In reverse-chronological order.
Date Speaker Title (click titles for abstract) June 26, 2012 Mike Zabrocki product and coproduct structure of k-Schur functions (with a t) June 12, 2012 Tom Denton progress and reflections May 25, 2012 Brant Jones Kazhdan-Lusztig theory May 18, 2012 Mike Zabrocki, Tom Denton Sage packages: Symmetric Functions, Quasi-symmetric Functions, Non-commutative symmetric functions, affine permuatations April 20, 2012 Samuele Giraudo Construction of combinatorial operads After quick recalls about operad theory, I present a construction which allows to obtain operads from monoids. I show that this construction enables to obtain new operads, for example on integer compositions and directed animals, and provides alternative constructions of the diassociative and triassociative operads of Loday.April 13, 2012 Nantel Bergeron k-Littlewood-Richardson rule, involving multiplying k-Schur functions indexed by a rectangle minus two boxes April 3, 2012 Nantel Bergeron k-Littlewood-Richardson rule, involving multiplying k-Schur functions indexed by a rectangle minus two boxes March 30, 2012 Nantel Bergeron k-Littlewood-Richardson rule, involving multiplying k-Schur functions indexed by a rectangle minus two boxes March 23, 2012 Tom Denton k-Castles and Canonical Factorizations in the Affine Symmetric Group We describe a new canonical factorization of affine permutations into products of cyclically decreasing elements. To this factorization, we associate a new combinatorial object, called a k-Castle, from which one may recover the factorization, as well as length and descent set of the permutation. These factorizations also have direct application to the study of k-Schur functions in non-commuting variables.March 16, 2012 Carolina Benedetti special case of the Littlewood Richardson rule for k-Schur functions March 15, 2012 Gregg Musiker Bases for cluster algebras from surfaces In previous work with Ralf Schiffler and Lauren Williams, we have provided combinatorial formulas, involving snake graphs, for Laurent expansions for cluster variables coming from a bordered surface. In this talk, I will discuss extensions of this work involving band graph formulas for closed loops, and how these formulas allow us to construct vector space bases for such algebras. This sequel is also joint work with Schiffler and Williams.March 9, 2012 Luis Serrano Pieri operators and dual graded graphs We present a pair of Fomin dual graded graphs for the elements of the nilCoxeter algebra, developed by Lam, Lapointe, Morse, and Shimozono. In this graph, we define down (Pieri) operators, and use them to prove several properties of strong Schur functions.March 7, 2012 Chris Berg Quasisymmetric and Noncommutative Affine Schur functions We define a new pair of dual Hopf algebras NSym_k and QSym^k. In analogy with the theory of k-Schur functions and Quasisymmetric Schur functions, we introduce Schur like bases NS^k and QS^k for NSym_k and QSym^k respectively. In particular, the Noncommutative Affine Schur functions NS^k lift the k-Schur functions and the Quasisymmetric Affine Schur functions QS^k decompose the dual k-Schur functions. This is an ongoing joint project with Franco Saliola and Luis Serrano.March 2, 2012 Holly Heglin Type C k-Schur functions February 17, 2012 Holly Heglin Type C k-Schur functions February 10, 2012 Mike Zabrocki k-Schur LR rule February 3, 2012 Tom Denton affine permutation factorization January 27, 2012 Tom Denton affine permutation factorization 4 November, 2011 Mike Zabrocki k-Shapes and Schur Positivity II Prof. Mike Zabrocki continues a presentation on the k-Shape poset, which can be used to give an expansion of the k-Schur functions in terms of the k+1 Schur functions.28 October, 2011 Mike Zabrocki k-Shapes and Schur Positivity I Prof. Mike Zabrocki began a presentation on the k-Shape poset, which can be used to give an expansion of the k-Schur functions in the k+1-Schur functions.21 October, 2011 Nantel Bergeron Dual Equivalence Graphs and Schur Positivity IV Prof. Bergeron completes the introduction to dual equivalence graphs.18 October, 2011 Andrew Rechnitzer Counting in Thompson's group F - enumeration and experimentation Richard Thompson's group F is a widely studied group which has provided examples of and counter-examples to a variety of conjectures in group theory. It is also an extremely combinatorially appealing object which has a beautiful description in terms of binary trees.
In this talk I will describe two important enumerative problems associated with F. The first is the problem of computing the growth-series of F - the number of elements with geodesic length n. I will describe the polynomial time algorithm that "solves" the problem and a couple of associated conjectures. The second problem is the cogrowth series - the number of words of length n equivalent to the identity. This second problem important because of its connections to the amenability of F and I will describe some of our recent experimental explorations of this problem using techniques from enumeration and statistical mechanics.
7 October, 2011 Nantel Bergeron Dual Equivalence Graphs and Schur Positivity III Part three of the continuing introduction to dual equivalence graphs and Schur positivity.30 September, 2011 Nantel Bergeron Dual Equivalence Graphs and Schur Positivity II Prof. Bergeron continued the discussion of dual equivalence graphs, giving a walkthrough of the proof that the axiomatic definition of dual equivalence graphs are indexed by partitions and are equivalent to the dual equivalence graphs obtained from tableaux. This is in preparation for demonstrating a proof of the Littlewood-Richardson rule using dual equivalence graphs.16 September, 2011 Nantel Bergeron Dual Equivalence Graphs and Schur Positivity I Nantel presented on Sami Assaf's construction of dual equivalence graphs to show Schur positivity of functions expressed as sums of quasisymmetric functions. The graphs were presented axiomatically, and their relation to the fundamental quasisymmetric functions was demonstrated. Here are some pictures of the final blackboard: [1] [2] [3] [4]
About the seminar. Every year we pick a new topic to explore. Nantel, Mike and Chris have suggested exploring the Littlewood Richardson rule for k-Schur functions, the shuffle conjecture, and the connection between noncommutative symmetric functions and supercharacter theory.