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64th Séminaire Lotharingien de Combinatoire
March
28-31,
2010
Institut Camille
Jordan - Bâtiment Braconnier
Université Claude Bernard Lyon 1
43, Bd du 11 novembre 1918
69622 Villeurbanne - France
The
invited
speakers of this session will be:
Alain Lascoux
(CNRS -
Université Marne la Vallée, France)
Key
polynomials (Demazure characters) for type A,B,C,D
Demazure
characters (key polynomials) are linear bases of the ring of (Laurent)
polynomials in x1,..., xn. They are defined as all
possible images of dominant monomials under products of
divided
differences for type A,B,C,D. For example, for the type A, one
has
two families of operators πi and ∑i
(i=1,...,
n-1),
which commute with the
multiplication by symmetric functions in xi,
xi+1, and are
defined by the rules:
πi sends 1
---> 1
and xi+1 ---> 0,
∑i sends 1
---> 0
and xi+1 ---> -xi+1.
It
is remarkable that such simple tools suffice to generate interesting
polynomials, which contain as a subfamily the Schur (symplectic-Schur,
orthogonal-Schur) functions and verify similar properties: Cauchy
kernels, adjoint bases, Pieri formulas. In
type A, vexillary key polynomials and vexillary Schubert
polynomials coincide. Still for this type, one can give a description
of key polynomials in terms of Young tableaux satisfying flag
conditions.
The
talks relying on great part on the article of Amy Fu and al., Non
symmetric Cauchy kernels for the classical Groups, JCTA 116 (2009)
903--917.
Arun Ram
(University of Melbourne, Australia)
Lyndon words, MV
polytopes and the Littelmann path model
Lecture 1:
Quantum groups and
Lyndon words
This
lecture will be a survey of the foundational work of Leclerc (following
Rosso and Green) who introduced Lyndon words into the canonical basis
theory and predicted the existence of the recently defined
Khovanov-Lauda-Rouquier graded quiver algebras.
Lecture 2:
Graded quiver
algebras and their representations
This
talk will be a survey of the recently defined Khovanov-Lauda-Rouquier
graded Hecke algebras, the role of quantum groups and Lyndon words in
the combinatorial representation theory of these algebras.
Lecture
3: Indexings of canonical bases: Lyndon words, MV
polytopes and the path model
This
talk will provide a comparison of three natural indexings of canonical
bases in quantum groups: indexings by products of Lyndon
words,
indexing by MV polytopes, and the Littelmann path (alcove walk)
indexings. The path model indexings can be viewed as generalizations of
Young tableau indexings.
The
other speakers are the following:
Arvind Ayyer
(CEA,
Saclay) and Volker
Strehl -
(Universität Erlangen-Nürnberg)
Combinatorial properties of an asymmetric exclusion process
Matthieu Deneufchâtel
(LIPN, Université Paris 13)
Asymptotic of Seilberg-like integrals: the unitary case and the
binomial transform
Thomas Feierl
-(INRIA Paris-Rocquencourt)
Asymptotics for reflectable lattice walks in a Weylchamber of type B
Valentin Féray-
(Université Bordeaux 1)
Asymptotic of characters of symmetric groups and limit shape of Young
diagrams
Roberta Fulci -
(Università di Bologna)
Models and refined models for involutory reflection groups and
classical Weyl groups
Florent Hivert
-
(LITIS Université de Rouen)
The Sage-Combinat Project: Status and future
Masao Ishikawa
-
(Tottori University)
A partial proof of Okada's conjecture for double-tailed diamonds
Matthieu
Josuat-Vergès
- (Université Paris XI)
Genocchi numbers and alternative tableaux
Jang Soo Kim -
(University
of Paris 7)
Some results on permutation tableaux
Ricardo Mamede
-
(Universidade de Coimbra)
On the full interval of linear expansion of a skew Schur function
Luca Moci -
(Università
di Roma 3)
Toric arrangements and Tutte polynomials
Philippe Nadeau -
(University of Vienna)
Polynomials around the Razumov Stroganov conjecture
Anne
Schilling -
(University of California at Davis) and Nicolas Thiery -
(Université Paris Sud)
Sorting monoids on Coxeter groups
Christian Stump -
(LaCIM,
Université du Québec à Montréal)
A cyclic sieving phenomenon in Catalan combinatorics
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