of the set {1,2,...,2n} with no
crossings or with no nestings is the Catalan number Cn. 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 fn(i,j) is the number of
complete matchings on {1,2,...,2n} with at most i mutually
crossing edges and with at most j mutually nested edges, then
fn(i,j) = fn(j,i). Moreover, the Gessel-Zeilberger reflection
principle allows the determination of the generating function
Fi(x) = ån ³ 0 fn(i,¥) [(x2n)/((2n)!)]. 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.
- JOHN STEMBRIDGE, University of Michigan, Ann Arbor MI 48109-1109, USA
Graded multiplicities in the Macdonald kernel and a bi-graded
co-invariant algebra
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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).
- STEPHANIE VAN WILLIGENBURG, University of British Columbia
Decomposable compositions and equality of ribbon Schur
functions
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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.
- MIKE ZABROCKI, University of York
Pattern avoidance and regular languages
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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.
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