Central elements in the $\mathrm{SL}_d$-skein algebra of a surface

TL;DR

This paper identifies a rich family of central elements in the $ ext{SL}_d$-skein algebra of a surface at roots of unity, linking quantum topology, symmetric functions, and representation theory.

Contribution

It introduces a new class of central elements in the $ ext{SL}_d$-skein algebra at roots of unity, constructed via threading framed links with symmetric function polynomials.

Findings

Central elements appear at roots of unity

Construction uses threading with symmetric function polynomials

Links to powers in $ ext{SL}_d$ and quantum invariants

Abstract

The $\mathrm{SL}_d$-skein algebra $\mathcal{S}_{\mathrm{SL}_d}^q(S)$ of a surface $S$ is a certain deformation of the coordinate ring of the character variety consisting of flat $\mathrm{SL}_d$-local systems over the surface. As a quantum topological object, $\mathcal{S}_{\mathrm{SL}_d}^q(S)$ is also closely related to the HOMFLYPT polynomial invariant of knots and links in $\mathbb{R}^3$. We exhibit a very rich family of central elements in this algebra $\mathcal{S}_{\mathrm{SL}_d}^q(S)$ that appear when the quantum parameter $q$ is a root of unity. These central elements are obtained by threading along framed links certain polynomials arising in the elementary theory of symmetric functions, and related to taking powers in $\mathrm{SL}_d$.