Jack polynomials and some identities for partitions

Abstract

Abstract. We prove an identity about partitions involving new combinatorial coefficients. The proof given is using a generating function. As an application we obtain the explicit expression of two shifted symmetric functions, related with Jack polynomials. These quantities are the moments of the "α-content" random variable with respect to some transition probability distributions.

Class expansion of some symmetric functions in Jucys–Murphy elements

“…It is known [20, Proposition 2] that the quantities θ λ µ (α) are shifted symmetric functions of λ, and form a basis of S * (α). Moreover [18,Lemma 7.1], given a symmetric function f , its α-content evaluation f (A Generalizing the method of Section 8, we have similar results for the Hall-Littlewood symmetric function P k (z). Its α-content evaluation may be written as…”

“…In analogy with symmetric functions, this defines S * (α), the algebra of shifted symmetric functions with coefficients in Q[α]. We refer to [32,33,34], or to [18,20] for a short survey.…”

Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures

“…µ (e k , n) for elementary symmetric functions e k . Also, by applying shifted symmetric function theory developed in [KOO,L2,L3,O], we prove that the A (α) µ (F, n) are polynomials in n. We could not obtain any strong results for A (α) µ (F, n). Our appoarch is experimental but the author believes that it is fascinating and applicable in futurer research.…”

“…Following to [KOO,L2], we review the theory of shifted symmetric functions related to Jack functions.…”

Superconformal blocks in diverse dimensions and $BC$ symmetric functions

“…Note that even the first one of these relations is quite non trivial, since the common denominator in the LHS has to cancel out in the end, and this only happens after taking the sum. These two relations are proven in a beautiful way in [67] by using the Lagrange Lemma for alphabets.…”