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Blocks with defect group [math]C_9[/math]

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

There are three [math]\mathcal{O}[/math]-Morita equivalence classes, accounting for all the possible Brauer trees.

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(9,1,1) [math]kC_9[/math] 9 1 [math]1[/math] [math]C_9 : C_6[/math] 1 1 M(9,1,1)tree.png
M(9,1,2) [math]kD_{18}[/math] 6 2 [math]C_2[/math] [math]C_2 \times C_3[/math] 1 1 M(9,1,2)tree.png
M(9,1,3) [math]B_0(kSL_2(8))[/math] 6 2 [math]C_2[/math] [math]C_3[/math] 1 1 M(9,1,3)tree.png

Blocks in M(9,1,2) are derived equivalent (over [math]\mathcal{O}[/math]) to those in M(9,1,3)