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Workshop "Representation Theory and Integrability"


Date: Wednesday April 17, 2019.
Time: 14:00-17:00.
Location: Room F3.20, KdV Institute for Mathematics, Science Park 105-107, Amsterdam (entrance via Nikhef).
Organizers: Kayed Al Qasimi, Bernard Nienhuis, Jasper Stokman.


Program

Time: 14:00-14:45
Speaker: Christian Korff (Univ. of Glasgow).
Title: The asymmetric six-vertex model, cylindric symmetric functions and virtual Hecke characters.

Abstract: The asymmetric six-vertex model describes ice and other ferroelectrics on a square lattice. In this talk we will use it in the infinite lattice limit as a combinatorial tool to describe Hecke characters of irreducible finite-dimensional modules. More precisely, we show that on the infinite square lattice a Hecke version of the celebrated boson-fermion correspondence ''diagonalises'' the transfer matrix. When specialising to periodic boundary conditions we show that one obtains from the six-vertex partition function on the infinite cylinder so-called cylindric symmetric functions whose expansions into monomials give rise to virtual Hecke characters. These virtual characters span an infinite-dimensional subcoalgebra in the Grothendieck ring of Hecke algebras with respect to the restriction functor. The structure constants of the subcoalgebra are the Gromov-Witten invariants of Grassmannians.

Time: 15:00-15:45
Speaker: Jesper Jacobsen (Ecole Normale Superieure & Sorbonne Universite).
Title: Topological defects in lattice models and affine Temperley-Lieb algebra.

Abstract: We define defects in critical lattice models that give rise to conformal field theory topological defects in the continuum limit. We focus on models based on the Temperley-Lieb algebra, in the case of generic q. Our algebraic approach considers the defects from two points of view: the ``crossed channel" where the defect is seen as an operator acting on the Hilbert space of the models, and the ``direct channel" where it corresponds to a modification of the basic Hamiltonian with an impurity. In the crossed channel, this leads to new results about the centre of the affine Temperley-Lieb algebra; in particular we identify a special subalgebra with non-negative integer structure constants that are interpreted as fusion rules of defects. In the direct channel, the construction leads to the introduction of fusion products and fusion quotients, which allow us to describe the representation content of the lattice model with a defect, and to describe its spectrum. This is joint work with J. Belletete, A.M. Gainutdinov, H. Saleur and T.S. Tavares.

Time 16:00-16:45
Speaker: Nicolai Reshetikhin (University of California, Berkeley & UvA).
Title: The statistics of irreducible components in large tensor products of finite dimensional representations of simple Lie algebras.

Abstract: The probability distribution of irreducible subrepresentations is computed for tensor products \otimes_k V_k^{\otimes N_k} in the limit when N_k\to \infty while N_1:N_2:\sdots N_m remain finite. For the character distribution, where the probability is proportional to the multiplicity of the irreducible representation time its charter computed at e^t where t is an element of principle Weyl chamber, the asymptotical distribution is universal and depends only on the stabilizer of t in the Weyl group, i.e. whether t is inside the Weyl chamber or on a startup of its boundary. This is a joint work with O. Postnova and V. Serganova.


We cordially invite you to attend the workshop.

The workshop is on the occasion of the PhD defence by Kayed Al Qasimi of his thesis entitled "An elevator ride with Knizhnik and Zamolodchikov" on April 18, 2019 at 10:00 (Agnietenkapel, Oudezijds Voorburgwal 231, Amsterdam). You are also most welcome to attend the defence.