Some combinatorial properties of skew Jack symmetric functions

TL;DR

This paper investigates combinatorial properties of skew Jack symmetric functions, proving identities related to their invariance under diagram transformations and establishing positivity of certain coefficients in specific cases.

Contribution

It introduces new identities for skew Jack symmetric functions and demonstrates nonnegativity of their coefficients in particular scenarios, advancing understanding of their combinatorial structure.

Findings

Skew Jack symmetric functions are semi-invariant under translation and rotation of the skew diagram.

Coefficients of skew Jack functions in the monomial basis are polynomials with nonnegative integer coefficients in some cases.

The results support Stanley’s conjecture on Jack symmetric functions.

Abstract

Motivated by Stanley's conjecture on the multiplication of Jack symmetric functions, we prove a couple of identities showing that skew Jack symmetric functions are semi-invariant up to translation and rotation of a $\pi$ angle of the skew diagram. It follows that, in some special cases, the coefficients of the skew Jack symmetric functions with respect to the basis of the monomial symmetric functions are polynomials with nonnegative integer coefficients.