In reverse-chronological order.
Date Speaker Title (click titles for abstract) 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.
