Combinatorial Invariance of Relative R-polynomials in the Hermitian Symmetric Case

TL;DR

This paper introduces a new combinatorial algorithm for computing relative R-polynomials in Hermitian symmetric spaces, demonstrating their invariance and providing a uniform computational approach.

Contribution

It develops a marking system and a closed-form algorithm for relative R-polynomials, establishing their combinatorial invariance in Hermitian symmetric cases.

Findings

Algorithm for computing relative R-polynomials

Proof of combinatorial invariance in Hermitian symmetric spaces

Unified approach applicable to Hermitian symmetric pairs

Abstract

We develop a marking system for an analog of Hasse diagrams of intervals $[u,v]$ with $u\leq v$ in a Hermitian symmetric pair $W/W_J$, and use this to create a closed form algorithm for computing relative R-polynomials. The uniform nature of this algorithm allows us to show combinatorial invariance of relative Kazhdan-Lusztig polynomials in the Hermitian symmetric space setting.