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

Abstract

The \(\textrm{SL}_d\)-skein algebra \(\mathcal {S}^q_{\textrm{SL}_d}(S)\) of a surface S is a certain deformation of the coordinate ring of the character variety consisting of flat \(\textrm{SL}_d\)-local systems over the surface. As a quantum topological object, \(\mathcal {S}^q_{\textrm{SL}_d}(S)\) is also closely related to the HOMFLYPT polynomial invariant of knots and links in \({\mathbb {R}}^3\). We exhibit a rich family of central elements in \(\mathcal {S}^q_{\textrm{SL}_d}(S)\) 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 the Lie group \(\textrm{SL}_d\).