Random 3-D young diagrams and representation theory

The topic of the thesis is related to statistical mechanics and probability theory from one side, and to the representation theory of ``big'' groups on the other side. A typical example of a ``big'' group is the union of unitary groups naturally embedded one into another; it is called the infinite--dimensional unitary group. In statistical mechanics we deal with models of random discrete surfaces in R^3 or, equivalently, random 3-D Young diagrams. The thesis starts from the study of the objects of purely combinatorial origin: we investigate various measures on 3-D Young diagrams confined to AxBxC box. We study a 2--parametric family of measures deforming the uniform probability measure on 3-D Young diagrams in a box and prove a variety of the results about these measures: we construct a family of Markov chains which agree with the measures and, in particular, lead to an efficient random sampling algorithm; we find the corresponding generalization of the well-known MacMahon product formula for the number of 3-D Young diagrams in a given box; we compute the correlation functions which describe the local geometry of random 3-D Young diagrams and study their asymptotics as A, B, C tend to infinity and q tends to 1. The second part of the thesis deals with combinatorial structures related to the infinite-dimensional unitary group. The characters (i.e. normalized central positive-definite functions on the group) of the infinite-dimensional unitary group are in bijection with probability measures on paths in the so-called Gelfand--Tsetlin graph possessing a certain Gibbs property. Through combinatorial bijections paths in Gelfand--Tsetlin graph can be viewed as stepped surfaces or 3-D Young diagrams (this time, of the infinite volume). Then the Gibbs property turns into the uniformity of certain conditional measures on 3-D Young diagrams. In the thesis we study a 1-parametric deformation of this Gibbs property. Our deformation is closely related to the measures on 3-D Young diagrams introduced in the first part of the thesis. Although the deformation is purely combinatorial, there are indications that these new q-Gibbs measures are still related to representation theory, but with unitary groups replaced by their q-deformations --- quantum groups. We prove a classification theorem for the q-Gibbs measures and discuss their relations with certain classes of matrices with non-negative minors and with limits of symmetric polynomials as the number of variable tens to infinite. Finally, in the last part of thesis we study a problem of purely representation--theoretic origin. We deal with a two-parametric family of representations of the infinite-dimensional unitary group introduced by Olshanski as a substitute of the non-existent regular representation of the infinite-dimensional unitary group. We seek for an answer to the question whether these representations are disjoint (i.e. contain no isomorphic subrepresentations). In order to prove the disjointness we reduce this problem to the analysis of certain measures on Gelfand--Tsetlin graph or, in other words, again to the study of random Young diagrams