Deformed Kazhdan-Lusztig elements and Macdonald polynomials

TL;DR

This paper introduces deformed Kazhdan-Lusztig elements and specialized nonsymmetric Macdonald polynomials, providing explicit formulas, transition matrices, and combinatorial interpretations within the Hecke algebra framework.

Contribution

It presents new deformations of key algebraic objects and explicit formulas, expanding understanding of their structure and combinatorial properties.

Findings

Explicit integral formulas for the deformed polynomials

Transition matrices between polynomial classes are explicitly described

Combinatorial interpretation of evaluations via Schubert polynomial expansions

Abstract

We introduce deformations of Kazhdan-Lusztig elements and specialised nonsymmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We give explicit integral formula for these polynomials, and explicitly describe the transition matrices between classes of polynomials. We further develop a combinatorial interpretation of homogeneous evaluations using an expansion in terms of Schubert polynomials in the deformation parameters.