Distribution of cycle lengths of infinite permutations

The aim of this paper is to show that the well-studied families of GEM and Poisson-Dirichlet measures may be obtained as distributions of normalized cycle lengths on the space of vitual pemutations (i.e., elements of a projective limit of symmetric groups). Two characterizations of Ewens distibutions are given. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1994, pp. 148–161.     

Infinite-Dimensional Schur–Weyl Duality and the Coxeter–Laplace Operator

We extend the classical Schur–Weyl duality between representations of the groups ${SL(n, \mathbb{C})}$ and ${\mathfrak{S}_N}$ to the case of ${SL(n, \mathbb{C})}$ and the infinite symmetric group ${\mathfrak{S}_\mathbb{N}}$ . Our construction is based on a “dynamic,” or inductive, scheme of Schur–Weyl dualities. It leads to a new class of representations of the infinite symmetric group, which has not appeared earlier. We describe these representations and, in particular, find their spectral types with respect to the Gelfand–Tsetlin algebra. The main example of such a representation acts in an incomplete infinite tensor product. As an important application, we consider the weak limit of the so-called Coxeter–Laplace operator, which is essentially the Hamiltonian of the XXX Heisenberg model, in these representations.     

The Schur–Weyl Graph and Thoma’s Theorem

Functional Analysis and Its Applications - We define a graded graph, called the Schur–Weyl graph, which arises naturally when one considers simultaneously the RSK algorithm and the classical...     

Acyclic analogs of nucleosides. Synthesis of 1,5-dihydroxy-3-oxa-2-pentyl derivatives of nucleic bases

A convenient method for the synthesis of 1,5-dihydroxy-3-oxa-2-pentyl derivatives of nucleic bases, which consists in the condensation of trimethylsilyl derivatives of nucleic bases with 1,2,5-triacetoxy-3-oxapentane in the presence of SnCl₄, is proposed. Optically active derivatives — (R)- and (S)-1-(1,5-dihydroxy-3-oxa-2-pentyl)uracil, respectively — were obtained by periodate oxidation of -L- and -D-arabinopyranosyluracil.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 2, pp. 223–228, February, 1988.     

Stick breaking process generated by virtual permutations with Ewens distribution

Given a sequence x of points in the unit interval, we associate with it a virtual permutation w=w(x) (that is, a sequence w of permutationsw _n such that for all n=1,2,..., w_n−1=w′_n is obtained from w_n by removing the last element n from its cycle). We introduce a detailed version of the well-known stick breaking process generating a random sequence x. It is proved that the associated random virtual permutation w(x) has a Ewens distribution. Up to subsets of zero measure, the space of virtual permutations is identified with the cube [0, 1]^∞. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 162–180.     

Quantum inverse scattering method for the q-boson model and symmetric functions

The purpose of this paper is to show that the quantum inverse scattering method for the so-called q-boson model has a nice interpretation in terms of the algebra of symmetric functions. In particular, in the case of the phase model (corresponding to q = 0) the creation operator coincides (modulo a scalar factor) with the operator of multiplication by the generating function of complete homogeneous symmetric functions, and the wave functions are expressed via the Schur functions s_λ(x). The general case of the q-boson model is related in a similar way to the Hall-Littlewood symmetric functions P_λ(x;q²).