SCIENTIFIC PROGRAMS AND ACTIVITIES

April 25, 2015

Algebraic Combinatorics Seminar 2006-07 held at the Fields Institute

The purpose of this seminar is to cover exposition on topics of algebraic combinatorics which are of interest to the people attending, so please feel free to come and participate. For more informations, contact Christophe Hohlweg - chohlweg (at) fields.utoronto.ca

We also organize special sessions jointly with the Applied Algebra Seminar (York University)

Fields Working Sessions Tuesdays 12:00 p.m. - 1:30 p.m., unless otherwise indicated.

 3-Apr-2007 12:00 p.m. Franco Saliola, UQAM Finding the Quiver of the Descent Algebras This will be a survey outlining the current status of my project to study of the representation theory of the descent algebra of a finite Coxeter group. It will concentrate on the descent algebra of the symmetric group because the current results are better in this case. The approach utilizes what I call the "geometric approach" to the descent algebra: there is a semigroup algebra constructed from the reflection arrangement of the Coxeter group that contains the descent algebra. The quiver of this semigroup algebra will be constructed and it will be used to get information about the quiver of the descent algebra. 27-Mar-2007 12:00 p.m. Nantel Bergeron, York University Working Session: Finding knuth relations for a type B Robinson Schensted corespondence invoving Domino tableaux 27-Feb-2007 12:00 p.m. Nantel Bergeron, York University Generalization of the weak order on tableaux to Coxeter groups and related problems II 20-Feb-2007 12:00 p.m. Nantel Bergeron, York University Working Session: Generalization of the weak order on tableaux to Coxeter groups and related problems 13-Feb-2007 12:00 p.m. Special Session Srecko Brlek, UQAM Variations on Cantor's celebrated diagonal argument Given a square $n \times n$ tableau $T=\left(a_i^j\right)$ on a finite alphabet $A$, let $L$ be the set of its row-words. The permanent $\Perm(T)$ is the set of words $a_{\pi(1)}^1a_{\pi(2)}^2\cdots a_{\pi(n)}^n$, where $\pi$ runs through the set of permutations of $n$ elements. Cantorian tableaux are those for which $\Perm(T)\cap L=\emptyset.$ Let $s=s(n)$ be the cardinality of $A$. We show in particular that for large $n$, if $s(n) <(1-\epsilon) n/\log n$ then most of the tableaux are non-Cantorian, whereas if $s(n) >(1+\epsilon) n/\log n$ then most of the tableaux are Cantorian. We conclude our article by the study of infinite tableaux. Consider for example the infinite tableaux whose rows are the binary expansions of the real algebraic numbers in the unit interval. We show that the permanent of this tableau contains exactly the set of binary expansions of all the transcendental numbers in the unit interval. 12-Feb-2007 12:00 p.m. Special Session Amy Glen, CRM-ISM-LaCIM, UQAM Characterizations of finite and infinite episturmian words via lexicographic orderings In this talk, I will present some new results arising from collaborative work with Jacques Justin (France) and Giuseppe Pirillo (Italy). This work, which extends previous results on extremal properties of infinite Sturmian and episturmian words, is purely combinatorial in nature. Specifically, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a wide sense (episturmian and episkew infinite words). That is, we characterize the set of all infinite words whose factors are (finite) episturmian. Similarly, we characterize by lexicographic order all balanced infinite words over a 2-letter alphabet; in other words, all Sturmian and skew infinite words, the factors of which are (finite) Sturmian. 06-Feb-2007 12:00 p.m. Muge Taskin, York University and Fields Problems on the weak order on tableaux, proofs II 30-Jan-2007 12:00 p.m. Muge Taskin, York University and Fields Problems on the weak order on tableaux, proofs I 23-Jan-2007 12:00 p.m. Hugh Thomas, U. New Brunswick Schiffler's formula for A_n cluster variables and some generalizations We will present Schiffler's formula for expanding a cluster variable as a sum of Laurent monomials in the seed variables in the A_n case (as presented in math.RT/0611956). We will discuss a generalization in the context of Fomin-Shapiro-Thurston cluster algebras associated to surfaces, and a related result in which, for an acyclic seed, we give a formula to expand a cluster variables as a sum of standard monomials in the lower cluster algebra. The new results we will be discussing are joint with Ralf Schiffler. 16-Jan-2007 12:00 p.m. Mike Zabrocki Decomposition of a Gl_n irrep into S_n irreps, Part 3 12-Dec-2006 12:00 p.m. Alejandra Premat Part II of Crystal Bases 05-Dec-2006 12:00 p.m. Mike Zabrocki Decomposition of a Gl_n irrep into S_n irreps (data for n=6) (some conjectures and results) 28-Nov-2006 12:00 p.m. Muge Taskin, York University and Fields Solution to open problems 21-Nov-2006 12:00 p.m. Muge Taskin, York University and Fields Open problems related to some orders on standard Young tableaux part 3 14-Nov-2006 12:00 p.m. Mike Zabrocki An explanation of a problem of computing the restriction of a Gl_n module to S_n. Remark: it is true that F_{S_n}( f g ) = F_{S_n}(f) * F_{S_n}(g) (example) 07-Nov-2006 12:00 p.m. Christophe Hohlweg, Fields Institute Permutahedra and Generalized Associahedra, isometry problem: proof Part 2 31-Oct-2006 12:00 p.m. Christophe Hohlweg, Fields Institute Permutahedra and Generalized Associahedra, isometry problem: proof Part 1 24-Oct-2006 12:00 p.m. Muge Taskin, York University and Fields Open problems related to some orders on standard Young tableaux Part 2 17-Oct-2006 12:00 p.m. Muge Taskin, York University and Fields Open problems related to some orders on standard Young tableaux Part 1 10-Oct-2006 12:00 p.m. Christophe Hohlweg, Fields Institute Permutahedra and Generalized Associahedra, open problems Part 2 03-Oct-2006 11:00 a.m. Special Session Christian Kassel, CNRS-Université Louis Pasteur, Strasbourg A Hall algebra based on the projective line (Joint work with Pierre Baumann) One can construct a Hall algebra out of the vector bundles over a projective curve on a finite field. I'll describe this algebra in the case when the curve is a projective line. This algebra is related to the quantum affine algebra associated to the Lie algebra sl(2). In my talk I'll insist on the combinatorial aspects of the subject. 26-Sep-2006 12:00 p.m. Alejandra Premat, York University and Fields Introduction to Crystal basis 19-Sep-2006 12:00 p.m. Christophe Hohlweg, Fields Institute Permutahedra and Generalized Associahedra, open problems