A Markov chain on the symmetric group and Jack symmetric functions

A Markov chain on the symmetric group and Jack symmetric functions

Hanlon, P., A Markov chain on the symmetric group and Jack symmetric functions, Discrete Mathematics 99 (1992) 123-140. Diaconis and Shahshahani studied a Markov chain Wf(l) whose states are the elements of the symmetric group S,. In W,(l), you move from a permutation n to any permutation of the form a(i, j) with equal probability. In this paper we study a deformation W,(a) of this Markov chain...

Anisotropic Young Diagrams and Jack Symmetric Functions

Λ:λրΛ(cα(b) + u)(cα(b) + v)κα(λ,Λ)φ(Λ) = (nα + uv) φ(λ), where cα(b) is the α-content of a new box b = Λ \ λ. If α = 1, this identity implies the existence of an interesting family of positive definite central functions on the infinite symmetric group. The approach is based on the interpretation of a Young diagram as a pair of interlacing sequences, so that analytic techniques may be used to so...

On a conjecture of Stanley on Jack symmetric functions

Koike, K., On a conjecture of Stanley on Jack symmetric functions, Discrete Mathematics 115 (1993) 211-216. The Jack symmetric function J,(x; G() is a symmetric function with interesting properties that J,(x; 2) is a spherical function of the symmetric pair (GL(n, FQ O(n, [w)) and that J,(x; 1) is the Schur function S,(x). Many interesting conjectures about the combinatorial properties of J,(x;...

Biorthogonal Expansion of Non-Symmetric Jack Functions⋆

We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner–Pollaczek type polynomials. This is done by computing the Cherednik–Opdam transform of the non-symmetric Jack polynomials multiplied by the exponential function.

The symmetric group - representations, combinatorial algorithms, and symmetric functions