In reverse-chronological order.
Date Speaker Title (click titles for abstract) 14 June, 2013 Mike Zabrocki combinatorial Hopf Algebras and sage I'm going to Sage Days 49 and giving a talk. I will give a short version of the presentation and I would like to have a discussion about what functionality would be good to take as a wish list.7 June, 2013 Franco Saliola The Spectrum of the random to random operator 31 May, 2013 Mike Zabrocki Murnaghan-Nakayama rules for immaculate and dual immaculate functions.(Part II) to work on on the dual Murnahgan-Nakayama rule on the immaculate basis. I spoke last week about the analogue of the Murnahgan-Nakayama rule and so after an introduction to the problem we will start working on a few examples and see if we can come up with a concrete formula.24 May, 2013 Mike Zabrocki Murnaghan-Nakayama rules for immaculate and dual immaculate functions. 17 May, 2013 Cesar Ceballos Hopf structure on subword complexes. 3 May, 2013 Juana Sanchez Ortega dual to multiplication by a fundamental on the immaculate basis Juana will present a rule for calculating dual to multiplication by a fundamental on the immaculate basis and we will apply this rule to calculating the immaculate Hall-Littlewood functions indexed by a partition.26 April, 2013 Carolina Benedetti COMBINATORIAL HOPF ALGEBRAS AND SUPERCHARACTERS dissertation colloquium1 March, 2013 Nantel Bergeron, Cesar Ceballos and Mike Zabrocki Various topic - Geometric non-commutative Littlewood-Richardson coefficient. [Saturation conjecture (Cesar Ceballos)] - Immaculate Hall-Littlewood [Mike Zabrocki] - Representation Theory of dual Immaculate [Nantel Bergeron]15 February, 2013 Nantel Bergeron and Mike Zabrocki Geometric non-commutative Littlewood-Richardson coefficient. We succeed into realizing the LR-coefficient as counting points inside a convex polytope. We will present this construction and continue our investigation of translating this construction to hives (Knutson-Tao analogue of the polytope) and BZ-triangle.8 February, 2013 Nantel Bergeron and Mike Zabrocki A Geometric interpretation of LR rule for immaculate (polytopes, Honeycomb, Hives) 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.1 February, 2013 Nantel Bergeron and Mike Zabrocki The LR rule for Immaculate again! (part II) 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)18 January, 2013 Nantel Bergeron and Mike Zabrocki The LR rule for Immaculate again!? and lets get back to noncommutative Heisenberg algebra! 14 December, 2012 Nantel Bergeron How many time will I try to prove the LR rule for Immaculate? 23 November, 2012 Juana Sanchez Ortega Creation Operators on the Immaculate basis Juana will revisit the defintions of the creation operators for the immaculate basis and derive formulas for the expansion of compositions of these elements when the index of the operator is 0. (Notes)
16 November, 2012 Nantel Bergeron A Noncommutative Littlewood-Richardson Rule Nantel will continue his talk from last time and show what is needed in order to prove the LR rule. The argument which shows that the LR rule is correct involves enumerating the terms of an algebraic expression with combinatoiral objects (immaculate tableaux) and showing that there is a sign reversing involution.
9 November, 2012 Nantel Bergeron Operators on Immaculate bases and a right Pieri 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.
2 November, 2012 Mike Zabrocki Commutation relations between NSym and QSym^* I will derive some of the commutation relations between multiplication by elements in NSym and dual to multiplication by elements in QSym. (Notes)26 October, 2012 Allen Knutson Combinatorial rules for branching to symmetric subgroups Given a pair G>K of compact connected Lie groups, and a dominant G-weight lambda, 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).12 October, 2012 Oded Jacobi Categorification of the Heisenberg algebra 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.5 October, 2012 Mike Zabrocki Operators on Immaculate bases of QSym and NSym I'll give an overview of how skewing and multiplication operators can be used to generate the endomorphism ring of Sym and what the corresponding statement is for NSym and QSym. Next I will show how these operators can be combined to define elements of End(NSym) which create bases.28 September, 2012 Nantel Bergeron An introduction to QSym and NSym II 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.21 September, 2012 Nantel Bergeron An introduction to QSym and NSym 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.14 September, 2012 Organizational meeting Discussion on open problems as possible topics for this seminar Oded Yacobi presented a summary of the Heisenberg algebra and proposed a similar construction for the construction of an analogous algebra related to the representation ring of the Hecke algebra at q=0 (NSym and/or QSym). Mike presented a conjecture about the evaluation of k-Schur functions at a root of unity.
About the seminar. Every year we pick a new topic to explore. Nantel, Mike and Oded have suggested exploring the 'categorification' of the Heisenberg-like algebra living in End(QSym) or End(NSym).