# Quantum symmetries through the lens of linear algebra

The McKay matrix \(M_V\) records the result of tensoring the simple modules with a finite-dimensional module \(V\). In the case of finite groups, the eigenvectors for \(M_V\) are the columns of the character table, and the eigenvalues come from evaluating the character of \(V\) on conjugacy class representatives.

In this talk, we will explore what can be said about such eigenvectors when the McKay matrix is determined by modules over an *arbitrary* finite-dimensional Hopf algebra \(H\). Here, the McKay matrix \(M_V\) encodes quantum symmetries coming from the actions of \(H\). We prove general results about \(M_V\) by using the coproduct and the characters of simple and projective \(H\)-modules, and also obtain results for a different matrix that encodes the fusion rules for Hopf algebra \(H\). We illustrate these results for the small quantum group \(u_q(\mathfrak{sl}_2)\), where \(q\) is a root of unity (and generally for the Drinfeld double \(D_n\) of the Taft algebra). In these examples, the eigenvalues and eigenvectors for these matrices can be described in terms of several kinds of Chebyshev polynomials.