** Speaker:** Aaron Lauve, Loyola University

** Title:** The characteristic polynomial of the antipode for combinatorial Hopf algebras

** Abstract: ** The Adams operators Phi_n on a Hopf algebra H are the convolution powers of the identity
of H. The antipode of H is the special case n = -1. We study the Adams operators when H is graded connected.
The main result is a complete description of the characteristic polynomial - both eigenvalues and their
multiplicities - for the action of the operator Phi_n on each homogeneous component of H. The eigenvalues are
powers of n. The multiplicities are independent of n, and in fact only depend on the dimension sequence of
H. We look at some combinatorial consequences of this result, such as closed-form generating function for
the trace of the antipode on H, and, time permitting, indicate extensions to Hopf monoids in species, q-Hopf
algebras, and cofree graded connected Hopf algebras.