C(3^n)

Blocks with defect group [math]C_{3^n}[/math]

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

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

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

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

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