# Integrable random systems, representation theory and geometry of Lie groups 2016

This workshop will take place in Les Diablerets during February 7-12, 2016.

Organizers: Anton Alekseev (UniGe), Maria Podkopaeva (UniGe), Reda Chhaibi (Toulouse).

**Program:**

**Arkady Berenstein** (Oregon), "Generalized RSK"

*The goal of my talk (based on joint work with Dima Grigoriev, Anatol Kirillov, and Gleb Koshevoy) is to generalize the celebrated Robinson-Schensted-Knuth (RSK) bijection between the set of matrices with nonnegative integer entries, and the set of the planar partitions. Namely, for any pair of injective valuations on an integral domain we construct a canonical bijection K, which we call the generalized RSK, between the images of the valuations, i.e., between certain ordered abelian monoids. Given a semisimple or Kac-Moody group, for each reduced word ii=(i_1,...,i_m) for a Weyl group element we produce a pair of injective valuations on C[x_1,...,x_m] and argue that the corresponding bijection K=K_ii, which maps the lattice points of the positive octant onto the lattice points of a convex polyhedral cone in R^m, is the most natural generalization of the classical RSK and, moreover, K_ii can be viewed as a bijection between Lusztig and Kashiwara parametrizations of the dual canonical basis in the corresponding quantum Schubert cell.*

*Generalized RSKs are abundant in ``nature", for instance, any pair of polynomial maps phi,psi:C^m-->C^m with dense images determines a pair of injective valuations on C[x_1,...,x_n] and thus defines a generalized RSK bijection K_{phi,psi} between two sub-monoids of Z_+^m.*

*When phi and psi are birational isomorphisms, we expect that K_{phi,psi} has a geometric ``mirror image", i.e., that there is a rational function f on C^m whose poles complement the image of phi and psi so that the tropicalization of the composition psi^{-1}phi along f equals to K_{phi,psi}. We refer to such a geometric data as a (generalized) geometric RSK, and view f as a ``super-potential." This fully applies to each ii-RSK situation, and we find a super-potential f=f_ii which helps to compute K_ii. While each K_ii has a ``crystal" flavor, its geometric (and mirror) counterpart f_ii emerges from the cluster twist of the relevant double Bruhat cell studied by Andrei Zelevinsky, David Kazhdan, and myself.*

**Philippe Biane** (Paris-Est), "A probabilistic view on Duistermaat-Heckman measure"

*I will explain how to obtain the Duistermaat-Heckman measure (projection of an orbital invariant measure onto a Cartan subalgebra) from path transofrmations on Brownian motion, and how this allows to extend it to situations outside of symplectic geometry.*

**Reda Chhaibi** (Toulouse), "The geometric Robinson-Schensted, quantum Toda and superpotentials, using a probabilistic perspective"

*The theory of crystal bases was started by Kashiwara after a fundamental paper of Jimbo, Date and Miwa (JDM). JDM understood that the relationship between the combinatorial game given by the Robinson-Schensted correspondence and tensor products in a quantum group, when q goes to zero. A crystal basis is a basis for a sublattice in the representations, which can be combinatorially described when q=0. Basically, it describes the skeleton of a representation and the theory generalizes to arbitrary type the combinarics of Young tableaux.*

*Since, a theory of geometric crystals has emerged by Berenstein and Kazhdan. The idea is that inside the group exist crystal-like structures. The aim of this talk is to introduce a geometric Robinson-Schensted correspondence, formulated in terms of paths and geometric crystals. In that framework, we recognize a remarkable path transform first introduced by Biane, Bougerol and O'Connell as the "highest path transform".*

*Upon taking a Brownian path as input, we will have a random dynamic by geometric tensor products. There are two aspects of the description of this dynamic:
- The highest weight will be a Markov process and its dynamic can be understood as an infinitesimal version of the Littlewood Richardson rule, which is known in the integrable systems litterature as the quantum Toda operator.
- The natural measure induced on geometric crystals will by the superpotential of flag manifolds.*

**Manon Defosseux** (Paris), "TBA"

*TBA*

**Vladimir Fock** (Strasbourg), "Cluster coordinates for integrable systems as tau-functions"

*TBA*

**Gleb Koshevoy** (Central Institute of Economics and Mathematics, Moscow) and **Leonid Parnovski** (University College London), "Horn problem and its ramifications"

*TBA*

**Thierry Lévy** (Paris 6), "Large N two-dimensional Yang-Mills"

*The Yang-Mills measure is a probability measure on the affine space of connections on a principal bundle over a manifold, which plays the role of the space-time. In the case where the space-time is a compact Riemannian and two-dimensional - a surface, it is possible to give a rigourous meaning to this measure, through the random holonomy which it induces along every smooth loop on the surface. This stochastic holonomy can be constructed on any principal bundle with any compact structure group, in particular on the trivial U(N) bundle for every integer N. As N tends to infinity and under appropriate scaling, the stochastic holonomy is expected to admit a non-trivial deterministic limit, called the master field.*

*I will describe the master field in the only case where it has been proved to exist, namely the case where the space-time is the Euclidean plane. This master field turns out to be entirely expressible in terms of a single real-valued function on the set of smooth loops on the plane. I will in particular discuss the Makeenko-Migdal equations, which are a set of differential equations on the set of loops of which this function is the unique solution.*

*I will also present more recent results which we obtained with Mylène Maïda in the case where the space-time is a sphere. In this case, the total area of the sphere is an important parameter of the measure (actually, it is essentially the only one) and it was discovered by Douglas and Kazakov that there is, in the large N limit, a third order phase transition as this area crosses the critical value pi squared. I will explain from two points of view why this phase transition takes place, and how we proved that it actually takes place. *

**Maria Podkopaeva** (Geneva & St. Petersburg), **Andras Szenes** (Geneva), "The Horn problem and planar networks"

*TBA*

**Konstanze Rietsch **(King's College London), "Mirror Symmetry for Grassmannians"

*I will report on joint work with Robert Marsh and joint work with Lauren Williams on mirror symmetry for Grassmannians. In toric geometry the mirror superpotental of a smooth toric Fano variety is a Laurent polynomial on a dual torus. While Grassmannians are not toric, they do have a 'special' anticanonical divisor whose complement can be understood as (nearly) covered by tori with certain nice coordinate tranformations: Namely the complement has a cluster structure in the sense of Fomin and Zelevinsky. In the work with Marsh we construct the superpotential of a Grassmannian X as a regular function W on the complement of the anticanonical divisor on a Langlands dual Grassmannian, X^, and prove that it encodes Gromov-Witten invariants of the original Grassmannian via an associated Gauss-Manin system. In the work with Williams we identify tori in the original Grassmannian X to which the tori on the mirror side are dual and show that this duality of tori is reflected in a duality of polytopes associated to X and to (X^,W), in analogy to what happens in the toric setting.*