Non-commutative resolutions of quotient singularities

In this paper we generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. We show in particular that quotient singularities for reductive groups always have non-commutative resolutions in an appropriate sense. Moreover we exhibit a large class of such singularities which have (twisted) non-commutative crepant resolutions. We discuss a number of examples, both new and old, that can be treated using our methods. Notably we prove that twisted non-commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew-symmetric matrices. In contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity.Comment: Final version. Many corrections by the referee implemente

Some algebraic structures in the KPZ universality

We review some algebraic and combinatorial structures that underlie models in the KPZ universality class. Emphasis is given on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov and we present how these are used to analyse the structure of solvable models in the KPZ class and lead to computation of their statistics via connecting to representation theoretic objects such as Schur, Macdonald and Whittaker functions. We also present how fundamental representation theoretic concepts, such as the Cauchy identity, the Pieri rule and the branching rule can be used, alongside RSK correspondences, and can be combined with probabilistic ideas, in order to construct stochastic dynamics on two dimensional arrays called Gelfand-Tsetlin patterns, in ways that couple different one dimensional stochastic processes. The goal of the notes is to expose some of the overarching principles, that have driven a significant number of developments in the field, as a unifying theme.Comment: 75 pages, several figures. This is a review / lecture notes material. Some references adde

Macdonald processes

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters q,t in [0,1). We prove several results about these processes, which include the following. (1) We explicitly evaluate expectations of a rich family of observables for these processes. (2) In the case t=0, we find a Fredholm determinant formula for a q-Laplace transform of the distribution of the last part of the Macdonald-random partition. (3) We introduce Markov dynamics that preserve the class of Macdonald processes and lead to new "integrable" 2d and 1d interacting particle systems. (4) In a large time limit transition, and as q goes to 1, the particles of these systems crystallize on a lattice, and fluctuations around the lattice converge to O'Connell's Whittaker process that describe semi-discrete Brownian directed polymers. (5) This yields a Fredholm determinant for the Laplace transform of the polymer partition function, and taking its asymptotics we prove KPZ universality for the polymer (free energy fluctuation exponent 1/3 and Tracy-Widom GUE limit law). (6) Under intermediate disorder scaling, we recover the Laplace transform of the solution of the KPZ equation with narrow wedge initial data. (7) We provide contour integral formulas for a wide array of polymer moments. (8) This results in a new ansatz for solving quantum many body systems such as the delta Bose gas.Comment: 175 pages (6 chapters, 24 page introduction, index, glossary), 6 figures; updated references and minor mistakes correcte