SCIENTIFIC PROGRAMS AND ACTIVITIES

Ongoing seminars

Nantel Bergeron and Mike Zabrocki
Algebraic Combinatorics working seminar at Fields

With Cesar Ceballos arriving (soon?) it might be a good occasion to shift our problematic a little bit. I suggest to try finding a natural polytopes that enumerate the non-commutative LR coefficient. The immaculate tableau conditions and yamanouchi conditions are a set of inequalities. Does this cut a nice polytopes? Is there better polytopes to use (like Bereinstein-Zelevinsky-Knutson-Tao-Buch etc) i.e. an weakening of the Hives model or Honeycomb or puzzle (whichever is more appropriate). I remember reading long time ago a nice account of many of those model in one paper... but I don't remember the author (I thought Posnikov... but no). Anyway, we will start slowly as Cesar is not here yet. So it is a good occasion to ask lots of questions.

Nantel Bergeron and Mike Zabrocki
Algebraic Combinatorics working seminar at Fields

We will conclude LR rule and start discussing were to go next. Cesar Ceballos will arrive next week and it might be a good occasion to shift our problematic a little bit. For example it might be a good time to see if there are natural polytopes that enumerate the non-commutative LR coefficient. The immaculate tableau coditions and yamanouchi conditions are a set of inequalities. Does this cut a nice polytopes? Is there better polytopes to use (like Zelevinsky-Fomin-Knutson-Tao etc).

Oded Yacobi

Juana Sanchez-Ortega
Straightening relations on the immaculate basis

Nantel Bergeron followed by Juana Sanchez-Ortega
Operators on Immaculate bases and noncommutative Littlewood-Richardson Rule.

The ideas is to slowly work our way (this semester) to better understand some subalgebra of the endomorphism of NSym At this point, we now have a description (generators and relations) of the subspace spanned by right-left multiplications and the dual operations (perp). Now we which to study the subspace spanned by endomorphisms of "creation" type (those are particular series in the operators above). The creation operators define (Create) particular basis (Immaculate basis) of NSym. To understand the relation among them, it seams we need to have the analogue of Littlewood-Richardson rule for the Immaculate basis.

Nantel will recall some notions related to the immaculate basis and then work his way to the LR rule. [That may take more then one session]
Juana will discuss the relation of the creation operators, and try to use the LR-rule to give the answer a better form.

The plan is to list the known commutation relations between the multiplication operators in NSym and QSym and their duals.

Allen Knutson (Cornell University)
Combinatorial rules for branching to symmetric subgroups

Given a pair G>K of compact connected Lie groups, and a dominant G-weightlambda, it is easy to use character theory to say how the irrep V_lambda decomposes as a K-representation. If G = K x K, this is tensor product decomposition, for which we have an enumerative formula: the constituents can be counted as a number of Littelmann paths or MV polytopes.
I'll give a positive formula in the more general case that K is a symmetric subgroup of G, i.e., the (identity component of) the fixed-point set of an involution. The combinatorics is controlled by the poset of K-orbits on the flag manifold G/B, which reduces to the Bruhat order in the case G = K x K. I can prove this formula in the asymptotic (or, symplectic) situation replacing lambda by a large multiple, and nonasymptotically for certain pairs (G,K).

This is a joint meeting with the Geometric Representation Theory Seminar

Nantel Bergeron (York University)
An introduction to QSym and NSym I

The ideas is to slowly work our way (this semester) to better understand some subalgebra of the endomorphism of QSym in ways similar to the way the Heisenberg algebra is understood as a subalgebra of the endomorphism of Sym.