S. V. Kerov

Combinatorics, Bethe Ansatz, and representations of the symmetric group

Techniques developed in the realms of the quantum method of the inverse problem are used to analyze combinatorial problems (Young diagrams and rigged configurations).

Harmonic analysis on the infinite symmetric group

AbstractThe infinite symmetric group S(∞), whose elements are finite permutations of {1,2,3,...}, is a model example of a “big” group. By virtue of an old result of Murray–von Neumann, the one–sided regular representation of S(∞) in the Hilbert space ℓ2(S(∞)) generates a type II1 von Neumann factor while the two–sided regular representation is irreducible. This shows that the conventional scheme of harmonic analysis is not applicable to S(∞): for the former representation, decomposition into irreducibles is highly non–unique, and for the latter representation, there is no need of any decomposition at all. We start with constructing a compactification

The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations

\mathfrak{S}\supset{S(\infty)}

Anisotropic young diagrams and jack symmetric functions

, which we call the space of virtual permutations. Although

Four drafts on the representation theory of the group of infinite matrices over a finite field

\mathfrak{S}

A q-Analog of the Hook Walk Algorithm for Random Young Tableaux

is no longer a group, it still admits a natural two–sided action of S(∞). Thus,

THE MARKOV-KREIN CORRESPONDENCE IN SEVERAL DIMENSIONS

\mathfrak{S}

Rooks on ferrers boards and matrix integrals

is a G–space, where G stands for the product of two copies of S(∞). On

A Characterization of GEM Distributions

\mathfrak{S}

The Martin Boundary of the Young-Fibonacci Lattice

, there exists a unique G-invariant probability measure μ1, which has to be viewed as a “true” Haar measure for S(∞). More generally, we include μ1 into a family {μt: t>0} of distinguished G-quasiinvariant probability measures on virtual permutations. By making use of these measures, we construct a family {Tz: z∈ℂ} of unitary representations of G, called generalized regular representations (each representation Tz with z≠=0 can be realized in the Hilbert space