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

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 $\mathrm{SL}_d$.