Ce mercredi, le séminaire Combi propose 2 exposés :
à 11h en salle Flajolet, Generalized Jucys-Murphy elements and canonical idempotents in towers of algebras par Aaron Lauve de l'université de Loyola
à 14h en salle Poincaré, séminaire joint avec Parsifal, A proof-theoretic analysis of the rotation lattice of binary trees par Noam Zeilberger, postdoc à Parsifal.
Les résumés sont disponibles ci-dessous.
Generalized Jucys-Murphy elements and canonical idempotents in towers of algebras
The collection of symmetric group algebras serves as a motivating example for what I'll call a multiplicity-free tower of finite dimensional algebras. Any such family has a canonical complete set of pairwise orthogonal primitive idempotents stemming from its representation theory. In the case of the symmetric group algebras, these idempotents were first given a tidy formula by Thrall (1941) using the infamous "Young symmetrizers." I'll provide other examples (Brauer algebras will feature prominently), then highlight several methods to compute these idempotents, extending work of Jucys, Murphy, and Vershik--Okunkov. Opportunities for SageMath projects will be littered throughout. (Based on work with S. Doty and G.H. Seelinger, arXiv:1606.08900.)
A proof-theoretic analysis of the rotation lattice of binary trees
The classical Tamari lattice Yn is defined as the set of binary trees with n internal nodes, with the partial ordering induced by the (right) rotation operation. It is not obvious why Yn is a lattice, but this was first proved by Haya Friedman and Dov Tamari in the late 1950s. More recently, Frédéric Chapoton discovered another surprising fact about the rotation ordering, namely that Yn contains exactly 2(4n + 1)!/((n + 1)!(3n + 2)!) pairs of related trees. (Even more surprising was how Chapoton discovered this formula: via the Online Encyclopedia of Integer Sequences, because the formula had already been computed in the early 1960s by Bill Tutte, but for a completely different family of objects!)
In the talk I will describe a new way of looking at the rotation ordering that is motivated by old ideas in proof theory. This will lead us to systematic ways of thinking about:
- the lattice property of Yn, and
- the Tutte-Chapoton formula for the number of intervals in Yn.
No advanced background in either proof theory or combinatorics will be assumed.