C(3^n)

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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(3^n,1,1)tree.png
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(3^n,1,2)tree.png
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 M(3^n,1,3)tree.png

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).