Interpolation polynomials, bar monomials, and their positivity

TL;DR

This paper proves a new positivity result for interpolation polynomials related to Jack and Macdonald polynomials, introduces bar monomials, and establishes a non-symmetric version of the Knop-Sahi conjecture, advancing understanding in algebraic combinatorics.

Contribution

It introduces a stronger positivity theorem for interpolation polynomials, generalizes Macdonald's positivity, and formulates the non-symmetric Knop-Sahi conjecture with novel concepts like bar monomials, bar order, and glissade.

Findings

Proved a positivity result for interpolation polynomials conjectured by Knop and Sahi.

Established a non-symmetric version of the Knop-Sahi conjecture.

Introduced bar monomials, bar order, and glissade operations, and derived a stronger positivity theorem.

Abstract

We prove a positivity result for interpolation polynomials that was conjectured by Knop and Sahi. These polynomials were first introduced by Sahi in the context of the Capelli eigenvalue problem for Jordan algebras, and were later shown to be related to Jack polynomials by Knop-Sahi and Okounkov-Olshanski. The positivity result proved here is an inhomogeneous generalization of Macdonald's positivity conjecture for Jack polynomials. We also formulate and prove the non-symmetric version of the Knop-Sahi conjecture, and in fact we deduce everything from an even stronger positivity result. This last result concerns certain inhomogeneous analogues of ordinary monomials that we call bar monomials. Their positivity involves in an essential way a new partial order on compositions that we call the bar order, and a new operation that we call a glissade.