  Sekine quantum groups are a family of finite quantum groups. The main result
of this paper is to compute all the idempotent states on Sekine quantum groups,
which completes the work of Franz and Skalski. This is achieved by solving a
complicated system of equations using linear algebra and basic number theory.
From this we discover a new class of non-Haar idempotent states. The order
structure of the idempotent states on Sekine quantum groups is also discussed.
Finally we give a sufficient condition for the convolution powers of states on
Sekine quantum group to converge.
