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 ${\mathrm{SL}}_d$.

19 pages