# C(3^n)

## Blocks with defect group $C_{3^n}$

These are blocks with cyclic defect groups and so they are described by Brauer trees.

For each $n\gt 1$ there are three $\mathcal{O}$-Morita equivalence classes, accounting for all the possible Brauer trees. For $n=1$ there are just two Morita equivalence classes (see $C_3$).

In the following $q_n$ is a prime power such that $(q_n+1)_3=3^n$. For example take $q_n=2^{3^{n-1}}$.

Class Representative $k(B)$ $l(B)$ Inertial quotients ${\rm Pic}_\mathcal{O}(B)$ ${\rm Pic}_k(B)$ ${\rm mf_\mathcal{O}(B)}$ ${\rm mf_k(B)}$ Notes
M($3^n$,1,1) $kC_{3^n}$ $3^n$ 1 $1$ $C_{3^n} : C_{2.3^{n-1}}$ 1 1
M($3^n$,1,2) $kD_{2.3^n}$ $\frac{(3^n+3)}{2}$ 2 $C_2$ 1 1
M($3^n$,1,3) $B_0(kPSL_2(q_n))$ $\frac{(3^n+3)}{2}$ 2 $C_2$ 1 1

Blocks in M($3^n$,1,2) are derived equivalent (over $\mathcal{O}$) to those in M($3^n$,1,3).