Conference

ARTA III Advances in Representation Theory of Algebras

The third Advances in Representation Theory of Algebras (ARTA) meeting will be held at the Université du Québec à Montréal, Montréal, Québec, Canada, from the 16th to the 20th of June 2014.

News

Friday, February 06, 2015

titre à venir

Mike Zabrocki, York University

résumé à venir

Friday, February 13, 2015

Weighted Hurwitz numbers, $\tau$ functions and symmetric polynomials (Part 1)

résumé: Part 1. Tau functions, integrable systems, and combinatorics. Tau functions were originally introduced in the context of classical integrable systems of PDE’ s or difference equations, and integrable quantum field theories. They serve the role, in the first context, of generating functions for commuting flows in the phase space, or in the quantum and statistical mechanical setting as partition functions. However, the simplest building blocks for their construction, Schur functions, already were understood as generating functions in a different sense, namely, for irreducible characters of the symmetric group, well before their appearance in the theory of solitons. This connection between integrable systems and problems in group theory, combinatorics and geometry has been constantly evolving; graphical enumeration, random matrix theory, counting of intersection indices on Riemann surfaces, enumerative theory of maps, counting of branched covers, and random partitions are just some of the applications. This talk will focus mainly on the notion of weighted Hurwitz numbers, which, in different variations, give weighted enumerations of branched coverings of the Riemann sphere. The equivalent combinatorial problem is the (weighted) enumeration of factorizations of elements in the symmetric group or, equivalent, weighted paths in the Cayley graph generated by transpositions. To the two standard constructions of $\tau$ functions - by either Bosonic or Fermionic methods, based on the geometry of flows on Grassmannians, and flag manifolds, together with Bose-Fermi equivalence, we add a third approach, based on the group algebra of the symmetric group, and the characteristic map from the center of the group algebra to the algebra of symmetric polynomials. A key element in the correspondence is the commuting algebra generated by the Jucys Murphy elements, and the fact that symmetric polynomials in these generate the full center of the algebra. A general approach to constructing tau functions of the generalized hypergeometric type in this setting will be related to their use as generating functions for weighted Hurwitz numbers, with the relation between the combinatorial and geometric notions following naturally from the use of dual bases in the algebra of symmetric functions. (The talk will be in a hybrid of French and English, with the written parts more in one and the oral part more in the other language. Questions are welcome in both.)

Friday, February 20, 2015

Weighted Hurwitz numbers, \tau functions and symmetric polynomials (Part 2)

résumé: Part 2. Quantum Hurwitz numbers and MacDonald Polynomials. This talk will focus on the most general version of weighted Hurwitz numbers known to date. These involve an infinity of parameters, of three different types: 1) a denumerable, possibly infinite set of weighting parameters $\{c_1, c_2,..\}$; 2) a finite set (possibly just one, or two) of quantum deformation parameters $(q,t)$, (which are those appearing in Macdonald polynomials); 3) a finite set of expansion parameters $(z_i, w_j)$ and an infinite set of flow parameters $(t_1, t_2, …)$, $(s_1, s_2,…)$, all of which serve just as bookkeeping parameters in the expansions of tau functions in suitable bases, in which the coefficients are the relevant weighted Hurwitz numbers. Everything follows from a simple general scheme, beginning with the double generating function for Macdonald polynomials, which generalizes the Cauchy-Littlewood generating function, evaluating one set of variables as the constants $\{c_i\}$, and the other set as the Jucys-Murphy elements. From there, everything follows automatically, including new theorems in combinatorics, expressing central elements of the group algebra associated to symmetric polynomials as cycle sums, with suitable, explicitly computable coefficients, and a lexicon of relations between the combinatorial and enumerative geometrical definitions of the various weighted Hurwitz numbers that appear.