# 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.