Class expansion of some symmetric functions in Jucys–Murphy elements

Abstract

We present a method to compute the class expansion of a symmetric function in the Jucys-Murphy elements of the symmetric group. We apply this method to onerow Hall-Littlewood symmetric functions, which interpolate between power sums and complete symmetric functions.

On Complete Functions in Jucys-Murphy Elements

“…In this paragraph, we give an elementary proof of the following theorem, which has been proved by M. Lassalle [Las10] using sophisticated algebraic tools.…”

“…Using equation (15), one can translate Theorems 2.4 and 2.6 into relations over the c's, but it is rather technical (see [Las10,section 12]). We prefer here to explain the combinatorial meaning of the c's and derive directly relations over the c's using this interpretation.…”

“…Recently, M. Lassalle [Las10] gave a unified method to obtain some induction relations for the coefficients of the class expansion of several families of symmetric functions in Jucys-Murphy elements. These induction relations allow to compute any coefficient quite quickly.…”

Polynomiality of shifted Plancherel averages and content evaluations

“…and obtains an alternative algebraic proof for the polynomiality of (1.2). Some explicit formulas for particular F are obtained in [5,6,14,15,18,19]. Just as an example, in [6] the identity λ∈Pn P Plan n (λ)…”

A new generalization of binomial coefficients

“…There are many generalizations of binomial coefficients, the most elementary of which are the Gaussian polynomials. In this note, we shall present another one-parameter extension, presumably new, encountered in the study of the symmetric groups [4]. This application will be described at the end.…”